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How to get amoxil without a doctor

About Insight how to get amoxil without a doctor Insight provides an in-depth look at health care issues in and affecting California.Have a story suggestion?. Let us how to get amoxil without a doctor know. This story was produced in partnership with PolitiFact. This story can be republished for free (details). President Donald Trump how to get amoxil without a doctor accepted the Republican Party’s nomination for president in a 70-minute speech from the South Lawn of the White House on Thursday night.Speaking to a friendly crowd that didn’t appear to be observing social distancing conventions, and with few participants wearing masks, he touched on a range of topics, including many related to the buy antibiotics amoxil and health care in general.Throughout, the partisan crowd applauded and chanted “Four more years!. € And, even as the nation’s buy antibiotics death toll exceeded 180,000, Trump was upbeat.

€œIn recent how to get amoxil without a doctor months, our nation and the entire planet has been struck by a new and powerful invisible enemy,” he said. €œLike those brave Americans before us, we are meeting this challenge.”At the end of the event, there were fireworks.Our partners at PolitiFact did an in-depth fact check on Trump’s entire acceptance speech. Here are the highlights related to the administration’s buy antibiotics response and other health policy issues:“We developed, from scratch, how to get amoxil without a doctor the largest and most advanced testing system in the world.” This is partially right, but it needs context.It’s accurate that the U.S. Developed its buy antibiotics testing system from scratch, because the government didn’t accept the World Health Organization’s testing recipe. But whether how to get amoxil without a doctor the system is the “largest” or “most advanced” is subject to debate.The U.S.

Has tested more individuals than any other country how to get amoxil without a doctor. But experts told us a more meaningful metric would be the percentage of positive tests out of all tests, indicating that not only sick people were getting tested. Another useful how to get amoxil without a doctor metric would be the percentage of the population that has been tested. The U.S. Is one of the most populous countries but has tested a lower percentage of its how to get amoxil without a doctor population than other countries.

Don't Miss A Story Subscribe to California Healthline’s free Weekly Edition newsletter. The how to get amoxil without a doctor U.S. Was also slower than other countries in rolling out tests and amping up testing capacity. Even now, many states are how to get amoxil without a doctor experiencing delays in reporting test results to positive individuals.As for “the most advanced,” Trump may be referring to new testing investments and systems, like Abbott’s recently announced $5, 15-minute rapid antigen test, which the company says will be about the size of a credit card, needs no instrumentation and comes with a phone app through which people can view their results. But Trump’s comment makes it sound as if these testing systems are already in place when they haven’t been distributed to the how to get amoxil without a doctor public.“The United States has among the lowest [buy antibiotics] case fatality rates of any major country in the world.

The European Union’s case fatality rate is nearly three times higher than ours.”The case fatality rate measures the known number of cases against the known number of deaths. The European Union has a rate how to get amoxil without a doctor that’s about 2½ times greater than the United States.But the source of that data, Oxford University’s Our World in Data project, reports that “during an outbreak of a amoxil, the case fatality rate is a poor measure of the mortality risk of the disease.”A better way to measure the threat of the amoxil, experts say, is to look at the number of deaths per 100,000 residents. Viewed that way, the U.S. Has the 10th-highest death rate in the world.“We will produce a treatment before the end of the year, or maybe even sooner.”It’s far from guaranteed that a antibiotics treatment will be ready before the end of the year.While how to get amoxil without a doctor researchers are making rapid strides, it’s not yet known precisely when the treatment will be available to the public, which is what’s most important. Six treatments are in the third phase of testing, which involves thousands of patients.

Like earlier phases, this one looks at the safety of a treatment but also examines its effectiveness how to get amoxil without a doctor and collects more data on side effects. Results of the third phase will be submitted to the Food and Drug Administration for approval.The government website Operation Warp Speed seems less optimistic than Trump, announcing it “aims to deliver 300 million doses of a safe, effective treatment for buy antibiotics by January 2021.”And federal health officials and other experts have generally predicted a treatment will be available in early 2021. Federal committees how to get amoxil without a doctor are working on recommendations for treatment distribution, including which groups should get it first. €œFrom everything we’ve seen now — in the animal data, as well as how to get amoxil without a doctor the human data — we feel cautiously optimistic that we will have a treatment by the end of this year and as we go into 2021,” said Dr. Anthony Fauci, the nation’s top infectious diseases expert.

€œI don’t think it’s dreaming.”“Last month, I took on how to get amoxil without a doctor Big Pharma. You think that is easy?. I signed how to get amoxil without a doctor orders that would massively lower the cost of your prescription drugs.”Quite misleading. Trump signed four executive orders on July 24 aimed at lowering prescription drug prices. But those orders haven’t taken effect yet — the text of one hasn’t even been made publicly available — and experts told us that, if implemented, the measures would be unlikely to result in significant drug how to get amoxil without a doctor price reductions for the majority of Americans.“We will always and very strongly protect patients with preexisting conditions, and that is a pledge from the entire Republican Party.”Trump’s pledge is undermined by his efforts to overturn the Affordable Care Act, the only law that guarantees people with preexisting conditions both receive health coverage and do not have to pay more for it than others do.

In 2017, Trump supported congressional efforts to repeal the ACA. The Trump administration is now backing GOP-led efforts to overturn the how to get amoxil without a doctor ACA through a court case. And Trump has also expanded short-term health plans that don’t have to comply with the ACA.“Joe Biden recently raised his hand on the debate stage and promised he was going to give it away, your health care dollars to illegal immigrants, which is going to bring a massive number of immigrants into our country.”This is misleading. During a June 2019 how to get amoxil without a doctor Democratic primary debate, candidates were asked. €œRaise your hand if your government plan would provide coverage for undocumented immigrants.” All candidates on stage, including Biden, raised how to get amoxil without a doctor their hands.

They were not asked if that coverage would be free or subsidized.Biden supports extending health care access to all immigrants, regardless of immigration status. A task force how to get amoxil without a doctor recommended that he allow immigrants who are in the country illegally to buy health insurance, without federal subsidies.“Joe Biden claims he has empathy for the vulnerable, yet the party he leads supports the extreme late-term abortion of defenseless babies right up to the moment of birth.”This mischaracterizes the Democratic Party’s stance on abortion and Biden’s position.Biden has said he would codify the Supreme Court’s ruling in Roe v. Wade and related precedents. This would generally limit abortions to the first 20 to 24 weeks of how to get amoxil without a doctor gestation. States are allowed under court rulings to ban abortion after the point at which a fetus can sustain life, usually considered to be between 24 and 28 weeks from the mother’s last menstrual period — and 43 states do.

But the rulings require how to get amoxil without a doctor states to make exceptions “to preserve the life or health of the mother.” Late-term abortions are very rare, about 1%.The Democratic Party platform holds that “every woman should have access to quality reproductive health care services, including safe and legal abortion — regardless of where she lives, how much money she makes, or how she is insured.” It does not address late-term abortion.PolitiFact’s Daniel Funke, Jon Greenberg, Louis Jacobson, Noah Y. Kim, Bill McCarthy, Samantha Putterman, Amy Sherman, Miriam Valverde and KHN reporter Victoria Knight contributed to this report. This how to get amoxil without a doctor story was produced by Kaiser Health News, an editorially independent program of the Kaiser Family Foundation. Related Topics Elections Health Industry how to get amoxil without a doctor Insight Pharmaceuticals Public Health The Health Law Abortion buy antibiotics Immigrants KHN &. PolitiFact HealthCheck Preexisting Conditions Trump Administration treatmentsAbout Insight Insight provides an in-depth look at health care issues in and affecting California.Have a story suggestion?.

Let us how to get amoxil without a doctor know. This story also ran on CNN. This story can be republished for free (details). Flu season will look different this year, as the country grapples with a antibiotics amoxil that has killed more than 172,000 people. Many Americans are reluctant to visit a doctor’s office and public health officials worry people will shy away from being immunized.Although how to get amoxil without a doctor sometimes incorrectly regarded as just another bad cold, flu also kills tens of thousands of people in the U.S. Each year, with the very young, the elderly and those with underlying conditions the most vulnerable. When coupled with the effects of buy antibiotics, public how to get amoxil without a doctor health experts say it’s more important than ever to get a flu shot.If enough of the U.S.

Population gets vaccinated — more than the 45% who did last flu season — it could help head off a nightmare scenario in the coming winter of hospitals stuffed with both buy antibiotics patients and those suffering from severe effects of influenza.Aside from the potential burden on hospitals, there’s the possibility people could get both amoxiles — and “no one knows what happens if you get influenza and buy antibiotics [simultaneously] because it’s never happened before,” Dr. Rachel Levine, Pennsylvania’s secretary of health, told reporters this month.In response, manufacturers are producing more treatment supply this year, between 194 million and 198 million doses, or about 20 million more than they distributed last season, according to the Centers for Disease how to get amoxil without a doctor Control and Prevention. Email Sign-Up Subscribe to California Healthline’s free Daily how to get amoxil without a doctor Edition. As flu season approaches, here are some answers to a few common questions:Q. When should I get my flu how to get amoxil without a doctor shot?.

Advertising has already begun, and some pharmacies and clinics have their supplies now. But, because the effectiveness of the treatment can wane over time, the CDC recommends against a shot in August.Many pharmacies and clinics will how to get amoxil without a doctor start immunizations in early September. Generally, influenza amoxiles start circulating in mid- to late October but become more widespread later, in the winter. It takes about two weeks after getting a shot for antibodies — which circulate in the blood and thwart how to get amoxil without a doctor s — to build up. €œYoung, healthy people can begin getting their flu shots in September, and elderly people and other vulnerable populations can begin in October,” said Dr.

Steve Miller, chief clinical officer for insurer Cigna.The CDC has recommended that people “get a flu treatment by the end of October,” but noted it’s not too late to how to get amoxil without a doctor get one after that because shots “can still be beneficial and vaccination should be offered throughout the flu season.”Even so, some experts say not to wait too long this year — not only because of buy antibiotics, but also in case a shortage develops because of overwhelming demand.Q. What are the reasons I should roll up my sleeve for this?. Get a shot because it protects you from catching the flu and spreading it to others, which may help lessen the burden on hospitals and medical staffs.And there’s another message that may resonate in this strange time.“It gives people a sense that there are some things you can control,” said Eduardo Sanchez, chief medical officer for prevention at the American Heart Association.While a flu shot won’t prevent buy antibiotics, he said, getting one could help your doctors differentiate between the diseases how to get amoxil without a doctor if you develop any symptoms — fever, cough, sore throat — they share.And even though flu shots won’t prevent all cases of the flu, getting vaccinated can lessen the severity if you do fall ill, he said.You cannot get influenza from having a flu treatment.All eligible people, especially essential workers, those with underlying conditions and those at higher risk — including very young children and pregnant women — should seek protection, the CDC said. It recommends that children over 6 months old get how to get amoxil without a doctor vaccinated.Q. What do we know about the effectiveness of this year’s treatment?.

Flu treatments — which must be developed anew each year because influenza amoxiles mutate — range in effectiveness annually, depending on how well they match the how to get amoxil without a doctor circulating amoxil. Last year’s formulation was estimated to be about 45% effective in preventing the flu overall, with about a 55% effectiveness in children. The treatments available in how to get amoxil without a doctor the U.S. This year are aimed at preventing at least three strains of the amoxil, and most cover four.It isn’t yet known how well this year’s supply will match the strains that will circulate in the U.S. Early indications from the Southern Hemisphere, which goes how to get amoxil without a doctor through its flu season during our summer, are encouraging.

There, people practiced social distancing, wore masks and got vaccinated in greater numbers this year — and global flu levels are lower than expected. Experts caution, however, not to count on a similarly mild season in the U.S., in how to get amoxil without a doctor part because masking and social distancing efforts vary widely.Q. What are insurance plans and health systems doing how to get amoxil without a doctor differently this year?. Insurers and health systems contacted by KHN say they will follow CDC guidelines, which call for limiting and spacing out the number of people waiting in lines and vaccination areas. Some are setting appointments for flu shots to help manage the flow.Health Fitness Concepts, a company that works with UnitedHealth Group and other businesses to set up flu shot clinics in the Northeast, said it is “encouraging smaller, more frequent events to support social distancing” and “requiring all forms to be completed and shirtsleeves rolled up before entering the flu shot area.” Everyone will be required to wear masks.Also, nationally, some physician groups contracted with UnitedHealth will set up tent areas so shots how to get amoxil without a doctor can be given outdoors, a spokesperson said.Kaiser Permanente plans drive-thru vaccinations at some of its medical facilities and is testing touch-free screening and check-in procedures at some locations.

