--- title: The Most Amazing Mathematical Discoveries Made by Amateurs description: "When it comes to math, most people either love it or hate it. But for those that love it, the concept of math as a recreational hobby rather than an academic pursuit dates back thousands of years, at least to the Ancient Egyptians.\n\nWhat makes math particularly interesting as a hobby compared to various sciences is how low the barrier of entry is. That's not to say that math is necessarily easy, but it doesn't require laboratories full of expensive, specialized equipment either. All a person needs to make the next big discovery in math is their imagination and ideally something to write on.\n\nAs such, it's no surprise that history is full of incredible mathematical discoveries made by amateurs. Today we'll be taking a look at some of these discoveries and the people who made them, or at least the people who didn't choose to remain anonymous.\n\n## Superpermutations\n\nIt began, as all great discoveries in math do, with a discussion about anime on 4chan. First airing in 2006, the anime *The Melancholy of Haruhi Suzumiya* was a show heavily featuring time travel. Because the story was about time travel, when the manga was adapted into an anime they decided not to air the episodes in chronological order. It was later rebroadcast in a different order, and was then released on DVD in a third order.\n\nThis got fans of the show wondering: since the episodes could be watched in any order, what is the fewest number of episodes a person would need to watch to have seen all 14 episodes in every order possible. Since there are 14 episodes, that means there are 14 factorial different permutations these episodes could appear in. That comes out to over 87 billion. If we multiply each permutation by its 14 episode length, we get over 1.2 trillion episodes.\n\nBut people knew there was a better answer than that, thanks to superpermutations. Let's say you have two objects, A and B. There are two different permutations of those objects, AB and BA. However, if we were to create a sequence of ABA, that single sequence contains all possible permutations of A and B. If we were to add a third object C, there would be six possible permutations. All six of those can be found in a superpermutation that is only 9 characters long: ABCABACBA.\n\nThese superpermutations can be worked out by hand when the number of objects is really low, but by the time you get to 5 objects it already starts getting really unwieldy. And how would you solve this for the general case of N objects? Fans of *Haruhi* didn't realize it, but they were actually trying to solve a math problem that professional mathematicians had been working on for decades. And on September 16, 2011, an anonymous 4chan user posted a proof.\n\nRather than trying to solve the problem specifically for the 14 episodes of *Haruhi*, this user had found a general solution for the lower bound of any superpermutation. They posted their proof to 4chan, where it went largely unnoticed. It wasn't until seven years later that a professional mathematician found the 4chan user's proof to the *Haruhi* Problem on a math and science fandom wiki. They checked the user's work, and everything they had done was correct. Despite having alluded others for decades, a proof for the lower bound had been discovered.\n\nSome mathematicians who were currently researching superpermutations published the paper *A lower bound on the length of the shortest superpattern*, with \"Anonymous 4chan Poster\" listed as the lead author. Essentially, their paper was just the 4chan post translated into more formal language and notation.\n\nThis story doesn't end there, though. Once this story blew up, it got others thinking about superpermutations. Among them was science fiction author Greg Egan, who came up with a proof for the upper bound of these superpermutations.\n\nFor anyone familiar with how this works, the 4chan user proved that the shortest superpermutation of N objects can't be shorter than the number given by his equation, but it might be longer. Author Greg Egan proved that the shortest one couldn't be longer than the number given by his equation, but it might be shorter. The goal is to get the upper and lower bounds closer and closer together until they finally meet, giving a definitive answer, and Egan's new upper bound was dramatically lower than the previously known upper bound.\n\nThis wasn't Egan's first foray as an amateur mathematician, either. He had coauthored two papers in 2002, and in 2014 he published what became known as Egan's conjecture, which related to spheres and higher dimensional geometry that we simply don't need to get into here.