(KHN is not affiliated with Kaiser Permanente.)Geisinger Health, a regional health provider in Pennsylvania and New Jersey, said it, too, would have outdoor flu vaccination programs at its facilities.Additionally, “Geisinger is making it mandatory for all employees to receive the flu treatment this year,” said Mark Shelly, the system’s director of prevention and control. €œBy taking this step, we hope to convey how to get amoxil without a doctor to our neighbors the importance of the flu treatment for everyone.”Q. Usually I get a flu shot at work. Will that be an option this year? how to get amoxil without a doctor. Aiming to avoid risky indoor gatherings, many employers are reluctant to sponsor the on-site flu clinics they’ve offered in years past.

And with how to get amoxil without a doctor so many people continuing to work from home, there’s less need to bring flu shots to employees on the job. Instead, many employers are encouraging workers to get shots from their primary care doctors, at pharmacies or in other community settings. Insurance will generally cover the cost of the treatment.Some employers are considering offering vouchers for flu shots to their uninsured workers or those who don’t participate in the company plan, said Julie Stone, managing director for health and benefits at Willis Towers Watson, a consulting firm. The vouchers could allow workers to get the shot at a particular lab at no cost, for example.Some employers are starting to think about how they might use their parking lots for administering drive-thru flu shots, said Dr. David Zieg, clinical services leader for benefits consultant Mercer.Although federal law allows employers to require employees to get flu shots, that step is typically taken only by health care facilities and some universities where people live and work closely together, Zieg said.Q.

What are pharmacies doing to encourage people to get flu shots?. Some pharmacies are making an extra push to get out into the community to offer flu shots.Walgreens, which has nearly 9,100 pharmacies nationwide, is continuing a partnership begun in 2015 with community organizations, churches and employers that has offered about 150,000 off-site and mobile flu clinics to date.The program places a special emphasis on working with vulnerable populations and in underserved areas, said Dr. Kevin Ban, chief medical officer for the drugstore chain.Walgreens began offering flu shots in mid-August and is encouraging people not to delay getting vaccinated.Both Walgreens and CVS are encouraging people to schedule appointments and do paperwork online this year to minimize time spent in the stores.At CVS MinuteClinic locations, once patients have checked in for their flu shot, they must wait outside or in their car, since the indoor waiting areas are now closed.“We don’t have tons of arrows in our quiver against buy antibiotics,” Walgreens’ Ban said. €œTaking pressure off the health care system by providing treatments in advance is one thing we can do.” This story was produced by Kaiser Health News, an editorially independent program of the Kaiser Family Foundation. Julie Appleby.

jappleby@kff.org, @julie_appleby Related Topics Insight Insurance Public Health buy antibiotics Insurers treatments.

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18 or amoxil for sale online <. 19 in school) 138% FPL*** Children <. 5 and pregnant women have HIGHER LIMITS than shown ESSENTIAL PLAN For MAGI-eligible people over MAGI income limit up to 200% FPL No long term care. See info here 1 2 1 2 3 1 2 Income $875 (up from $859 in 201) $1284 (up from $1,267 in 2019) $1,468 $1,983 $2,498 $2,127 $2,873 Resources $15,750 (up from $15,450 in 2019) $23,100 (up from $22,800 amoxil for sale online in 2019) NO LIMIT** NO LIMIT SOURCE for 2019 figures is GIS 18 MA/015 - 2019 Medicaid Levels and Other Updates (PDF). All of the attachments with the various levels are posted here.

NEED TO KNOW PAST MEDICAID INCOME AND RESOURCE LEVELS?. Which household amoxil for sale online size applies?. The rules are complicated. See rules here. On the HRA Medicaid Levels chart - Boxes 1 and 2 are NON-MAGI Income and Resource levels -- Age 65+, Blind or Disabled and amoxil for sale online other adults who need to use "spend-down" because they are over the MAGI income levels.

Box 10 on page 3 are the MAGI income levels -- The Affordable Care Act changed the rules for Medicaid income eligibility for many BUT NOT ALL New Yorkers. People in the "MAGI" category - those NOT on Medicare -- have expanded eligibility up to 138% of the Federal Poverty Line, so may now qualify for Medicaid even if they were not eligible before, or may now be eligible for Medicaid without a "spend-down." They have NO resource limit. Box 3 on page 1 is Spousal Impoverishment levels for Managed Long Term Care & amoxil for sale online. Nursing Homes and Box 8 has the Transfer Penalty rates for nursing home eligibility Box 4 has Medicaid Buy-In for Working People with Disabilities Under Age 65 (still 2017 levels til April 2018) Box 6 are Medicare Savings Program levels (will be updated in April 2018) MAGI INCOME LEVEL of 138% FPL applies to most adults who are not disabled and who do not have Medicare, AND can also apply to adults with Medicare if they have a dependent child/relative under age 18 or under 19 if in school. 42 C.F.R.

§ 435.4 amoxil for sale online. Certain populations have an even higher income limit - 224% FPL for pregnant women and babies <. Age 1, 154% FPL for children age 1 - 19. CAUTION amoxil for sale online. What is counted as income may not be what you think.

For the NON-MAGI Disabled/Aged 65+/Blind, income will still be determined by the same rules as before, explained in this outline and these charts on income disregards. However, for the MAGI population - which is virtually everyone under age 65 who is not on Medicare - their income will now be amoxil for sale online determined under new rules, based on federal income tax concepts - called "Modifed Adjusted Gross Income" (MAGI). There are good changes and bad changes. GOOD. Veteran's amoxil for sale online benefits, Workers compensation, and gifts from family or others no longer count as income.

BAD. There is no more "spousal" or parental refusal for this population (but there still is for the Disabled/Aged/Blind.) and some other rules. For all of the rules see amoxil for sale online. ALSO SEE 2018 Manual on Lump Sums and Impact on Public Benefits - with resource rules The income limits increase with the "household size." In other words, the income limit for a family of 5 may be higher than the income limit for a single person. HOWEVER, Medicaid rules about how to calculate the household size are not intuitive or even logical.

There are different rules depending on the "category" of the person seeking Medicaid. Here are amoxil for sale online the 2 basic categories and the rules for calculating their household size. People who are Disabled, Aged 65+ or Blind - "DAB" or "SSI-Related" Category -- NON-MAGI - See this chart for their household size. These same rules apply to the Medicare Savings Program, with some exceptions explained in this article. Everyone else -- MAGI - All children amoxil for sale online and adults under age 65, including people with disabilities who are not yet on Medicare -- this is the new "MAGI" population.

Their household size will be determined using federal income tax rules, which are very complicated. New rule is explained in State's directive 13 ADM-03 - Medicaid Eligibility Changes under the Affordable Care Act (ACA) of 2010 (PDF) pp. 8-10 of the amoxil for sale online PDF, This PowerPoint by NYLAG on MAGI Budgeting attempts to explain the new MAGI budgeting, including how to determine the Household Size. See slides 28-49. Also seeLegal Aid Society and Empire Justice Center materials OLD RULE used until end of 2013 -- Count the person(s) applying for Medicaid who live together, plus any of their legally responsible relatives who do not receive SNA, ADC, or SSI and reside with an applicant/recipient.

Spouses or legally amoxil for sale online responsible for one another, and parents are legally responsible for their children under age 21 (though if the child is disabled, use the rule in the 1st "DAB" category. Under this rule, a child may be excluded from the household if that child's income causes other family members to lose Medicaid eligibility. See 18 NYCRR 360-4.2, MRG p. 573, NYS GIS 2000 MA-007 CAUTION. Different people in the same household may be in different "categories" and hence have different household sizes AND Medicaid income and resource limits.

If a man is age 67 and has Medicare and his wife is age 62 and not disabled or blind, the husband's household size for Medicaid is determined under Category 1/ Non-MAGI above and his wife's is under Category 2/MAGI. The following programs were available prior to 2014, but are now discontinued because they are folded into MAGI Medicaid. Prenatal Care Assistance Program (PCAP) was Medicaid for pregnant women and children under age 19, with higher income limits for pregnant woman and infants under one year (200% FPL for pregnant women receiving perinatal coverage only not full Medicaid) than for children ages 1-18 (133% FPL). Medicaid for adults between ages 21-65 who are not disabled and without children under 21 in the household. It was sometimes known as "S/CC" category for Singles and Childless Couples.

This category had lower income limits than DAB/ADC-related, but had no asset limits. It did not allow "spend down" of excess income. This category has now been subsumed under the new MAGI adult group whose limit is now raised to 138% FPL. Family Health Plus - this was an expansion of Medicaid to families with income up to 150% FPL and for childless adults up to 100% FPL. This has now been folded into the new MAGI adult group whose limit is 138% FPL.

For applicants between 138%-150% FPL, they will be eligible for a new program where Medicaid will subsidize their purchase of Qualified Health Plans on the Exchange. PAST INCOME &. RESOURCE LEVELS -- Past Medicaid income and resource levels in NYS are shown on these oldNYC HRA charts for 2001 through 2019, in chronological order. These include Medicaid levels for MAGI and non-MAGI populations, Child Health Plus, MBI-WPD, Medicare Savings Programs and other public health programs in NYS.

A Gateway to Coverage for Immigrants The report how to get amoxil without a doctor includes a new tool -- Immigrant Eligibility Crosswalk -- Eligibility by Immigration https://mytutorlab.com/students-with-special-needs/ Status-- designed to help advocates and policymakers sort through the tangle of immigrant eligibility categories to determine who is eligible for which health care programs in 2014 and beyond. The report was made possible with support from the United Hospital Fund and benefited from the advice and input from many of our national partners in the effort to ensure maximum participation of immigrants in the nation's healthcare system as well as experts from the New York State Department of Health and the Centers for Medicare and Medicaid Services. SEE more about "PRUCOL" immigrant eligibility for Medicaid in this article.

"Undocumented" immigrants are, with some exceptions for pregnant women and Child Health Plus, only eligible for "emergency Medicaid."NYS announced the 2020 Income and Resource levels in GIS 19 MA/12 – 2020 how to get amoxil without a doctor Medicaid Levels and Other Updates ) and levels based on the Federal Poverty Level are in GIS 20 MA/02 – 2020 Federal Poverty Levels Here is the 2020 HRA Income and Resources Level Chart Non-MAGI - 2020 Disabled, 65+ or Blind ("DAB" or SSI-Related) and have Medicare MAGI (2020) (<. 65, Does not have Medicare)(OR has Medicare and has dependent child <. 18 or <.

19 in school) 138% FPL*** Children < how to get amoxil without a doctor. 5 and pregnant women have HIGHER LIMITS than shown ESSENTIAL PLAN For MAGI-eligible people over MAGI income limit up to 200% FPL No long term care. See info here 1 2 1 2 3 1 2 Income $875 (up from $859 in 201) $1284 (up from $1,267 in 2019) $1,468 $1,983 $2,498 $2,127 $2,873 Resources $15,750 (up from $15,450 in 2019) $23,100 (up from $22,800 in 2019) NO LIMIT** NO LIMIT SOURCE for 2019 figures is GIS 18 MA/015 - 2019 Medicaid Levels and Other Updates (PDF).

All of the attachments with the how to get amoxil without a doctor various levels are posted here. NEED TO KNOW PAST MEDICAID INCOME AND RESOURCE LEVELS?. Which household size applies?.

The rules are complicated how to get amoxil without a doctor. See rules here. On the HRA Medicaid Levels chart - Boxes 1 and 2 are NON-MAGI Income and Resource levels -- Age 65+, Blind or Disabled and other adults who need to use "spend-down" because they are over the MAGI income levels.

Box 10 on page 3 are how to get amoxil without a doctor the MAGI income levels -- The Affordable Care Act changed the rules for Medicaid income eligibility for many BUT NOT ALL New Yorkers. People in the "MAGI" category - those NOT on Medicare -- have expanded eligibility up to 138% of the Federal Poverty Line, so may now qualify for Medicaid even if they were not eligible before, or may now be eligible for Medicaid without a "spend-down." They have NO resource limit. Box 3 on page 1 is Spousal Impoverishment levels for Managed Long Term Care &.

Nursing Homes and Box 8 has the Transfer Penalty rates for nursing home eligibility Box 4 has Medicaid Buy-In for Working People with Disabilities Under Age 65 (still 2017 levels til April 2018) Box 6 are Medicare Savings Program levels (will be updated in April 2018) MAGI INCOME LEVEL of 138% FPL applies to most adults who are not disabled and who do not have Medicare, AND can also apply to adults with Medicare if they have a how to get amoxil without a doctor dependent child/relative under age 18 or under 19 if in school. 42 C.F.R. § 435.4.

Certain populations have an how to get amoxil without a doctor even higher income limit - 224% FPL for pregnant women and babies <. Age 1, 154% FPL for children age 1 - 19. CAUTION.

What is counted as income how to get amoxil without a doctor may not be what you think. For the NON-MAGI Disabled/Aged 65+/Blind, income will still be determined by the same rules as before, explained in this outline and these charts on income disregards. However, for the MAGI population - which is virtually everyone under age 65 who is not on Medicare - their income will now be determined under new rules, based on federal income tax concepts - called "Modifed Adjusted Gross Income" (MAGI).