\n\n## Srinivasa Ramanujan\n\nThere is some debate over whether or not Srinivasa Ramanujan should be considered an amateur mathematician. Born in India in 1887, it was clear from a young age that Ramanujan was a math prodigy. He excelled in all subjects in school, but math was his clear passion. By the time he was 11, he was reading advanced books on math designed for students in their final year of undergraduate studies. He reportedly understood these books completely and began developing his own theorems.\n\nAfter graduating high school, Ramanujan was awarded a scholarship to Government Arts College in Kumbakonam. However, he only cared about math so he failed his other subjects, causing him to lose his scholarship. After failing out he ran away from home and enrolled at Pachaiyappa's College in Madras. Once again he only cared about his math courses, but he struggled to pass those as well since he only bothered to answer test questions that he found interesting.\n\nAgain Ramanujan failed out of college, and for a period of time he lived in extreme poverty. He continued his work in math while also attempting to find any sort of clerical or accounting work that he could. Despite attempting to go to college twice, nearly everything Ramanujan knew had been self taught. He had devoured the most advanced materials available at a young age and worked in isolation.\n\nThis lack of formal education in math is why many consider Ramanujan to be an amateur. Unfortunately this meant that, like many amateurs, Ramanujan wasn't good at formally proving his ideas. He filled notebooks with theorems and equations, largely based on his own intuition. This became a problem when he tried to get the attention of Indian mathematicians.\n\nRamanujan would send letters to people containing his work, but to no avail. Though it was clear that his work was impressive, the lack of an ability to prove his equations led many to believe that the work wasn't truly his own. And even if they did believe it was his work, there was a good chance that they wouldn't understand it anyway because it was so advanced.\n\nOver the course of several years, he eventually began making a name for himself in Indian mathematics circles, and once they were able to recognize his brilliance they became extremely supportive of his work. Some of them sent Ramanujan's work to British mathematicians in London, but it did not go as expected. They essentially rejected Ramanujan for being too much of an amateur, stating that his work was full of holes and that he lacked the formal education needed for anybody to take him seriously.\n\nRamanujan then personally sent letters with his work to a few professors at Cambridge. Two returned his work with no reply, but he had caught the eye of G. H. Hardy. Hardy wanted Ramanujan to come to Cambridge, but Ramanujan was hesitant to abandon his family and travel to a foreign land. After a few years of correspondence with Hardy he eventually made the trip to Cambridge, but Ramanujan and Hardy's working relationship was as complicated as it was short. After only a few years in England, Ramanujan had to return to India.\n\nHe had suffered from health problems for his entire life, and Ramanujan died when he was only 32 years old. Despite dying so young and having no formal training in math, to the point that many of the top mathematicians of the day wouldn't even look at his work, Ramanujan left behind notebooks filled with over 4,000 different theorems.\n\nA few of them turned out to be incorrect, and because of his lack of formal education some of Ramanujan's theorems were things he was unaware other mathematicians had already proven. But the vast majority of his work was novel and accurate, and he even provided solutions to questions in math that were not only unsolved, but were considered unsolvable at the time. His contemporaries often struggled to understand what his work even meant, let alone how he could have come up with the ideas for his theorems.\n\nBecause of the volume of his work, it's impossible to point to a single discovery made by Ramanujan as being his most important. What we can say is that his understanding of math was so far ahead of its time that his notebooks have kept mathematicians busy for the past century. Though most of his theorems have now been proven, many still remain unsolved as professional mathematicians attempt to catch up to the level of understanding the self taught genius possessed over 100 years ago.\n\n## Tiling the Plane\n\nTiling the plane, or tessellation, has been of interest to mathematicians, artists, and sculptors for thousands of years. Tessellation is the process of covering a plane with one or more geometric shapes, like a mosaic. Of particular interest to geometers are monohedral tessellations, in which an infinite plane can be completely covered with no gaps or overlap using tiles that are all of a single size and shape.\n\nSome of these are really easy to find. For example every triangle and rectangle can tile the plane seamlessly. However, things get interesting once you try the same thing with pentagons. Many might expect that this would be easiest with a regular polygon, where the angles are all the same size and the sides are all the same length. But because the 108 degree angles of a regular pentagon do not divide evenly into 360 degrees, this is actually impossible. There is no way to tile the plane using regular pentagons without either leaving gaps or overlapping the tiles.\n\nThis led to the question of how you could design a pentagonal tile that would form a tessellation, with the first such shape being identified in 1918 by Karl Reinhardt. He went on to find four more pentagonal tiles, with Richard Kershner finding three more in 1968. When Kershner published his findings, he declared that the eight pentagons discovered were a comprehensive list of all pentagons that could tile the plane.\n\nFast forward eight years, when a 52 year old homemaker and mother of five from San Diego, Marjorie Rice, rushed to her mailbox to get the latest issue of *Scientific American*. Rice was a huge fan of the column \"Mathematical Games\" by Martin Gardner, and she enjoyed reading his articles every month. In the July issue, Gardner wrote about tiling the plane with various polygons, and in it he mentioned Kershner's claim that all pentagonal tiles had been discovered.\n\nHowever, when the December issue was published, Gardner revealed that a reader had submitted a then unknown ninth pentagonal tile. Inspired by this new revelation, Rice decided to experiment with the problem herself. She created her own system of notation to work through the possible relationships between the angles and lengths of the sides, and spent the holiday season drawing diagrams at the kitchen table.\n\nWithin just two months Rice discovered a new pentagon as well as 58 dihedral tessellations that each used two pentagons. She mailed her work to Gardner, who in turn shared it with Doris Schattschneider, an expert in tessellation. As is common when examining work from amateurs, it took Schattschneider a while to figure out what the Hell Rice's notation actually meant. She was eventually able to validate the results, but Rice wasn't done yet.\n\nBy the end of 1977, Rice had discovered a few more new monohedral pentagon tilings. With a total of four new tiles discovered, Rice had now found more tiles than Kershner did back when he claimed that there were no more left to find. Though her discoveries weren't published in Gardner's *Scientific American* column, only being published as an addendum in a 1988 compilation of his articles, Rice still received considerable recognition for her discoveries.\n\nIn 1999, the Mathematical Association of America even used one of Rice's pentagons to tile the foyer of their headquarters in Washington D.C.\n\n## Pierre de Fermat\n\nBorn in southern France in 1607, Pierre de Fermat was referred to as the \"prince of amateurs\" by famed mathematician E.T. Bell. Fermat's contributions to mathematics are so numerable than many even argued he should count as a professional. But despite how prolific Fermat's work was, he really was just a hobbyist.\n\nFermat earned his law degree from the University of Orleans in 1623, and he spent the remainder of his life working as a lawyer and magistrate. In his free time he loved to study math, and he would send letters to his friends and associates detailing his work. Through these letters, he made major contributions to number theory, probability, and analytic geometry. Some of his work even laid the foundation for Newton and Leibniz to discover calculus.\n\nHowever, there are two theorems for which Fermat is most known: Fermat's little theorem and Fermat's last theorem. His little theorem states that for any integer A and any prime number P, A^(P-1) – 1 will be an integer multiple of P.\n\nAs a simple example, let's use 2 for A and 5 for our prime number. Five minus one is four, so we get 2^4 = 16 – 1 = 15, and 15 is a multiple of 5. This simple theorem was developed during Fermat's work with number theory, and it was the basis of his primality test to identify probable prime numbers. It could only find probable primes because sometimes this equation will still work even if P isn't a prime number, but in the days before computers little tricks like these were the best tool mathematicians had to find new prime numbers.\n\nThat just brings us to Fermat's last theorem, which looks something like this:\n\n∄a, b, c, n ∈ Z\\{0}; n>2 | aⁿ+bⁿ = cⁿ\n\nWhile that probably looks really complicated, you might also notice that the bit at the end is reminiscent of the Pythagorean Theorem, A squared plus B squared equals C squared. What Fermat was saying was that there are no integer solutions to Pythagoras's formula if N is greater than 2. This was something that mathematicians had mulled over for thousands of years and were pretty sure was true, but nobody had ever been able to prove it so they never wrote it down as fact.