There are good changes and bad changes how to get amoxil without a doctor. GOOD. Veteran's benefits, Workers compensation, and gifts from family or others no longer count as income.

BAD. There is how to get amoxil without a doctor no more "spousal" or parental refusal for this population (but there still is for the Disabled/Aged/Blind.) and some other rules. For all of the rules see.

ALSO SEE 2018 Manual on Lump Sums and Impact on Public Benefits - with resource rules The income limits increase with the "household size." In other words, the income limit for a family of 5 may be higher than the income limit for a single person. HOWEVER, how to get amoxil without a doctor Medicaid rules about how to calculate the household size are not intuitive or even logical. There are different rules depending on the "category" of the person seeking Medicaid.

Here are the 2 basic categories and the rules for calculating their household size. People who are Disabled, Aged 65+ or Blind - "DAB" or "SSI-Related" Category -- NON-MAGI - See this chart for their how to get amoxil without a doctor household size. These same rules apply to the Medicare Savings Program, with some exceptions explained in this article.

Everyone else -- MAGI - All children and adults under age 65, including people with disabilities who are not yet on Medicare -- this is the new "MAGI" population. Their household size will how to get amoxil without a doctor be determined using federal income tax rules, which are very complicated. New rule is explained in State's directive 13 ADM-03 - Medicaid Eligibility Changes under the Affordable Care Act (ACA) of 2010 (PDF) pp.

8-10 of the PDF, This PowerPoint by NYLAG on MAGI Budgeting attempts to explain the new MAGI budgeting, including how to determine the Household Size. See slides how to get amoxil without a doctor 28-49. Also seeLegal Aid Society and Empire Justice Center materials OLD RULE used until end of 2013 -- Count the person(s) applying for Medicaid who live together, plus any of their legally responsible relatives who do not receive SNA, ADC, or SSI and reside with an applicant/recipient.

Spouses or legally responsible for one another, and parents are legally responsible for their children under age 21 (though if the child is disabled, use the rule in the 1st "DAB" category. Under this rule, a child may be excluded from the household if that child's income causes other family how to get amoxil without a doctor members to lose Medicaid eligibility. See 18 NYCRR 360-4.2, MRG p.

573, NYS GIS 2000 MA-007 CAUTION. Different people in the same household may be in different "categories" and how to get amoxil without a doctor hence have different household sizes AND Medicaid income and resource limits. If a man is age 67 and has Medicare and his wife is age 62 and not disabled or blind, the husband's household size for Medicaid is determined under Category 1/ Non-MAGI above and his wife's is under Category 2/MAGI.

The following programs were available prior to 2014, but are now discontinued because they are folded into MAGI Medicaid. Prenatal Care Assistance Program (PCAP) was Medicaid for pregnant women and children under age 19, with higher income limits for pregnant woman and infants under one year (200% FPL for pregnant women receiving perinatal coverage only not full Medicaid) than for children ages 1-18 (133% FPL). Medicaid for adults between ages 21-65 who are not disabled and without children under 21 in the household.

It was sometimes known as "S/CC" category for Singles and Childless Couples. This category had lower income limits than DAB/ADC-related, but had no asset limits. It did not allow "spend down" of excess income.

This category has now been subsumed under the new MAGI adult group whose limit is now raised to 138% FPL. Family Health Plus - this was an expansion of Medicaid to families with income up to 150% FPL and for childless adults up to 100% FPL.

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We can’t actually Where can i buy cipro over the counter usa cultivate disgust buy amoxil with free samples in the ways that its advocates presume. That said, and in contrast with the skeptics’ assessment, we can improve our ability to control when and how we feel our disgust. This difference between cultivating and controlling disgust is subtle but important.

And once we recognize it, we’re forced to rethink not only our assessments of buy amoxil with free samples disgust’s moral value but also more fundamental questions about what becoming a more virtuous person involves. Let’s start by considering disgust’s virtues. Not only do we tend to experience disgust toward moral wrongs like hypocrisy and exploitation, but the shunning and social excluding that disgust brings seems a fitting response to those who pollute the moral fabric in these ways.

Moreover, in the face of worries about morally problematic disgust—disgust felt at the wrong buy amoxil with free samples time or in the wrong way—advocates respond that it’s an emotion we can substantively change for the better. On this front, disgust’s advocates point to exposure and habituation. Just like I might overcome the disgust I feel about exotic foods by trying them, I can overcome the disgust I feel about same-sex marriage by spending more time with gay couples.

Moreover, work in psychology appears to buy amoxil with free samples support this picture. Medical school students, for instance, lose their disgust about touching dead bodies after a few months of dissecting corpses, and new mothers quickly become less disgusted by the smell of soiled diapers. But these findings may be deceptive.

For starters, when we look more closely at the results of the diaper buy amoxil with free samples experiment, we see that a mother’s reduced disgust sensitivity is most pronounced with regard to her own baby’s diapers, and additional research indicates that mothers have a general preference for the smell of their own children. This combination suggests, contra the disgust advocates, that a mother’s disgust is not being eliminated. Rather, her disgust at the soiled diapers is still there.

It’s just being masked by the positive feelings that she’s getting from the smell of buy amoxil with free samples her newborn. Similarly, when we look carefully at the cadaver study, we see that while the disgust of medical students toward touching the cold bodies of the dissection lab is reduced with exposure, the disgust they feel toward touching the warm bodies of the recently deceased remained unchanged. All this may seem like fodder for the skeptic’s claim that disgust is morally problematic.

After all, it seems there’s little we buy amoxil with free samples can do to shape our disgust for the better. But that would be too quick. While there may not be much we can do to substantively change what we’re disgusted by, we may be able to improve our ability to control when and how we feel our disgust.

More specifically, even if disgust itself is too rigid to be buy amoxil with free samples changed, it appears there are other psychological mechanisms associated with disgust—things like our attentional systems and cognitive processing routines—that are more malleable. So, focusing on these mechanisms may offer a better strategy for addressing morally problematic disgust. We get a hint of this in the diaper experiment, where it appears that mothers’ disgust responses are canceled out by the positive feelings they experience through mother-child bonding processes.

And this buy amoxil with free samples picture finds further support in research highlighting the effectiveness of “implementation intentions” for our ability to control problematic disgust. At a gloss, implementation intentions are the if-then rules that guide our actions. Importantly, strategies that appeal to them aren’t trying to directly change a person’s disgust.

Rather, they’re buy amoxil with free samples aiming to develop people’s (nondisgust) attentional capacities. Allowing them to better recognize situations where disgust response may misfire, so that they can better control the resulting disgust. For instance, someone disgusted by the sight of blood might adopt an implementation intention like “If I see blood, I’ll adopt the perspective of a physician,” or “If I see blood, I’ll stay calm and relaxed,” in order to moderate both their assessment of how disgusting the blood is and their subsequent reactions to it.

While researchers have yet to investigate the effectiveness of implementation intentions as correctives for morally problematic disgust, multiple studies have found the technique effective as a way buy amoxil with free samples of combating excessive disgust experienced in nonmoral situations (e.g., seeing bodily fluids). So where does all of this leave us regarding the question of disgust’s moral value?. For starters, we can see that advocates are right that disgust is a morally powerful response to hypocrites, cheaters and the like.

Without disgust, we’d lack an important way of responding to those who take advantage of buy amoxil with free samples others. But advocates are wrong in thinking disgust is a malleable emotion that we can substantively change for the better. In the other direction, we also see that skeptics overstate their concerns.

Though we cannot buy amoxil with free samples substantively change morally problematic disgust, we can learn to effectively control it through the use of implementation intentions. To see what this might look like, consider someone who is strongly disgusted by members of a particular minority group (let’s call that group the “Gs”). Such an individual would be well-served to adopt implementation intentions aimed at helping him control his disgust—something like, “If I see Gs, I’ll adopt the perspective of Martin Luther King, Jr.” or “If I see Gs, I’ll relax and be friendly.” As suggested above, deploying such a strategy should allow them to better recognize situations where their disgust response may misfire, so that they can engage implementation intentions that will help them control their reaction.

But there may be a buy amoxil with free samples further lesson here. The dominant philosophical view of moral development, one that has roots in Aristotle, sees becoming virtuous as a process whereby one transforms problematic emotions. The cowardly person’s fear is transformed into the courageous person’s emotional attunement to danger.

But looking closely at the science of disgust reveals that not all emotions are like this buy amoxil with free samples. Some emotions resist our efforts to substantively change them for the better. So, in these cases, becoming a more virtuous person is not a matter of seeking emotional transformation.

Rather, it’s the process by which buy amoxil with free samples we improve our emotional self-awareness and self-control. This is an opinion and analysis article.This spring, the 17-year cicadas of Brood X will emerge from underground, climb tree trunks and molt, leaving their crunchy shells behind. Soon after, the males will join together in a droning chorus to the delight (or consternation) of their human neighbors.

Those with a keen ear might detect that there are several buzzy songs buy amoxil with free samples occurring at once. This is not because the cicadas have a large repertoire. Rather, there are few different cicada species, including the Magicicada septendecim and the Magicicada cassini, each with a different tune.

In the early 19th century this was still a mystery, but the entomologist buy amoxil with free samples Margaretta Hare Morris had suspicions. Ever since she had been a teenager, she had carefully observed the emergence of cicadas. She had heard the different cicada songs in 1817 and again 1834.

It was in 1846, though, when she was 49, that Morris felt confident enough buy amoxil with free samples to announce that she had discovered a new species. Digging down beneath her fruit trees, Morris had found cicada larvae sucking at the roots, five years before they were scheduled to emerge. Even just determining how the cicadas subsisted underground for 17 years was a breakthrough in 1846.

She also found buy amoxil with free samples something else. Some were significantly smaller than others. Morris sent a report of her discoveries to the Academy of Natural Sciences of Philadelphia in 1846, one of the leading scientific organizations at the time.

€œI am inclined to believe that there are two species, differing sufficiently in size,” buy amoxil with free samples she wrote. Morris described the larger cicadas as sluggish with a droning song that sounded like “Faaaa ROO.” The smaller, unnamed species was “extremely active, springing backwards with a sudden motion” with a tune “sharp and shrill, like the noise made by the loom of a stocking weaver.” For those of us unfamiliar with the sound of 19th-century stocking weavers, the modern entomologist Gene Kritsky has described their calls as “a short series of rattling, rapid clicks followed by a longer buzzing or ‘swishing’ sound.” Given that Morris wasn’t a member of the Academy of Natural Sciences, she had to have a male scientist read and present her report and specimens to her peers. Still, she persisted.

She published articles in popular journals and invited the country’s leading scientists to come to her garden and witness her discoveries, creating a large network of supporters ready to buy amoxil with free samples endorse her methods. It was due to these efforts that Morris was one of the first women elected to the American Association for the Advancement of Science in 1850, alongside the astronomer Maria Mitchell. Still, you’ve probably never heard her name.

We have buy amoxil with free samples forgotten Morris and her work for several reasons, one of which involves this cicada species. In 1851 after that year’s cicada season was over, John Cassin (an ornithologist) and James Coggswell Fisher (a geologist), members of the Academy of Natural Science who had read Morris’s reports, proclaimed to have discovered a new species of cicada. It was much smaller and shriller than its better-known relative (this sounds familiar, yes?.

). They named it for themselves. Cicada Cassinii (Fisher, 1851).

What, ultimately, is in a name?. Why does this matter?. Given what I know about her, Margaretta Morris likely would not care that the insect she discovered was named for someone else.

She wasn’t particularly interested in scientific fame. She was hoping to better understand the wonders of the insect world. Still, the fact that Cassini and Fisher jumped at the chance to name the bug for themselves means that when the excitement about Brood X’s emergence this spring is covered in newspapers, on podcasts, and in books, we will hear Cassin’s name over and over again.

He has been immortalized while Morris has been erased. If Morris’s name had been tied to this tiny shrill creature as well as the other insects she discovered, she could have served as a model for others like herself who felt a passion for studying insects. Women entomologists were hard to come by in the 19th century, and they continue to be significantly outnumbered today.

When the cracks in the glass ceiling are obscured, it can feel daunting for the next person looking up at that ceiling, trying to determine how to break through. It is for this reason that it’s important to remember pioneers who pursued their passions no matter how lonely it had been and no matter how many times someone else took credit for their work. It is no less important to give credit where credit is due.

So, when you hear that shrill cicada sound this spring, whether you find it obnoxious or wondrous, know that there was a wily 19th-century scientist named Margaretta Morris who was entranced by the mystery of these creatures. She would have awaited the emergence of the 17-year cicadas with glee.On April 11, 2020, John Horton Conway died of buy antibiotics at the age of 82 in New Brunswick, N.J. The areas of research covered by this remarkable mathematician included group theory, node theory, geometry, analysis, combinatorial game theory, algebra, algorithmics and even theoretical physics.

Conway’s inclinations and talent led him to invent a remarkable cellular automaton called the Game of Life, which continues to fascinate after 50 years. Conway also devised elegant puzzles for packing boxes of blocks that can only be solved efficiently with clever reasoning. According to Conway, his most important contribution was his conceptualization of a marvelous system of numbers called surreal numbers.

This class comprises integers, real numbers, transfinite numbers and infinitesimals—a structure that no one previously imagined was possible in which everything can be added, multiplied, and so on. People who worked with Conway report that he thought so fast that no sooner did he hear a problem stated than he often already had a solution. Conway’s commitment to mathematics for everyone led him to work on puzzles that delight fans of recreational mathematics, such as the famous Collatz conjecture (which is discussed toward the end of this article).

Conway was the mathematician most cited (more then 30 times!. ) in my columns in Pour la Science, the French edition of Scientific American. In preparing the articles, I would often come across a result he had demonstrated or an important idea that he had been the first to propose on the topic, much to my surprise.

Conway liked simple, practical mathematical problems that called on his creativity. The best way to honor him, it seems to me, is to highlight the kind of mathematics that amazed and fascinated him. I am going to cover several different topics to which Conway contributed.

But it would take several volumes to do justice to this exceptional mind. He was able to invent objects and problems in many different domains and to solve the most recalcitrant puzzles, conceiving methods no one could have imagined. The Irrationality of √2 One of the most surprising and important mathematical findings is the irrationality of the square root of 2 (√2)—the length of the diagonal of a square with sides that are one unit long.

It cannot be expressed by the quotient of positive integers n and m, or n/m. The discovery of the irrationality of √2 is credited to Pythagoras or one of his disciples, although we do not know whether the reasoning behind it was arithmetic or geometric. The discovery and its proof were profoundly unsettling for mathematicians.

This first negative finding in mathematics showed that humans do not create the laws governing numbers but rather uncover them as they explore uncharted mathematical territory. Though there are many proofs of this theorem, the most intuitive is a very simple little drawing that Conway included in a lecture published in a book in 2005. He attributed the creation of the proof to mathematician Stanley Tennenbaum, who, according to Conway, had abandoned mathematics.

You might ask whether it was Conway himself who formulated the proof. But that does not matter. Because even if he did not create it, the proof offers a perfect example of Conway’s approach to mathematics, which he demonstrated in 100 different ways.

It also shows that it is wrong to believe that everything simple has already been discovered. Brilliant yet astonishingly simple ideas are still waiting to be revealed. Say that √2 is the quotient of n and m—that is, 2 = n2/m2, or 2m2 = n2.

If so, there exists a square, with sides equal to n, whose area equals twice that of a square with sides equal to m [see part A of “An Irrational Square Root”]. We assume that in our drawing, m is the smallest positive integer satisfying this equation. The assumption would be valid only if there is no smaller positive integer that satisfies it.

Credit. Pour la Science Wedging the two blue squares into two diagonally opposite corners of the red square produces a new shape. The two red squares in that shape must have the same area as the central purple square that is created where the two blue squares overlap [see Part B of “An Irrational Square Root”].

Our reasoning requires that the area of the two blue squares must be equal to the area of the red one (A). Consequently, the area not covered by the two blue squares is equal to twice the area covered by both of them. In other words, there are two equal and smaller squares (red, B) that together have the same area as the larger square (purple, B).

That means each side of the new small squares is equal to the integer n – m, and each side of the large purple square is equal to the integer n – 2(n – m) = 2m – n. In short, the initial square of side m was not the smallest possible one that satisfies the geometric equation. The result is a contradiction, and thus the assumption is false.

ˆš2 is not a quotient of two integers, which means it is an irrational number. In the same way, you can demonstrate that the square root of 3 is irrational [see part C of “An Irrational Square Root”]. Three Cube-Packing Puzzles Many puzzles consist of putting a small number of blocks in a box (say 10).

Both the blocks and the box are typically rectangular parallelepipeds. The solution is usually arrived at by trial and error. With a problem like this one, unless you increase the number of blocks and their variety of shapes, it is hard to conceive a challenging puzzle.

Moreover, although it can be fun to manipulate the blocks until the solution is found, doing so involves only a little reasoning about shape and thus does not really reduce the solution time. Conway reimagined the problem by creating puzzles that can potentially lead you in circles while you try to figure out how to fill the box. But a few astute and orderly considerations will quickly guide you to the solution.

The simplest of these three puzzles consists of filling a 3 x 3 x 3 box with three 1 x 1 x 1 cubes and six 1 x 2 x 2 parallelepipeds [see part 1 of “Blocks in a Box”]. Credit. Pour la Science The reasoning consists of taking parity into account.

The 3 x 3 x 3 box is composed of three horizontal 3 x 3 x 1 layers, each with a volume of 9. The 3 x 3 x 3 box can also be decomposed into three vertical 3 x 3 x 1 parallelepipeds, each parallel to a vertical face of the box. Considering the perpendicular vertical face produces a third decomposition of the box into three 3 x 3 x 1 parallelepipeds.

Let’s examine these nine parallelepipeds. When a 1 x 2 x 2 block is positioned in the box, it can only fill an even number of the nine cells in each 3 x 3 x 1 layer. So each layer must contain one—and only one—of the three small 1 x 1 x 1 blocks (otherwise you would have to place all three of them in a single 3 x 3 x 1 layer, which would leave none in the other two layers, where they are needed).

The upper layer of the box must therefore contain one 1 x 1 x 1 block. If we place it in the center of the layer, we quickly reach a dead end, because we are obliged to place another small block just below it. That puts two of them in a single 3 x 3 x 1 vertical layer.

Placing the small block on the top face in the middle of any side also results in an impasse. Consequently, the small block on the top layer must go in a corner. The same reasoning applies to the bottom layer, where a small block must be placed in a corner.

That corner must be diametrically opposite to the small block on the top layer. The third small block must therefore go in the center of the middle layer. The remaining steps become quickly obvious.

Note that this reasoning not only produces a solution but also shows that, except for symmetries, there is only one solution. The solution to another of Conway’s packing problems is shown in part 2 of “Blocks in a Box.” To complete the packing, all you have to do is to lower the two upper structures and place the green layer (b) at the side on top of the pink layer (a). Reasoning based on parity forces us to place the unit cubes along the diagonal.

The Riddle of the Two Wizards In the 1960s Conway devised a fiendishly tricky puzzle that, until recently, prompted much debate and the publication of a paper by Tanya Khovanova of the Massachusetts Institute of Technology in 2013. Here is the puzzle as it appears in that paper. Last night I sat behind two wizards on a bus, and overheard the following:A.

€œI have a positive integral number of children, whose ages are positive integers, the sum of which is the number of this bus, while the product is my own age.”B. €œHow interesting!. Perhaps if you told me your age and the number of your children, I could work out their individual ages?.

AT LAST I know how old you are!. €Now what was the number of the bus?. Note that in the exchange, the “no” uttered by Wizard A does not mean that he refuses to answer but that the knowledge of his age and the number of his children does not make it possible to know their individual ages.

Of course, you have to assume that Wizard B knows the number of the bus. Note, too, that the wizards can be very young or very old. Wizard A could well be two or 20,000 years old.

Here is the solution, which I was only able to fully understand and verify with the help of a program. I can’t reproduce all the calculations required for the conclusion, but you can take my word for it —or redo all the calculations I’ve left out. Let us denote the age of Wizard A as a, the number of the bus as b and the number of Wizard A’s children as c.

Suppose, for example, that the number of the bus is b = 5. The following are the options for the number of children, the distribution of their ages and the age of Wizard A. C = 5 of ages 1, 1, 1, 1, 1.

Thus, a = 1;c = 4 of ages 1, 1, 1, 2. Thus, a = 2;c = 3 of ages 1, 1, 3. Thus, a = 3;c = 3 of ages 1, 2, 2.

Thus, a = 4;c = 2 of ages 1, 4. Thus, a = 4;c = 2 of ages 2, 3. Thus, a = 6;c = 1 of age 5.

Thus, a = 5. In each case, knowing the age of Wizard A and the number of children indicates the possible ages of the latter. Because Wizard A answered “no” and Wizard B knows the number of the bus, this means b is not equal to 5.

Similarly, you can solve the problem by examining the possible bus numbers one by one to find those for which knowing the age of the wizard and the number of his children will not enable you to know the ages of each of the children (a property we denote as P). Calculating b = 1, 2, 3, ..., 12 (as we have just done for b = 5) shows that b = 12 is the smallest number possessing P. Indeed, for b = 12 and c = 4, there are two sets of possible ages of the four children—(2, 2, 2, 6) and (1, 3, 4, 4)—which both give the same age for Wizard A.

A = 48. No other two sets of the same length with the same product for b = 12 exist. Therefore, for b = 12, even knowing that c = 4 and a = 48, it is impossible to deduce the ages of the four children.

Does this mean that b = 12 is the solution?. Unfortunately, not yet. For bus number b = 13, for example, because the two possible sets of the three children’s ages—(1, 6, 6) and (2, 2, 9)—are both compatible with a = 36, Wizard B cannot derive the ages of Wizard A’s children from the knowledge of either his age or the number of his children.

Knowing that b = 12 is no more conclusive about the ages of the children than knowing that b = 13. When confronted with the puzzle, most people often answer “b = 12,” as if the riddle somehow implies that the smallest possible solution for b is the right one. But the puzzle does not make that assertion.

Moreover, without further reasoning, you cannot choose between b = 12 and b = 13, nor can you choose among other values of b, as my further calculations show. Yet b = 12 is the correct answer, and the reason for that is the most interesting and unexpected part of the riddle. Conway crafted his puzzle carefully, and you must consider the final statement of Wizard B.

Following Wizard A’s “no,” Wizard B responds, “Aha!. AT LAST I know how old you are!. € That eliminates b = 13.

In fact, for b = 13, there are two additional sets of the children’s ages—(1, 2, 2, 2, 6) and (1, 1, 3, 4, 4)—that give a = 48. In other words, if the bus number were 13, Wizard B could not deduce the age of Wizard A from his negative answer because his age could be 36 or 48. Thus, b = 13 should be eliminated.

Yet eliminating b = 13 leads to the elimination of b = 14 when we consider the age sequences found for b = 13 and add 1. Doing so shows that the product of two sets of the children’s ages—(1, 1, 6, 6) and (1, 2, 2, 9)—is a = 36, whereas the product of another two sets—(1, 1, 2, 2, 6) and (1, 1, 1, 3, 4, 4)—is a = 48. The same process eliminates b = 15 and, one by one, all of the b’s greater than 12.

Consequently, it is only bus number 12 (b = 12), with two sets of the children’s ages—(2, 2, 2, 6) and (1, 3, 4, 4)—that uniquely determines the age of Wizard A. A = 48. Given all the calculations you’d have to do to arrive at the solution—the details of which I have not reproduced here and which take up several pages—I confess that I don’t understand how Conway managed to conceive this incredible puzzle!.

Paving a Plane with Line Segments Paving a plane with squares is easy. It is just as easy to do so using equilateral triangles or hexagons. It is also possible to pave the plane with infinite straight lines.

Just place them all side by side and parallel. There will be an infinite number of them, but the paving will be perfectly satisfactory because each point of the plane will belong to one —and only one—line of the paving. Here are a few trickier questions.

Can the plane be paved with closed segments—i.e., straight line segments [A, B], including their endpoints?. Can the plane be paved with open segments—i.e., segments ]A, B[, without their endpoints?. Can the plane be paved with semi-open segments—i.e., segments ]A, B], with only one of their endpoints?.

Think about these unusual yet perfectly natural questions. They are not so simple. Conway and a colleague discussed them and other types of problems in a splendid paper published in 1964, proving once again that very simple questions that no one thinks about are great opportunities to do mathematics that is not so obvious.

In that paper, Conway and mathematician Hallard Croft devised a way to fill a plane with straight line segments, which can be seen in “Fine Paving” below. Credit. Pour la Science Panel a shows how to pave a plane with semi-open segments in which one point is included and another is excluded.

Doing so is easy because placing them together, head to tail, reproduces the obvious paving with straight lines. Panel b shows the solution for equal closed segments in which both points of each segment are included. Pillar 1 is placed first.

Pillar 2 is then added, but of course the leftmost segment of this stack is not retained. For pillar 3, the rightmost segment is not retained. Ever finer slanted pillars are added successively, omitting the “first” segment for each.

Panel c shows a solution for differently sized open segments (with their endpoints excluded). The segments create a central square open on all sides—i.e., neither the sides nor the top or bottom of the square are covered. Each successively added rectangle is also open.