\n\nOf course, Fermat didn't actually prove this either. Or at the very least if he did, he never bothered to write it down. This was a common trend in his career, as his letters frequently included advanced discoveries in math with no proof whatsoever. Indeed, Fermat's little theorem wasn't proven until nearly 100 years after his death when Leonhard Euler became the first to create a successful proof.\n\nIn the letter in which Fermat presented his little theorem, he wrote \"I would send you a demonstration of it, if I did not fear going on too long.\" Similarly, Fermat's last theorem was written inside the margin of his personal copy of *Diophantus of Alexandria's Arithmetica*. Fermat wrote his theorem claiming that he had a proof, but that it was too big to fit in the margin. This was a running theme in his work, and perhaps if Fermat had a little more time and some more paper he could have solved all of the great mysteries of the universe.\n\nUnsurprisingly, many mathematicians believe that Fermat never actually had proofs for these things. It took over 300 years before anybody could solve his last theorem, and it involved advanced branches of math that simply didn't exist at the time. Still, his instincts appear to always be spot on because whether he could prove it or not, the ideas Fermat came up with inevitably turned out to be true.\n\nAnd because of his notoriety and the confidence with which he claimed to have a proof, the belief that a proof for Fermat's last theorem must surely be just over the horizon led to many important advancements in number theory as people tried to find the answer.\n\n## Key Takeaways\n\n- Math has been a recreational hobby since ancient times, requiring minimal equipment.\n- Amateur mathematicians have made significant discoveries, such as the superpermutation proof.\n- Srinivasa Ramanujan, despite limited formal education, contributed over 4,000 theorems.\n- Marjorie Rice, a homemaker, discovered new pentagonal tiling patterns challenging experts.\n- Pierre de Fermat, a lawyer, made foundational contributions to number theory and calculus.\n\n## Frequently Asked Questions\n\n### What is a superpermutation?\n\nA superpermutation is a sequence that contains all possible permutations of a set of objects. For example, the sequence 'ABCABACBA' contains all permutations of the objects A, B, and C.\n\n### How did the Haruhi Problem originate?\n\nThe Haruhi Problem originated from discussions about the anime 'The Melancholy of Haruhi Suzumiya' on 4chan. Fans wondered about the fewest number of episodes needed to watch all 14 episodes in every possible order.\n\n### Who was the first to solve the Haruhi Problem?\n\nAn anonymous 4chan user posted a proof for the lower bound of any superpermutation on September 16, 2011, effectively solving the Haruhi Problem.\n\n### What is the significance of Srinivasa Ramanujan's work?\n\nSrinivasa Ramanujan left behind notebooks filled with over 4,000 theorems, many of which were novel and accurate. His work has kept mathematicians busy for over a century, with many of his theorems still being studied and proven today.\n\n### What is a monohedral tessellation?\n\nA monohedral tessellation is the process of covering an infinite plane with no gaps or overlap using tiles that are all of a single size and shape.\n\n### Who was Marjorie Rice and what did she discover?\n\nMarjorie Rice was a homemaker who discovered several new pentagonal tiles that could tile the plane. She found a total of four new tiles, more than Richard Kershner who had previously claimed to have found all possible pentagonal tiles.\n\n### What is Fermat's Little Theorem?\n\nFermat's Little Theorem states that for any integer A and any prime number P, A^(P-1) - 1 will be an integer multiple of P. It was the basis of Fermat's primality test to identify probable prime numbers.\n\n### What is Fermat's Last Theorem?\n\nFermat's Last Theorem states that there are no integer solutions to the equation a^n + b^n = c^n for any integer value of n greater than 2. Fermat claimed to have a proof but never wrote it down.\n\n### Why is Pierre de Fermat referred to as the 'prince of amateurs'?\n\nPierre de Fermat is referred to as the 'prince of amateurs' because despite his prolific contributions to mathematics, he was primarily a lawyer and magistrate who studied math as a hobby.\n\n### How did Marjorie Rice's work gain recognition?\n\nMarjorie Rice's work gained recognition after she mailed her discoveries to Martin Gardner, who shared them with Doris Schattschneider, an expert in tessellation. Her discoveries were eventually validated and recognized by the mathematical community.