For open segments of the same length, there is no solution for paving the plane. The Collatz Conjecture Another puzzle is as interesting to amateur mathematicians as it is to professionals. Consider a function (f) that gives n/2 for positive integers (n) that are even and 3n + 1 for those that are odd.

For example, starting with 7 and applying f, you get 22, then 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, and so on. Once you get to 1, you go around in circles. Whatever integer n you start with, you always seem to end up at 1 and then get stuck in a loop.

Computer calculations have tested this property, which is true for all n’s up to at least 87 x 260 (about 1020). It has not been proved, however, that this is always the case, nor has an initial n been found that would continue to infinity or lead to a cycle other than 4, 2, 1. This problem is called the Collatz conjecture or the Syracuse conjecture.

Much work has been devoted to it, some of which has been compiled in a book. The great simplicity of the problem’s statement attracts amateurs, and I regularly receive proposed solutions that, to date, are either wrong or incomprehensible. Faced with such a challenge—so simply stated and yet unsolved, which is still the case after more than 80 years—Conway could not resist.

In 1972 he published his first paper on the subject, in which he proposed similarly formulated problems and demonstrated their undecidability. For certain values of the starting integer, the sequence generated by the variant of f does not end up arriving at 1, but set theory cannot demonstrate this. More generally, for any logical system of demonstration (S), there is a problem of this category and a starting point for a sequence that does not end up at 1, which cannot be demonstrated by S.

In 2013 Conway returned to the problem with probabilistic arguments suggesting that the Collatz conjecture itself is unprovable with the axiom systems that are usually used in mathematics. This was not a definitive demonstration of the undecidability of the Collatz conjecture. But the types of arguments he proposed seemed to strongly indicate that it is no accident that everyone stumbles in their attempt to prove the conjecture.

All mathematicians encounter problems they cannot solve, and Conway was no exception to this general rule. Nevertheless, his skill as a logician may have afforded some comfort by enabling him to demonstrate undecidability and to elaborate arguments suggesting that this simple yet intractable problem would continue to challenge mathematicians indefinitely. A New Pattern in the Game of Life The Game of Life, first introduced in Martin Gardner’s Mathematical Games column in Scientific American in 1970, is still being studied, and all of the puzzles that it poses have not been solved.

A square cell in an infinite, two-dimensional rectangular grid may be alive or dead. Or the cell may go from live (black) to dead (white), or vice versa, from one generation to another. Or the cell may remain stable, depending on whether its eight nearest neighbor cells are dead or alive.

The rule of evolution can be expressed in 12 words. Birth if three live neighbors. Survival if two or three live neighbors.

Some patterns with few live cells at the initial state grow to infinity. Ideally, the number of live cells increases proportionately with n2, where n is the number of the generation. The optimal density of living cells in a stable part of the grid is 1/2.

Noam Elkies of Harvard University demonstrated the proof in 29 pages in 1997. Credit. Pour la Science The extraordinary pattern shown at the top of “Game of Life Redux” was discovered in April 2020 by computer scientist Mateusz Naściszewski.

It is the smallest known pattern that, once launched, grows quadratically to cover the grid of a stable live cell population of density 1/2. (This pattern is thus the best possible one in terms of speed and density.) The configuration begins with a population of 183 cells. We drew generation 0 and two other generations at different scales [see bottom patterns in “Game of Life Redux”].

It is believed to be impossible to do better than 183, but that has not been demonstrated. Note, too, that there are patterns that calculate the prime numbers or that even display graphic images of decimal digits of π = 3.14159... One after the other.

This article originally appeared in Pour la Science and was reproduced with permission.Peru Scientists declared a Liolaemus lizard the world's highest-altitude reptile after a population was spotted at 5,400 meters in the Andes. These lizards must endure frigid temperatures, a particular challenge for cold-blooded animals, as well as reduced oxygen and increased ultraviolet radiation. Algeria Analysis suggests a meteorite from the Sahara Desert contains material as old as, or older than, Earth itself.

The meteorite holds the oldest-known sample of magma from space and most likely came from a protoplanet forming in the early solar system. Egypt An excavation on the Red Sea coast revealed what seems to be a pet cemetery from nearly 2,000 years ago, the earliest yet identified. Nearly 600 cats, dogs and monkeys—mostly cats—were carefully buried, many with textiles, pottery or ornate collars.

China Researchers tested a new soft-body swimming robot first in a lake, next in the South China Sea, and finally in the Mariana Trench—almost 11,000 meters down—to prove it can flap its fins in extreme pressure as it explores the depths. Borneo Scientists captured dung beetles in a forest in Sabah, dissected them, then sequenced the DNA in their guts to find matches with nearby wildlife. Because dung lasts within the beetles for about 48 hours, this method can reveal a location's recent visitors and inhabitants.

New Zealand Conservation rangers worked with hundreds of volunteers to “refloat” 40 stranded long-finned pilot whales, returning them to open water. Nine more of the beached whales died.Picture the scene. A small drone the size of a suitcase descends into a dark Martian crevasse—perhaps a lava tube that was formed billions of years ago by volcanic activity on the Red Planet.

The drone illuminates its surroundings, recording views never seen before by human eyes as its suite of instruments seeks out signs of past or present alien biology. Finally, its reconnaissance complete, the drone flies back to a landing zone on the surface to transmit invaluable data back to Earth. After soaking up the Martian sunlight to recharge its batteries, it continues its explorations of terrain inaccessible to any other machine.

Far from being some starry-eyed flight of fancy, such a mission could soon become a reality thanks to the resounding success of NASA’s Ingenuity rotorcraft, sometimes referred to as a helicopter or drone—a technology demonstration that has taken place on Mars over the past few weeks. Carried to the planet by NASA’s Perseverance rover, which touched down on February 18, this small machine, weighing a paltry 1.8 kilograms, was the first attempt at controlled aerial flight on another world—more than a century after that same feat was mastered on Earth by the Wright brothers. €œWe can now say that human beings have flown a rover craft on another planet,” said MiMi Aung, project manager of Ingenuity at NASA’s Jet Propulsion Laboratory (JPL), in a speech to her team from mission control following the successful first flight on April 19.

€œWe together now have our Wright brothers moment.” With Ingenuity’s success, space scientists are contemplating the roles that aerial vehicles might play in our exploration of the solar system. Few worlds possess the necessary conditions for powered aerodynamic flight, namely an atmosphere and rocky surface like that of Mars or Earth, but there are two others of note. €œThe general technique of aerial flight is applicable to places like [Saturn’s moon] Titan and Venus,” said Bob Balaram, chief engineer of the Ingenuity team, in a press briefing following the first flight.

The latter’s exceedingly high temperatures and pressures pose some unique challenges. €œNear the surface it’s closer to swimming,” says Paul Byrne, a planetary scientist at North Carolina State University. Yet flight there is not impossible, which was proved by the Soviet Union’s Vega balloons in 1985.

With a rotorcraft called Dragonfly already being developed to visit Titan in the next decade and work continuing on a conceptual successor to Ingenuity, the future looks bright for aerial exploration of alien worlds. €œThis could be the start of a new era,” Byrne says. Ingenuity’s first flight, from a strip of land on Mars’s Jezero Crater that is now dubbed “Wright Brothers Field,” was modest but impressive.

The planet’s atmosphere is incredibly thin, just 1 percent that of Earth, so generating lift is exceedingly difficult. €œIt’s similar to Earth at about 100,000 feet above the ground,” says Ben Pipenberg, an engineer at defense contractor AeroVironment, who helped build Ingenuity. With Perseverance watching from a safe distance, Ingenuity spun its blades at 2,500 revolutions per minute (rpm) to rise to an altitude of three meters, where it hovered for 30 seconds and performed a 96-degree rotation.

Then it descended back to the ground, landing on its four legs, with a total flight time of 39.1 seconds. From there, things got more complex. The second flight lasted 51.9 seconds, reaching a height of five meters.

And it included a lateral movement of about two meters—something not attempted in the confines of the test chamber on Earth where Ingenuity first flew in simulated Mars conditions. Flight three saw Ingenuity travel half the length of a football field, some 50 meters, reaching a top speed of just more than two meters per second. The fourth flight on April 30 pushed the envelope once again, with Ingenuity remaining airborne for nearly two minutes—117 seconds—and reaching an impressive speed of 3.5 meters per second as it scouted a potential future landing zone over a round trip of more than 260 meters.

Ingenuity’s fifth flight—completed on May 7 and initially planned to be its last—sent it on a one-way trip to the new landing zone to await the arrival of Perseverance, its mothership. Now, this wildly successful technology demonstration drone is entering a new phase of its mission—a second month-long set of more ambitious operational tests. These tests are meant to show how airborne drones “could play an active role in a future rover science mission,” says Dave Lavery, the program executive for Ingenuity at NASA headquarters in Washington, DC.

Although Ingenuity will not directly support the science objectives of Perseverance, namely looking for signs of past life on Mars, it will help scout out the rover’s potential route ahead as the team plans their optimal path through Jezero Crater’s riches, or even photograph nearby locations not in the rover’s planned path. There is even a slim chance Ingenuity could support the rover’s later mission too, if it survives. €œWe might see about potentially looking over the rim of the crater,” Lavery says.

Much has been made of how these vehicles might one day support human missions, acting as reconnaissance drones for humans to scout out regions of interest near a landing site or carrying tools between locations. In the near-term, prospects of more exciting robotic science are on the horizon—perhaps in the same way that the Sojourner rover in 1997, itself a prototype of wheeled exploration and part of NASA’s Pathfinder mission, paved the way for its successors Spirit, Opportunity, Curiosity and now Perseverance. €œI do think we’re going to see some flying vehicles in the future,” says Michael Meyer, lead scientist of NASA’s Mars Exploration Program at the agency’s headquarters in Washington, D.C.

€œIt will now be part of our portfolio of methods that we use for exploration. There are things you can do with a helicopter that you can’t do with other platforms.” Examples could include exploring the aforementioned lava tubes or perhaps approaching crater walls—too high and steep for a rover to scale—where a helicopter could take images and perform some up-close analysis as well. Another example could be studying recurring slope lineae, dark flows on Mars that have arguably been linked to liquid water flowing on the surface.

Perversely, it is this possibility of water—and the accompanying risk of contamination with bacteria imported from Earth—that essentially prohibits anyone or anything from setting foot (or wheel) there to seek out signs of native Martian life. But a hovering drone could look without touching, offering a novel route of exploration. €œA rotorcraft would give us the ability to go and look up close at something that we would otherwise deem not suitable for a rover,” Byrne says, “either because of planetary protection issues or because it’s too dangerous.” One concept for a possible aerial vehicle beyond Ingenuity is already being investigated.

Known as the Mars Science Helicopter, this six-bladed hexacopter would weigh nearly 30 kilograms. And it would be equipped with several kilograms worth of instruments to analyze different regions of the Martian surface and would have the ability to fly for minutes at a time over several kilometers. €œWe’re trying to learn from Ingenuity and ask ourselves, ‘What could we accomplish if we push it further?.

€™â€ says Theodore Tzanetos of JPL, who is part of the Mars Science Helicopter concept team. The science such traits would afford would be tremendous, bringing large swathes of the Martian surface suddenly within reach. The current distance record on Mars is held by NASA’s Opportunity rover, which traveled more than 42 kilometers in a little more than 11 years.

A helicopter could achieve the same feat in weeks. Other ideas involve using rotorcraft to perform surveys of exposed water ice on regions of the Martian surface inaccessible to rovers. Drones could dive into Martian valleys like the two-kilometer-deep Mawrth Vallis, looking for evidence of clays linked to astrobiology, or perhaps use instruments to probe the lower reaches of the Martian atmosphere says Shannah Withrow-Maser, the Mars Science Helicopter vehicle systems lead at NASA’s Ames Research Center in California.

And this could all be done either alongside a bigger rover mission or as more cost-effective and much lighter standalone missions, enabling more widespread exploration of a variety of Martian locales. €œI personally would love that,” says Withrow-Maser. Elsewhere in the solar system, flight options are more limited.

One could imagine a rotorcraft in the atmosphere of one of the gas giants such as Jupiter or Saturn, where theoretically flight would be possible. But actually getting there would be an issue. €œThe problem, of course, is slowing down and the amount of energy that would take” on arrival at the planet, Byrne says.

But Titan, Saturn’s intriguing moon with an incredibly thick atmosphere and lakes of hydrocarbons on its surface, is a very tantalizing prospect. In 2019 NASA selected a mission that would attempt to deploy the rotorcraft Dragonfly on the moon. Dragonfly is intended to launch as early as 2026 and arrive in 2034, and its team has been watching Ingenuity’s successes very closely.

€œWe’ve been following with great interest,” says Elizabeth Turtle, lead of the Dragonfly mission at the Johns Hopkins University Applied Physics Laboratory. €œWe’re very anxious to see what lessons we can take forward to Dragonfly.” Like Ingenuity, Dragonfly will be flying autonomously, so it will make use of similar onboard image processing capabilities to decide where to land on the Titanian surface. (Ingenuity performs terrain mapping by taking 30 images of the ground per second.) But Dragonfly is a mammoth compared to Ingenuity, weighing nearly half a metric ton and powered by plutonium.

And it is a standalone mission rather than a ride along like Ingenuity. €œIt’s like Perseverance [in scale], except we fly instead of drive across the surface,” Turtle says. Despite Titan being a much more distant alien world than Mars—with a light travel time from Earth of about an hour, compared with up to about 20 minutes for the Red Planet—flight is relatively easier there.

Titan’s gravity is only 14 percent that of Earth and much less than that of Mars, while the moon’s much thicker atmosphere makes generating lift a comparative breeze. €œA person could put wings on and soar over Titan’s surface,” Turtle says. Winds on Titan are also much slower, barely more than a kilometer an hour versus tens of kilometers an hour on Mars.

And whereas Ingenuity’s blades require 2,500 rpm to lift its fragile 1.8-kg body off the surface, Dragonfly’s half-metric-ton bulk can be lofted just by its rotors spinning at 800 rpm. Titan’s major challenge is its temperature, which averages only about –180 degrees Celsius—hence the need for a long-lived, heat-generating plutonium power source. €œIt’s certainly cold,” Turtle says.

Some heap praise, others scorn, on the idea how to get amoxil without a doctor that disgust is morally valuable. For the advocates, disgust is a powerful and malleable emotion, one that we can shape in order to guard ourselves against morally polluting behavior. Hypocrisy, betrayal, cruelty and the like.

Skeptics, by contrast, maintain that disgust is a misleading and troublingly rigid how to get amoxil without a doctor response. As they see it, we’re too easily disgusted by the morally innocuous and too powerless to keep ourselves from demonizing those we’re disgusted by. Yet, until recently, few have noticed that these assessments of disgust’s moral value turn on empirical questions about what we can do to shape disgust for the better.

Moreover, when we look at what recent research in cognitive science tell us about this question—can how to get amoxil without a doctor we cultivate disgust?. €”we see that both sides are mistaken. We can’t actually cultivate disgust in the ways that its advocates presume.

That said, and in contrast with the skeptics’ assessment, we can improve our how to get amoxil without a doctor ability to control when and how we feel our disgust. This difference between cultivating and controlling disgust is subtle but important. And once we recognize it, we’re forced to rethink not only our assessments of disgust’s moral value but also more fundamental questions about what becoming a more virtuous person involves.

Let’s start how to get amoxil without a doctor by considering disgust’s virtues. Not only do we tend to experience disgust toward moral wrongs like hypocrisy and exploitation, but the shunning and social excluding that disgust brings seems a fitting response to those who pollute the moral fabric in these ways. Moreover, in the face of worries about morally problematic disgust—disgust felt at the wrong time or in the wrong way—advocates respond that it’s an emotion we can substantively change for the better.

On this front, disgust’s advocates point how to get amoxil without a doctor to exposure and habituation. Just like I might overcome the disgust I feel about exotic foods by trying them, I can overcome the disgust I feel about same-sex marriage by spending more time with gay couples. Moreover, work in psychology appears to support this picture.

Medical school students, for instance, lose their disgust about how to get amoxil without a doctor touching dead bodies after a few months of dissecting corpses, and new mothers quickly become less disgusted by the smell of soiled diapers. But these findings may be deceptive. For starters, when we look more closely at the results of the diaper experiment, we see that a mother’s reduced disgust sensitivity is most pronounced with regard to her own baby’s diapers, and additional research indicates that mothers have a general preference for the smell of their own children.

This combination suggests, contra the disgust advocates, that a mother’s disgust is not being how to get amoxil without a doctor eliminated. Rather, her disgust at the soiled diapers is still there. It’s just being masked by the positive feelings that she’s getting from the smell of her newborn.

Similarly, when we look how to get amoxil without a doctor carefully at the cadaver study, we see that while the disgust of medical students toward touching the cold bodies of the dissection lab is reduced with exposure, the disgust they feel toward touching the warm bodies of the recently deceased remained unchanged. All this may seem like fodder for the skeptic’s claim that disgust is morally problematic. After all, it seems there’s little we can do to shape our disgust for the better.

But that would how to get amoxil without a doctor be too quick. While there may not be much we can do to substantively change what we’re disgusted by, we may be able to improve our ability to control when and how we feel our disgust. More specifically, even if disgust itself is too rigid to be changed, it appears there are other psychological mechanisms associated with disgust—things like our attentional systems and cognitive processing routines—that are more malleable.

So, focusing on these mechanisms may offer a better strategy for how to get amoxil without a doctor addressing morally problematic disgust. We get a hint of this in the diaper experiment, where it appears that mothers’ disgust responses are canceled out by the positive feelings they experience through mother-child bonding processes. And this picture finds further support in research highlighting the effectiveness of “implementation intentions” for our ability to control problematic disgust.

At a gloss, implementation intentions are the if-then rules that how to get amoxil without a doctor guide our actions. Importantly, strategies that appeal to them aren’t trying to directly change a person’s disgust. Rather, they’re aiming to develop people’s (nondisgust) attentional capacities.

Allowing them to better recognize situations where disgust response may misfire, so how to get amoxil without a doctor that they can better control the resulting disgust. For instance, someone disgusted by the sight of blood might adopt an implementation intention like “If I see blood, I’ll adopt the perspective of a physician,” or “If I see blood, I’ll stay calm and relaxed,” in order to moderate both their assessment of how disgusting the blood is and their subsequent reactions to it. While researchers have yet to investigate the effectiveness of implementation intentions as correctives for morally problematic disgust, multiple studies have found the technique effective as a way of combating excessive disgust experienced in nonmoral situations (e.g., seeing bodily fluids).

So where does all of this leave us regarding the question of disgust’s moral value? how to get amoxil without a doctor. For starters, we can see that advocates are right that disgust is a morally powerful response to hypocrites, cheaters and the like. Without disgust, we’d lack an important way of responding to those who take advantage of others.

But advocates are how to get amoxil without a doctor wrong in thinking disgust is a malleable emotion that we can substantively change for the better. In the other direction, we also see that skeptics overstate their concerns. Though we cannot substantively change morally problematic disgust, we can learn to effectively control it through the use of implementation intentions.

To see how to get amoxil without a doctor what this might look like, consider someone who is strongly disgusted by members of a particular minority group (let’s call that group the “Gs”). Such an individual would be well-served to adopt implementation intentions aimed at helping him control his disgust—something like, “If I see Gs, I’ll adopt the perspective of Martin Luther King, Jr.” or “If I see Gs, I’ll relax and be friendly.” As suggested above, deploying such a strategy should allow them to better recognize situations where their disgust response may misfire, so that they can engage implementation intentions that will help them control their reaction. But there may be a further lesson here.

The dominant philosophical how to get amoxil without a doctor view of moral development, one that has roots in Aristotle, sees becoming virtuous as a process whereby one transforms problematic emotions. The cowardly person’s fear is transformed into the courageous person’s emotional attunement to danger. But looking closely at the science of disgust reveals that not all emotions are like this.

Some emotions resist our efforts to substantively change them for the how to get amoxil without a doctor better. So, in these cases, becoming a more virtuous person is not a matter of seeking emotional transformation. Rather, it’s the process by which we improve our emotional self-awareness and self-control.

This is an opinion and analysis article.This spring, how to get amoxil without a doctor the 17-year cicadas of Brood X will emerge from underground, climb tree trunks and molt, leaving their crunchy shells behind. Soon after, the males will join together in a droning chorus to the delight (or consternation) of their human neighbors. Those with a keen ear might detect that there are several buzzy songs occurring at once.

This is not how to get amoxil without a doctor because the cicadas have a large repertoire. Rather, there are few different cicada species, including the Magicicada septendecim and the Magicicada cassini, each with a different tune. In the early 19th century this was still a mystery, but the entomologist Margaretta Hare Morris had suspicions.

Ever since she had been a teenager, how to get amoxil without a doctor she had carefully observed the emergence of cicadas. She had heard the different cicada songs in 1817 and again 1834. It was in 1846, though, when she was 49, that Morris felt confident enough to announce that she had discovered a new species.

Digging down beneath her fruit trees, Morris had found cicada larvae sucking how to get amoxil without a doctor at the roots, five years before they were scheduled to emerge. Even just determining how the cicadas subsisted underground for 17 years was a breakthrough in 1846. She also found something else.

Some were significantly smaller than others how to get amoxil without a doctor. Morris sent a report of her discoveries to the Academy of Natural Sciences of Philadelphia in 1846, one of the leading scientific organizations at the time. €œI am inclined to believe that there are two species, differing sufficiently in size,” she wrote.

Morris described the larger cicadas as sluggish with a droning song that sounded like “Faaaa ROO.” The smaller, unnamed species was “extremely active, springing backwards with a sudden motion” with a tune “sharp and shrill, like the noise made by the loom of a stocking weaver.” For those of us unfamiliar with the sound of 19th-century stocking weavers, the modern entomologist Gene Kritsky has described their calls as “a short how to get amoxil without a doctor series of rattling, rapid clicks followed by a longer buzzing or ‘swishing’ sound.” Given that Morris wasn’t a member of the Academy of Natural Sciences, she had to have a male scientist read and present her report and specimens to her peers. Still, she persisted. She published articles in popular journals and invited the country’s leading scientists to come to her garden and witness her discoveries, creating a large network of supporters ready to endorse her methods.

It was due to these efforts that Morris was one of the first women elected to the American Association for the Advancement of Science how to get amoxil without a doctor in 1850, alongside the astronomer Maria Mitchell. Still, you’ve probably never heard her name. We have forgotten Morris and her work for several reasons, one of which involves this cicada species.

In 1851 after that year’s cicada season was over, John Cassin (an ornithologist) and how to get amoxil without a doctor James Coggswell Fisher (a geologist), members of the Academy of Natural Science who had read Morris’s reports, proclaimed to have discovered a new species of cicada. It was much smaller and shriller than its better-known relative (this sounds familiar, yes?. ).

They named how to get amoxil without a doctor it for themselves. Cicada Cassinii (Fisher, 1851). What, ultimately, is in a name?.

Why does this matter? how to get amoxil without a doctor. Given what I know about her, Margaretta Morris likely would not care that the insect she discovered was named for someone else. She wasn’t particularly interested in scientific fame.

She was hoping to better understand the how to get amoxil without a doctor wonders of the insect world. Still, the fact that Cassini and Fisher jumped at the chance to name the bug for themselves means that when the excitement about Brood X’s emergence this spring is covered in newspapers, on podcasts, and in books, we will hear Cassin’s name over and over again. He has been immortalized while Morris has been erased.

If Morris’s name had been tied to this tiny shrill creature as well as the other insects she discovered, she could have served as a how to get amoxil without a doctor model for others like herself who felt a passion for studying insects. Women entomologists were hard to come by in the 19th century, and they continue to be significantly outnumbered today. When the cracks in the glass ceiling are obscured, it can feel daunting for the next person looking up at that ceiling, trying to determine how to break through.

It is for this reason that it’s important to remember pioneers who pursued their passions no matter how lonely it had been and no matter how many times someone else took credit how to get amoxil without a doctor for their work. It is no less important to give credit where credit is due. So, when you hear that shrill cicada sound this spring, whether you find it obnoxious or wondrous, know that there was a wily 19th-century scientist named Margaretta Morris who was entranced by the mystery of these creatures.

She would have awaited the emergence of the 17-year cicadas with glee.On April 11, 2020, John Horton Conway died of buy antibiotics at the age of 82 in New how to get amoxil without a doctor Brunswick, N.J. The areas of research covered by this remarkable mathematician included group theory, node theory, geometry, analysis, combinatorial game theory, algebra, algorithmics and even theoretical physics. Conway’s inclinations and talent led him to invent a remarkable cellular automaton called the Game of Life, which continues to fascinate after 50 years.

Conway also devised elegant puzzles for packing boxes of blocks that can only be solved efficiently how to get amoxil without a doctor with clever reasoning. According to Conway, his most important contribution was his conceptualization of a marvelous system of numbers called surreal numbers. This class comprises integers, real numbers, transfinite numbers and infinitesimals—a structure that no one previously imagined was possible in which everything can be added, multiplied, and so on.

People who worked with Conway report that he thought so fast that no sooner how to get amoxil without a doctor did he hear a problem stated than he often already had a solution. Conway’s commitment to mathematics for everyone led him to work on puzzles that delight fans of recreational mathematics, such as the famous Collatz conjecture (which is discussed toward the end of this article). Conway was the mathematician most cited (more then 30 times!.

) in my columns in Pour la Science, how to get amoxil without a doctor the French edition of Scientific American. In preparing the articles, I would often come across a result he had demonstrated or an important idea that he had been the first to propose on the topic, much to my surprise. Conway liked simple, practical mathematical problems that called on his creativity.