\n\n## Sources\n\n- [Original Side Projects video: The Most Amazing Mathematical Discoveries Made by Amateurs](https://www.youtube.com/watch?v=1ocStzx72yw)\n\n## Related Coverage" url: https://sideprojects.pub/article/most-amazing-mathematical-discoveries-made-by-amateurs.md canonical: https://sideprojects.pub/article/most-amazing-mathematical-discoveries-made-by-amateurs datePublished: 2024-09-17 dateModified: 2026-07-28 author: - name: Simon Whistler url: https://sideprojects.pub/author/simon-whistler publisher: Side Projects image: "https://media.sideprojects.pub/cdn-cgi/image/width=1600,height=900,fit=cover,quality=80,format=auto/articles/1ocStzx72yw/hero.jpg" type: NewsArticle contentHash: 40124eddcd67d55734e8b3ec5f79509e151e6e18a005ead956cd97ee282ce8b0 tokens: 4853 summaryUrl: https://sideprojects.pub/article/most-amazing-mathematical-discoveries-made-by-amateurs.md.summary.md --- When it comes to math, most people either love it or hate it. But for those that love it, the concept of math as a recreational hobby rather than an academic pursuit dates back thousands of years, at least to the Ancient Egyptians. What makes math particularly interesting as a hobby compared to various sciences is how low the barrier of entry is. That's not to say that math is necessarily easy, but it doesn't require laboratories full of expensive, specialized equipment either. All a person needs to make the next big discovery in math is their imagination and ideally something to write on. As such, it's no surprise that history is full of incredible mathematical discoveries made by amateurs. Today we'll be taking a look at some of these discoveries and the people who made them, or at least the people who didn't choose to remain anonymous. ## Superpermutations It began, as all great discoveries in math do, with a discussion about anime on 4chan. First airing in 2006, the anime *The Melancholy of Haruhi Suzumiya* was a show heavily featuring time travel. Because the story was about time travel, when the manga was adapted into an anime they decided not to air the episodes in chronological order. It was later rebroadcast in a different order, and was then released on DVD in a third order. This got fans of the show wondering: since the episodes could be watched in any order, what is the fewest number of episodes a person would need to watch to have seen all 14 episodes in every order possible. Since there are 14 episodes, that means there are 14 factorial different permutations these episodes could appear in. That comes out to over 87 billion. If we multiply each permutation by its 14 episode length, we get over 1.2 trillion episodes. But people knew there was a better answer than that, thanks to superpermutations. Let's say you have two objects, A and B. There are two different permutations of those objects, AB and BA. However, if we were to create a sequence of ABA, that single sequence contains all possible permutations of A and B. If we were to add a third object C, there would be six possible permutations. All six of those can be found in a superpermutation that is only 9 characters long: ABCABACBA. These superpermutations can be worked out by hand when the number of objects is really low, but by the time you get to 5 objects it already starts getting really unwieldy. And how would you solve this for the general case of N objects? Fans of *Haruhi* didn't realize it, but they were actually trying to solve a math problem that professional mathematicians had been working on for decades. And on September 16, 2011, an anonymous 4chan user posted a proof. Rather than trying to solve the problem specifically for the 14 episodes of *Haruhi*, this user had found a general solution for the lower bound of any superpermutation. They posted their proof to 4chan, where it went largely unnoticed. It wasn't until seven years later that a professional mathematician found the 4chan user's proof to the *Haruhi* Problem on a math and science fandom wiki. They checked the user's work, and everything they had done was correct. Despite having alluded others for decades, a proof for the lower bound had been discovered. Some mathematicians who were currently researching superpermutations published the paper *A lower bound on the length of the shortest superpattern*, with "Anonymous 4chan Poster" listed as the lead author. Essentially, their paper was just the 4chan post translated into more formal language and notation. This story doesn't end there, though. Once this story blew up, it got others thinking about superpermutations. Among them was science fiction author Greg Egan, who came up with a proof for the upper bound of these superpermutations. For anyone familiar with how this works, the 4chan user proved that the shortest superpermutation of N objects can't be shorter than the number given by his equation, but it might be longer. Author Greg Egan proved that the shortest one couldn't be longer than the number given by his equation, but it might be shorter. The goal is to get the upper and lower bounds closer and closer together until they finally meet, giving a definitive answer, and Egan's new upper bound was dramatically lower than the previously known upper bound. This wasn't Egan's first foray as an amateur mathematician, either. He had coauthored two papers in 2002, and in 2014 he published what became known as Egan's conjecture, which related to spheres and higher dimensional geometry that we simply don't need to get into here. ## Srinivasa Ramanujan There is some debate over whether or not Srinivasa Ramanujan should be considered an amateur mathematician. Born in India in 1887, it was clear from a young age that Ramanujan was a math prodigy. He excelled in all subjects in school, but math was his clear passion. By the time he was 11, he was reading advanced books on math designed for students in their final year of undergraduate studies. He reportedly understood these books completely and began developing his own theorems. After graduating high school, Ramanujan was awarded a scholarship to Government Arts College in Kumbakonam. However, he only cared about math so he failed his other subjects, causing him to lose his scholarship. After failing out he ran away from home and enrolled at Pachaiyappa's College in Madras. Once again he only cared about his math courses, but he struggled to pass those as well since he only bothered to answer test questions that he found interesting. Again Ramanujan failed out of college, and for a period of time he lived in extreme poverty. He continued his work in math while also attempting to find any sort of clerical or accounting work that he could. Despite attempting to go to college twice, nearly everything Ramanujan knew had been self taught. He had devoured the most advanced materials available at a young age and worked in isolation. This lack of formal education in math is why many consider Ramanujan to be an amateur. Unfortunately this meant that, like many amateurs, Ramanujan wasn't good at formally proving his ideas. He filled notebooks with theorems and equations, largely based on his own intuition. This became a problem when he tried to get the attention of Indian mathematicians. Ramanujan would send letters to people containing his work, but to no avail. Though it was clear that his work was impressive, the lack of an ability to prove his equations led many to believe that the work wasn't truly his own. And even if they did believe it was his work, there was a good chance that they wouldn't understand it anyway because it was so advanced. Over the course of several years, he eventually began making a name for himself in Indian mathematics circles, and once they were able to recognize his brilliance they became extremely supportive of his work. Some of them sent Ramanujan's work to British mathematicians in London, but it did not go as expected. They essentially rejected Ramanujan for being too much of an amateur, stating that his work was full of holes and that he lacked the formal education needed for anybody to take him seriously. Ramanujan then personally sent letters with his work to a few professors at Cambridge. Two returned his work with no reply, but he had caught the eye of G. H. Hardy. Hardy wanted Ramanujan to come to Cambridge, but Ramanujan was hesitant to abandon his family and travel to a foreign land. After a few years of correspondence with Hardy he eventually made the trip to Cambridge, but Ramanujan and Hardy's working relationship was as complicated as it was short. After only a few years in England, Ramanujan had to return to India. He had suffered from health problems for his entire life, and Ramanujan died when he was only 32 years old. Despite dying so young and having no formal training in math, to the point that many of the top mathematicians of the day wouldn't even look at his work, Ramanujan left behind notebooks filled with over 4,000 different theorems. A few of them turned out to be incorrect, and because of his lack of formal education some of Ramanujan's theorems were things he was unaware other mathematicians had already proven. But the vast majority of his work was novel and accurate, and he even provided solutions to questions in math that were not only unsolved, but were considered unsolvable at the time. His contemporaries often struggled to understand what his work even meant, let alone how he could have come up with the ideas for his theorems. Because of the volume of his work, it's impossible to point to a single discovery made by Ramanujan as being his most important. What we can say is that his understanding of math was so far ahead of its time that his notebooks have kept mathematicians busy for the past century. Though most of his theorems have now been proven, many still remain unsolved as professional mathematicians attempt to catch up to the level of understanding the self taught genius possessed over 100 years ago. ## Tiling the Plane Tiling the plane, or tessellation, has been of interest to mathematicians, artists, and sculptors for thousands of years. Tessellation is the process of covering a plane with one or more geometric shapes, like a mosaic. Of particular interest to geometers are monohedral tessellations, in which an infinite plane can be