The best way to honor him, it how to get amoxil without a doctor seems to me, is to highlight the kind of mathematics that amazed and fascinated him. I am going to cover several different topics to which Conway contributed. But it would take several volumes to do justice to this exceptional mind.

He was able to invent objects and problems in many different domains and to solve the most how to get amoxil without a doctor recalcitrant puzzles, conceiving methods no one could have imagined. The Irrationality of √2 One of the most surprising and important mathematical findings is the irrationality of the square root of 2 (√2)—the length of the diagonal of a square with sides that are one unit long. It cannot be expressed by the quotient of positive integers n and m, or n/m.

The discovery of the irrationality of √2 is credited to Pythagoras or one of his disciples, although we do not know whether the reasoning behind it was arithmetic or how to get amoxil without a doctor geometric. The discovery and its proof were profoundly unsettling for mathematicians. This first negative finding in mathematics showed that humans do not create the laws governing numbers but rather uncover them as they explore uncharted mathematical territory.

Though there are many proofs of this theorem, the most intuitive is a very simple little drawing that Conway included in how to get amoxil without a doctor a lecture published in a book in 2005. He attributed the creation of the proof to mathematician Stanley Tennenbaum, who, according to Conway, had abandoned mathematics. You might ask whether it was Conway himself who formulated the proof.

But that does how to get amoxil without a doctor not matter. Because even if he did not create it, the proof offers a perfect example of Conway’s approach to mathematics, which he demonstrated in 100 different ways. It also shows that it is wrong to believe that everything simple has already been discovered.

Brilliant yet how to get amoxil without a doctor astonishingly simple ideas are still waiting to be revealed. Say that √2 is the quotient of n and m—that is, 2 = n2/m2, or 2m2 = n2. If so, there exists a square, with sides equal to n, whose area equals twice that of a square with sides equal to m [see part A of “An Irrational Square Root”].

We assume that in our drawing, m is the smallest positive integer satisfying how to get amoxil without a doctor this equation. The assumption would be valid only if there is no smaller positive integer that satisfies it. Credit.

Pour la Science Wedging the two blue squares into two diagonally opposite corners of the red square produces how to get amoxil without a doctor a new shape. The two red squares in that shape must have the same area as the central purple square that is created where the two blue squares overlap [see Part B of “An Irrational Square Root”]. Our reasoning requires that the area of the two blue squares must be equal to the area of the red one (A).

Consequently, the area not covered by the two blue squares is equal to twice the area how to get amoxil without a doctor covered by both of them. In other words, there are two equal and smaller squares (red, B) that together have the same area as the larger square (purple, B). That means each side of the new small squares is equal to the integer n – m, and each side of the large purple square is equal to the integer n – 2(n – m) = 2m – n.

In short, how to get amoxil without a doctor the initial square of side m was not the smallest possible one that satisfies the geometric equation. The result is a contradiction, and thus the assumption is false. ˆš2 is not a quotient of two integers, which means it is an irrational number.

In the same way, you can demonstrate that the square root of 3 is irrational [see part C of “An Irrational Square how to get amoxil without a doctor Root”]. Three Cube-Packing Puzzles Many puzzles consist of putting a small number of blocks in a box (say 10). Both the blocks and the box are typically rectangular parallelepipeds.

The solution is usually arrived at by trial and how to get amoxil without a doctor error. With a problem like this one, unless you increase the number of blocks and their variety of shapes, it is hard to conceive a challenging puzzle. Moreover, although it can be fun to manipulate the blocks until the solution is found, doing so involves only a little reasoning about shape and thus does not really reduce the solution time.

Conway reimagined the problem by creating puzzles that can potentially lead how to get amoxil without a doctor you in circles while you try to figure out how to fill the box. But a few astute and orderly considerations will quickly guide you to the solution. The simplest of these three puzzles consists of filling a 3 x 3 x 3 box with three 1 x 1 x 1 cubes and six 1 x 2 x 2 parallelepipeds [see part 1 of “Blocks in a Box”].

Credit how to get amoxil without a doctor. Pour la Science The reasoning consists of taking parity into account. The 3 x 3 x 3 box is composed of three horizontal 3 x 3 x 1 layers, each with a volume of 9.

The 3 x 3 how to get amoxil without a doctor x 3 box can also be decomposed into three vertical 3 x 3 x 1 parallelepipeds, each parallel to a vertical face of the box. Considering the perpendicular vertical face produces a third decomposition of the box into three 3 x 3 x 1 parallelepipeds. Let’s examine these nine parallelepipeds.

When a 1 x 2 x 2 block is positioned in the box, it can only fill an even number of the nine how to get amoxil without a doctor cells in each 3 x 3 x 1 layer. So each layer must contain one—and only one—of the three small 1 x 1 x 1 blocks (otherwise you would have to place all three of them in a single 3 x 3 x 1 layer, which would leave none in the other two layers, where they are needed). The upper layer of the box must therefore contain one 1 x 1 x 1 block.

If we place how to get amoxil without a doctor it in the center of the layer, we quickly reach a dead end, because we are obliged to place another small block just below it. That puts two of them in a single 3 x 3 x 1 vertical layer. Placing the small block on the top face in the middle of any side also results in an impasse.

Consequently, the small block how to get amoxil without a doctor on the top layer must go in a corner. The same reasoning applies to the bottom layer, where a small block must be placed in a corner. That corner must be diametrically opposite to the small block on the top layer.

The third small block must how to get amoxil without a doctor therefore go in the center of the middle layer. The remaining steps become quickly obvious. Note that this reasoning not only produces a solution but also shows that, except for symmetries, there is only one solution.

The solution how to get amoxil without a doctor to another of Conway’s packing problems is shown in part 2 of “Blocks in a Box.” To complete the packing, all you have to do is to lower the two upper structures and place the green layer (b) at the side on top of the pink layer (a). Reasoning based on parity forces us to place the unit cubes along the diagonal. The Riddle of the Two Wizards In the 1960s Conway devised a fiendishly tricky puzzle that, until recently, prompted much debate and the publication of a paper by Tanya Khovanova of the Massachusetts Institute of Technology in 2013.

Here is the puzzle as it appears in that how to get amoxil without a doctor paper. Last night I sat behind two wizards on a bus, and overheard the following:A. €œI have a positive integral number of children, whose ages are positive integers, the sum of which is the number of this bus, while the product is my own age.”B.

€œHow interesting! how to get amoxil without a doctor. Perhaps if you told me your age and the number of your children, I could work out their individual ages?. €A.

€œNo.”B. €œAha!. AT LAST I know how old you are!.

€Now what was the number of the bus?. Note that in the exchange, the “no” uttered by Wizard A does not mean that he refuses to answer but that the knowledge of his age and the number of his children does not make it possible to know their individual ages. Of course, you have to assume that Wizard B knows the number of the bus.

Note, too, that the wizards can be very young or very old. Wizard A could well be two or 20,000 years old. Here is the solution, which I was only able to fully understand and verify with the help of a program.

I can’t reproduce all the calculations required for the conclusion, but you can take my word for it —or redo all the calculations I’ve left out. Let us denote the age of Wizard A as a, the number of the bus as b and the number of Wizard A’s children as c. Suppose, for example, that the number of the bus is b = 5.

The following are the options for the number of children, the distribution of their ages and the age of Wizard A. C = 5 of ages 1, 1, 1, 1, 1. Thus, a = 1;c = 4 of ages 1, 1, 1, 2.

Thus, a = 2;c = 3 of ages 1, 1, 3. Thus, a = 3;c = 3 of ages 1, 2, 2. Thus, a = 4;c = 2 of ages 1, 4.

Thus, a = 4;c = 2 of ages 2, 3. Thus, a = 6;c = 1 of age 5. Thus, a = 5.

In each case, knowing the age of Wizard A and the number of children indicates the possible ages of the latter. Because Wizard A answered “no” and Wizard B knows the number of the bus, this means b is not equal to 5. Similarly, you can solve the problem by examining the possible bus numbers one by one to find those for which knowing the age of the wizard and the number of his children will not enable you to know the ages of each of the children (a property we denote as P).

Calculating b = 1, 2, 3, ..., 12 (as we have just done for b = 5) shows that b = 12 is the smallest number possessing P. Indeed, for b = 12 and c = 4, there are two sets of possible ages of the four children—(2, 2, 2, 6) and (1, 3, 4, 4)—which both give the same age for Wizard A. A = 48.

No other two sets of the same length with the same product for b = 12 exist. Therefore, for b = 12, even knowing that c = 4 and a = 48, it is impossible to deduce the ages of the four children. Does this mean that b = 12 is the solution?.

Unfortunately, not yet. For bus number b = 13, for example, because the two possible sets of the three children’s ages—(1, 6, 6) and (2, 2, 9)—are both compatible with a = 36, Wizard B cannot derive the ages of Wizard A’s children from the knowledge of either his age or the number of his children. Knowing that b = 12 is no more conclusive about the ages of the children than knowing that b = 13.

When confronted with the puzzle, most people often answer “b = 12,” as if the riddle somehow implies that the smallest possible solution for b is the right one. But the puzzle does not make that assertion. Moreover, without further reasoning, you cannot choose between b = 12 and b = 13, nor can you choose among other values of b, as my further calculations show.

Yet b = 12 is the correct answer, and the reason for that is the most interesting and unexpected part of the riddle. Conway crafted his puzzle carefully, and you must consider the final statement of Wizard B. Following Wizard A’s “no,” Wizard B responds, “Aha!.

AT LAST I know how old you are!. € That eliminates b = 13. In fact, for b = 13, there are two additional sets of the children’s ages—(1, 2, 2, 2, 6) and (1, 1, 3, 4, 4)—that give a = 48.

In other words, if the bus number were 13, Wizard B could not deduce the age of Wizard A from his negative answer because his age could be 36 or 48. Thus, b = 13 should be eliminated. Yet eliminating b = 13 leads to the elimination of b = 14 when we consider the age sequences found for b = 13 and add 1.

Doing so shows that the product of two sets of the children’s ages—(1, 1, 6, 6) and (1, 2, 2, 9)—is a = 36, whereas the product of another two sets—(1, 1, 2, 2, 6) and (1, 1, 1, 3, 4, 4)—is a = 48. The same process eliminates b = 15 and, one by one, all of the b’s greater than 12. Consequently, it is only bus number 12 (b = 12), with two sets of the children’s ages—(2, 2, 2, 6) and (1, 3, 4, 4)—that uniquely determines the age of Wizard A.

A = 48. Given all the calculations you’d have to do to arrive at the solution—the details of which I have not reproduced here and which take up several pages—I confess that I don’t understand how Conway managed to conceive this incredible puzzle!. Paving a Plane with Line Segments Paving a plane with squares is easy.

It is just as easy to do so using equilateral triangles or hexagons. It is also possible to pave the plane with infinite straight lines. Just place them all side by side and parallel.

There will be an infinite number of them, but the paving will be perfectly satisfactory because each point of the plane will belong to one —and only one—line of the paving. Here are a few trickier questions. Can the plane be paved with closed segments—i.e., straight line segments [A, B], including their endpoints?.

Can the plane be paved with open segments—i.e., segments ]A, B[, without their endpoints?. Can the plane be paved with semi-open segments—i.e., segments ]A, B], with only one of their endpoints?. Think about these unusual yet perfectly natural questions.

They are not so simple. Conway and a colleague discussed them and other types of problems in a splendid paper published in 1964, proving once again that very simple questions that no one thinks about are great opportunities to do mathematics that is not so obvious. In that paper, Conway and mathematician Hallard Croft devised a way to fill a plane with straight line segments, which can be seen in “Fine Paving” below.

Credit. Pour la Science Panel a shows how to pave a plane with semi-open segments in which one point is included and another is excluded. Doing so is easy because placing them together, head to tail, reproduces the obvious paving with straight lines.

Panel b shows the solution for equal closed segments in which both points of each segment are included. Pillar 1 is placed first. Pillar 2 is then added, but of course the leftmost segment of this stack is not retained.

For pillar 3, the rightmost segment is not retained. Ever finer slanted pillars are added successively, omitting the “first” segment for each. Panel c shows a solution for differently sized open segments (with their endpoints excluded).

The segments create a central square open on all sides—i.e., neither the sides nor the top or bottom of the square are covered. Each successively added rectangle is also open. For open segments of the same length, there is no solution for paving the plane.

The Collatz Conjecture Another puzzle is as interesting to amateur mathematicians as it is to professionals. Consider a function (f) that gives n/2 for positive integers (n) that are even and 3n + 1 for those that are odd. For example, starting with 7 and applying f, you get 22, then 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, and so on.

Once you get to 1, you go around in circles. Whatever integer n you start with, you always seem to end up at 1 and then get stuck in a loop. Computer calculations have tested this property, which is true for all n’s up to at least 87 x 260 (about 1020).

It has not been proved, however, that this is always the case, nor has an initial n been found that would continue to infinity or lead to a cycle other than 4, 2, 1. This problem is called the Collatz conjecture or the Syracuse conjecture. Much work has been devoted to it, some of which has been compiled in a book.

The great simplicity of the problem’s statement attracts amateurs, and I regularly receive proposed solutions that, to date, are either wrong or incomprehensible. Faced with such a challenge—so simply stated and yet unsolved, which is still the case after more than 80 years—Conway could not resist. In 1972 he published his first paper on the subject, in which he proposed similarly formulated problems and demonstrated their undecidability.

For certain values of the starting integer, the sequence generated by the variant of f does not end up arriving at 1, but set theory cannot demonstrate this. More generally, for any logical system of demonstration (S), there is a problem of this category and a starting point for a sequence that does not end up at 1, which cannot be demonstrated by S. In 2013 Conway returned to the problem with probabilistic arguments suggesting that the Collatz conjecture itself is unprovable with the axiom systems that are usually used in mathematics.

This was not a definitive demonstration of the undecidability of the Collatz conjecture. But the types of arguments he proposed seemed to strongly indicate that it is no accident that everyone stumbles in their attempt to prove the conjecture. All mathematicians encounter problems they cannot solve, and Conway was no exception to this general rule.

Nevertheless, his skill as a logician may have afforded some comfort by enabling him to demonstrate undecidability and to elaborate arguments suggesting that this simple yet intractable problem would continue to challenge mathematicians indefinitely. A New Pattern in the Game of Life The Game of Life, first introduced in Martin Gardner’s Mathematical Games column in Scientific American in 1970, is still being studied, and all of the puzzles that it poses have not been solved. A square cell in an infinite, two-dimensional rectangular grid may be alive or dead.

Or the cell may go from live (black) to dead (white), or vice versa, from one generation to another. Or the cell may remain stable, depending on whether its eight nearest neighbor cells are dead or alive. The rule of evolution can be expressed in 12 words.

Birth if three live neighbors. Survival if two or three live neighbors. Some patterns with few live cells at the initial state grow to infinity.

Ideally, the number of live cells increases proportionately with n2, where n is the number of the generation. The optimal density of living cells in a stable part of the grid is 1/2. Noam Elkies of Harvard University demonstrated the proof in 29 pages in 1997.

Credit. Pour la Science The extraordinary pattern shown at the top of “Game of Life Redux” was discovered in April 2020 by computer scientist Mateusz Naściszewski. It is the smallest known pattern that, once launched, grows quadratically to cover the grid of a stable live cell population of density 1/2.

(This pattern is thus the best possible one in terms of speed and density.) The configuration begins with a population of 183 cells. We drew generation 0 and two other generations at different scales [see bottom patterns in “Game of Life Redux”]. It is believed to be impossible to do better than 183, but that has not been demonstrated.

Note, too, that there are patterns that calculate the prime numbers or that even display graphic images of decimal digits of π = 3.14159... One after the other. This article originally appeared in Pour la Science and was reproduced with permission.Peru Scientists declared a Liolaemus lizard the world's highest-altitude reptile after a population was spotted at 5,400 meters in the Andes.

These lizards must endure frigid temperatures, a particular challenge for cold-blooded animals, as well as reduced oxygen and increased ultraviolet radiation. Algeria Analysis suggests a meteorite from the Sahara Desert contains material as old as, or older than, Earth itself. The meteorite holds the oldest-known sample of magma from space and most likely came from a protoplanet forming in the early solar system.

Egypt An excavation on the Red Sea coast revealed what seems to be a pet cemetery from nearly 2,000 years ago, the earliest yet identified. Nearly 600 cats, dogs and monkeys—mostly cats—were carefully buried, many with textiles, pottery or ornate collars. China Researchers tested a new soft-body swimming robot first in a lake, next in the South China Sea, and finally in the Mariana Trench—almost 11,000 meters down—to prove it can flap its fins in extreme pressure as it explores the depths.

Borneo Scientists captured dung beetles in a forest in Sabah, dissected them, then sequenced the DNA in their guts to find matches with nearby wildlife. Because dung lasts within the beetles for about 48 hours, this method can reveal a location's recent visitors and inhabitants. New Zealand Conservation rangers worked with hundreds of volunteers to “refloat” 40 stranded long-finned pilot whales, returning them to open water.

Nine more of the beached whales died.Picture the scene. A small drone the size of a suitcase descends into a dark Martian crevasse—perhaps a lava tube that was formed billions of years ago by volcanic activity on the Red Planet. The drone illuminates its surroundings, recording views never seen before by human eyes as its suite of instruments seeks out signs of past or present alien biology.

Finally, its reconnaissance complete, the drone flies back to a landing zone on the surface to transmit invaluable data back to Earth. After soaking up the Martian sunlight to recharge its batteries, it continues its explorations of terrain inaccessible to any other machine. Far from being some starry-eyed flight of fancy, such a mission could soon become a reality thanks to the resounding success of NASA’s Ingenuity rotorcraft, sometimes referred to as a helicopter or drone—a technology demonstration that has taken place on Mars over the past few weeks.

Carried to the planet by NASA’s Perseverance rover, which touched down on February 18, this small machine, weighing a paltry 1.8 kilograms, was the first attempt at controlled aerial flight on another world—more than a century after that same feat was mastered on Earth by the Wright brothers. €œWe can now say that human beings have flown a rover craft on another planet,” said MiMi Aung, project manager of Ingenuity at NASA’s Jet Propulsion Laboratory (JPL), in a speech to her team from mission control following the successful first flight on April 19. €œWe together now have our Wright brothers moment.” With Ingenuity’s success, space scientists are contemplating the roles that aerial vehicles might play in our exploration of the solar system.

Few worlds possess the necessary conditions for powered aerodynamic flight, namely an atmosphere and rocky surface like that of Mars or Earth, but there are two others of note. €œThe general technique of aerial flight is applicable to places like [Saturn’s moon] Titan and Venus,” said Bob Balaram, chief engineer of the Ingenuity team, in a press briefing following the first flight. The latter’s exceedingly high temperatures and pressures pose some unique challenges.

€œNear the surface it’s closer to swimming,” says Paul Byrne, a planetary scientist at North Carolina State University. Yet flight there is not impossible, which was proved by the Soviet Union’s Vega balloons in 1985. With a rotorcraft called Dragonfly already being developed to visit Titan in the next decade and work continuing on a conceptual successor to Ingenuity, the future looks bright for aerial exploration of alien worlds.

€œThis could be the start of a new era,” Byrne says. Ingenuity’s first flight, from a strip of land on Mars’s Jezero Crater that is now dubbed “Wright Brothers Field,” was modest but impressive. The planet’s atmosphere is incredibly thin, just 1 percent that of Earth, so generating lift is exceedingly difficult.

€œIt’s similar to Earth at about 100,000 feet above the ground,” says Ben Pipenberg, an engineer at defense contractor AeroVironment, who helped build Ingenuity. With Perseverance watching from a safe distance, Ingenuity spun its blades at 2,500 revolutions per minute (rpm) to rise to an altitude of three meters, where it hovered for 30 seconds and performed a 96-degree rotation. Then it descended back to the ground, landing on its four legs, with a total flight time of 39.1 seconds.

From there, things got more complex. The second flight lasted 51.9 seconds, reaching a height of five meters. And it included a lateral movement of about two meters—something not attempted in the confines of the test chamber on Earth where Ingenuity first flew in simulated Mars conditions.

Flight three saw Ingenuity travel half the length of a football field, some 50 meters, reaching a top speed of just more than two meters per second. The fourth flight on April 30 pushed the envelope once again, with Ingenuity remaining airborne for nearly two minutes—117 seconds—and reaching an impressive speed of 3.5 meters per second as it scouted a potential future landing zone over a round trip of more than 260 meters. Ingenuity’s fifth flight—completed on May 7 and initially planned to be its last—sent it on a one-way trip to the new landing zone to await the arrival of Perseverance, its mothership.

Now, this wildly successful technology demonstration drone is entering a new phase of its mission—a second month-long set of more ambitious operational tests. These tests are meant to show how airborne drones “could play an active role in a future rover science mission,” says Dave Lavery, the program executive for Ingenuity at NASA headquarters in Washington, DC. Although Ingenuity will not directly support the science objectives of Perseverance, namely looking for signs of past life on Mars, it will help scout out the rover’s potential route ahead as the team plans their optimal path through Jezero Crater’s riches, or even photograph nearby locations not in the rover’s planned path.

There is even a slim chance Ingenuity could support the rover’s later mission too, if it survives. €œWe might see about potentially looking over the rim of the crater,” Lavery says. Much has been made of how these vehicles might one day support human missions, acting as reconnaissance drones for humans to scout out regions of interest near a landing site or carrying tools between locations.

In the near-term, prospects of more exciting robotic science are on the horizon—perhaps in the same way that the Sojourner rover in 1997, itself a prototype of wheeled exploration and part of NASA’s Pathfinder mission, paved the way for its successors Spirit, Opportunity, Curiosity and now Perseverance. €œI do think we’re going to see some flying vehicles in the future,” says Michael Meyer, lead scientist of NASA’s Mars Exploration Program at the agency’s headquarters in Washington, D.C. €œIt will now be part of our portfolio of methods that we use for exploration.

There are things you can do with a helicopter that you can’t do with other platforms.” Examples could include exploring the aforementioned lava tubes or perhaps approaching crater walls—too high and steep for a rover to scale—where a helicopter could take images and perform some up-close analysis as well. Another example could be studying recurring slope lineae, dark flows on Mars that have arguably been linked to liquid water flowing on the surface. Perversely, it is this possibility of water—and the accompanying risk of contamination with bacteria imported from Earth—that essentially prohibits anyone or anything from setting foot (or wheel) there to seek out signs of native Martian life.

But a hovering drone could look without touching, offering a novel route of exploration. €œA rotorcraft would give us the ability to go and look up close at something that we would otherwise deem not suitable for a rover,” Byrne says, “either because of planetary protection issues or because it’s too dangerous.” One concept for a possible aerial vehicle beyond Ingenuity is already being investigated. Known as the Mars Science Helicopter, this six-bladed hexacopter would weigh nearly 30 kilograms.

And it would be equipped with several kilograms worth of instruments to analyze different regions of the Martian surface and would have the ability to fly for minutes at a time over several kilometers. €œWe’re trying to learn from Ingenuity and ask ourselves, ‘What could we accomplish if we push it further?. €™â€ says Theodore Tzanetos of JPL, who is part of the Mars Science Helicopter concept team.

The science such traits would afford would be tremendous, bringing large swathes of the Martian surface suddenly within reach. The current distance record on Mars is held by NASA’s Opportunity rover, which traveled more than 42 kilometers in a little more than 11 years. A helicopter could achieve the same feat in weeks.

Other ideas involve using rotorcraft to perform surveys of exposed water ice on regions of the Martian surface inaccessible to rovers. Drones could dive into Martian valleys like the two-kilometer-deep Mawrth Vallis, looking for evidence of clays linked to astrobiology, or perhaps use instruments to probe the lower reaches of the Martian atmosphere says Shannah Withrow-Maser, the Mars Science Helicopter vehicle systems lead at NASA’s Ames Research Center in California. And this could all be done either alongside a bigger rover mission or as more cost-effective and much lighter standalone missions, enabling more widespread exploration of a variety of Martian locales.

€œI personally would love that,” says Withrow-Maser. Elsewhere in the solar system, flight options are more limited. One could imagine a rotorcraft in the atmosphere of one of the gas giants such as Jupiter or Saturn, where theoretically flight would be possible.

But actually getting there would be an issue. €œThe problem, of course, is slowing down and the amount of energy that would take” on arrival at the planet, Byrne says. But Titan, Saturn’s intriguing moon with an incredibly thick atmosphere and lakes of hydrocarbons on its surface, is a very tantalizing prospect.

In 2019 NASA selected a mission that would attempt to deploy the rotorcraft Dragonfly on the moon. Dragonfly is intended to launch as early as 2026 and arrive in 2034, and its team has been watching Ingenuity’s successes very closely. €œWe’ve been following with great interest,” says Elizabeth Turtle, lead of the Dragonfly mission at the Johns Hopkins University Applied Physics Laboratory.

€œWe’re very anxious to see what lessons we can take forward to Dragonfly.” Like Ingenuity, Dragonfly will be flying autonomously, so it will make use of similar onboard image processing capabilities to decide where to land on the Titanian surface. (Ingenuity performs terrain mapping by taking 30 images of the ground per second.) But Dragonfly is a mammoth compared to Ingenuity, weighing nearly half a metric ton and powered by plutonium. And it is a standalone mission rather than a ride along like Ingenuity.

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