completely covered with no gaps or overlap using tiles that are all of a single size and shape. Some of these are really easy to find. For example every triangle and rectangle can tile the plane seamlessly. However, things get interesting once you try the same thing with pentagons. Many might expect that this would be easiest with a regular polygon, where the angles are all the same size and the sides are all the same length. But because the 108 degree angles of a regular pentagon do not divide evenly into 360 degrees, this is actually impossible. There is no way to tile the plane using regular pentagons without either leaving gaps or overlapping the tiles. This led to the question of how you could design a pentagonal tile that would form a tessellation, with the first such shape being identified in 1918 by Karl Reinhardt. He went on to find four more pentagonal tiles, with Richard Kershner finding three more in 1968. When Kershner published his findings, he declared that the eight pentagons discovered were a comprehensive list of all pentagons that could tile the plane. Fast forward eight years, when a 52 year old homemaker and mother of five from San Diego, Marjorie Rice, rushed to her mailbox to get the latest issue of *Scientific American*. Rice was a huge fan of the column "Mathematical Games" by Martin Gardner, and she enjoyed reading his articles every month. In the July issue, Gardner wrote about tiling the plane with various polygons, and in it he mentioned Kershner's claim that all pentagonal tiles had been discovered. However, when the December issue was published, Gardner revealed that a reader had submitted a then unknown ninth pentagonal tile. Inspired by this new revelation, Rice decided to experiment with the problem herself. She created her own system of notation to work through the possible relationships between the angles and lengths of the sides, and spent the holiday season drawing diagrams at the kitchen table. Within just two months Rice discovered a new pentagon as well as 58 dihedral tessellations that each used two pentagons. She mailed her work to Gardner, who in turn shared it with Doris Schattschneider, an expert in tessellation. As is common when examining work from amateurs, it took Schattschneider a while to figure out what the Hell Rice's notation actually meant. She was eventually able to validate the results, but Rice wasn't done yet. By the end of 1977, Rice had discovered a few more new monohedral pentagon tilings. With a total of four new tiles discovered, Rice had now found more tiles than Kershner did back when he claimed that there were no more left to find. Though her discoveries weren't published in Gardner's *Scientific American* column, only being published as an addendum in a 1988 compilation of his articles, Rice still received considerable recognition for her discoveries. In 1999, the Mathematical Association of America even used one of Rice's pentagons to tile the foyer of their headquarters in Washington D.C. ## Pierre de Fermat Born in southern France in 1607, Pierre de Fermat was referred to as the "prince of amateurs" by famed mathematician E.T. Bell. Fermat's contributions to mathematics are so numerable than many even argued he should count as a professional. But despite how prolific Fermat's work was, he really was just a hobbyist. Fermat earned his law degree from the University of Orleans in 1623, and he spent the remainder of his life working as a lawyer and magistrate. In his free time he loved to study math, and he would send letters to his friends and associates detailing his work. Through these letters, he made major contributions to number theory, probability, and analytic geometry. Some of his work even laid the foundation for Newton and Leibniz to discover calculus. However, there are two theorems for which Fermat is most known: Fermat's little theorem and Fermat's last theorem. His little theorem states that for any integer A and any prime number P, A^(P-1) – 1 will be an integer multiple of P. As a simple example, let's use 2 for A and 5 for our prime number. Five minus one is four, so we get 2^4 = 16 – 1 = 15, and 15 is a multiple of 5. This simple theorem was developed during Fermat's work with number theory, and it was the basis of his primality test to identify probable prime numbers. It could only find probable primes because sometimes this equation will still work even if P isn't a prime number, but in the days before computers little tricks like these were the best tool mathematicians had to find new prime numbers. That just brings us to Fermat's last theorem, which looks something like this: ∄a, b, c, n ∈ Z\{0}; n>2 | aⁿ+bⁿ = cⁿ While that probably looks really complicated, you might also notice that the bit at the end is reminiscent of the Pythagorean Theorem, A squared plus B squared equals C squared. What Fermat was saying was that there are no integer solutions to Pythagoras's formula if N is greater than 2. This was something that mathematicians had mulled over for thousands of years and were pretty sure was true, but nobody had ever been able to prove it so they never wrote it down as fact. Of course, Fermat didn't actually prove this either. Or at the very least if he did, he never bothered to write it down. This was a common trend in his career, as his letters frequently included advanced discoveries in math with no proof whatsoever. Indeed, Fermat's little theorem wasn't proven until nearly 100 years after his death when Leonhard Euler became the first to create a successful proof. In the letter in which Fermat presented his little theorem, he wrote "I would send you a demonstration of it, if I did not fear going on too long." Similarly, Fermat's last theorem was written inside the margin of his personal copy of *Diophantus of Alexandria's Arithmetica*. Fermat wrote his theorem claiming that he had a proof, but that it was too big to fit in the margin. This was a running theme in his work, and perhaps if Fermat had a little more time and some more paper he could have solved all of the great mysteries of the universe. Unsurprisingly, many mathematicians believe that Fermat never actually had proofs for these things. It took over 300 years before anybody could solve his last theorem, and it involved advanced branches of math that simply didn't exist at the time. Still, his instincts appear to always be spot on because whether he could prove it or not, the ideas Fermat came up with inevitably turned out to be true. And because of his notoriety and the confidence with which he claimed to have a proof, the belief that a proof for Fermat's last theorem must surely be just over the horizon led to many important advancements in number theory as people tried to find the answer. ## Key Takeaways - Math has been a recreational hobby since ancient times, requiring minimal equipment. - Amateur mathematicians have made significant discoveries, such as the superpermutation proof. - Srinivasa Ramanujan, despite limited formal education, contributed over 4,000 theorems. - Marjorie Rice, a homemaker, discovered new pentagonal tiling patterns challenging experts. - Pierre de Fermat, a lawyer, made foundational contributions to number theory and calculus. ## Frequently Asked Questions ### What is a superpermutation? A superpermutation is a sequence that contains all possible permutations of a set of objects. For example, the sequence 'ABCABACBA' contains all permutations of the objects A, B, and C. ### How did the Haruhi Problem originate? The Haruhi Problem originated from discussions about the anime 'The Melancholy of Haruhi Suzumiya' on 4chan. Fans wondered about the fewest number of episodes needed to watch all 14 episodes in every possible order. ### Who was the first to solve the Haruhi Problem? An anonymous 4chan user posted a proof for the lower bound of any superpermutation on September 16, 2011, effectively solving the Haruhi Problem. ### What is the significance of Srinivasa Ramanujan's work? Srinivasa Ramanujan left behind notebooks filled with over 4,000 theorems, many of which were novel and accurate. His work has kept mathematicians busy for over a century, with many of his theorems still being studied and proven today. ### What is a monohedral tessellation? A monohedral tessellation is the process of covering an infinite plane with no gaps or overlap using tiles that are all of a single size and shape. ### Who was Marjorie Rice and what did she discover? Marjorie Rice was a homemaker who discovered several new pentagonal tiles that could tile the plane. She found a total of four new tiles, more than Richard Kershner who had previously claimed to have found all possible pentagonal tiles. ### What is Fermat's Little Theorem? Fermat's Little Theorem states that for any integer A and any prime number P, A^(P-1) - 1 will be an integer multiple of P. It was the basis of Fermat's primality test to identify probable prime numbers. ### What is Fermat's Last Theorem? Fermat's Last Theorem states that there are no integer solutions to the equation a^n + b^n = c^n for any integer value of n greater than 2. Fermat claimed to have a proof but never wrote it down. ### Why is Pierre de Fermat referred to as the 'prince of amateurs'? Pierre de Fermat is referred to as the 'prince of amateurs' because despite his prolific contributions to mathematics, he was primarily a lawyer and magistrate who studied math as a hobby. ### How did Marjorie Rice's work gain recognition? Marjorie Rice's work gained recognition after she mailed her discoveries to Martin Gardner, who shared them with Doris Schattschneider, an expert in tessellation. Her discoveries were eventually validated and recognized by the mathematical community. ## Sources - [Original Side Projects video: The Most Amazing Mathematical Discoveries Made by Amateurs](https://www.youtube.com/watch?v=1ocStzx72yw) ## Related Coverage