An interactive field guide to patterns, proofs and open questions.
Mathomaly is an interactive field guide to surprising mathematical behaviour, from the first visible pattern to what is proved, what is only observed, and what remains unknown. Change the numbers or the rule and follow what happens.
Watch No. 03: Why does chance fool us? 9:13Write a whole number from 0 to 9999 using four digits, adding leading zeroes if needed. Arrange its digits from largest to smallest and from smallest to largest, then subtract the smaller arrangement from the larger and repeat. For instance, 3524 gives 5432 − 2345 = 3087. Unless all four digits are identical, this reaches 6174 within seven steps, and the same rule leaves 6174 unchanged.
Begin with the primes in order: 2, 3, 5, 7, 11 and so on. Make a new row from the absolute differences between neighbours, subtracting the smaller from the larger: the first differences are 1, 2, 2 and 4. Repeat with each new row. Every computed difference row begins with 1, but whether this always happens remains unproved.
Start with a positive whole number written in base 10, square each digit and add the squares, then repeat with the total. For example, 19 becomes 82 because 1² + 9² = 82. Every start eventually reaches 1 or the same eight-number cycle; a number that reaches 1 is called happy.
Start at 0 and use jump lengths 1, 2, 3 and so on. For each jump, subtract its length if the result is positive and has not appeared before; otherwise add it, even if that repeats a value. The sequence begins 0, 1, 3, 6, 2, 7, and drawing an arc for each jump reveals its overlapping structure.
Build Pascal’s triangle from a top 1, placing 1 at each edge of a new row and adding neighbouring entries to fill the inside. Colour odd entries and leave even ones blank: the row 1, 2, 1 becomes coloured, blank, coloured. As you add rows, repeated triangular gaps form larger finite pieces of the Sierpiński gasket.
For any positive whole number, the Collatz rule gives the next number by halving it if it is even, or multiplying it by three and adding one if it is odd. Repeating this from 6 gives 6 → 3 → 10 → 5 → 16 → 8 → 4 → 2 → 1. Whether every positive start eventually reaches 1 is still unproved.
Start with a positive whole number in base 10, reverse its digits and add that reversal to the original, repeating with the result. Stop if the result is a palindrome, meaning it reads the same forwards and backwards: 59 reaches 1111 after three additions. No palindrome has been found from 196, but whether it ever reaches one remains unresolved.
Place numbered points evenly around a circle and draw a straight line from each label to twice that label, wrapping around when you pass the last point. With 10 points labelled 0 to 9, label 6 connects to 2 because 12 wraps to 2. This doubles labels, not line lengths; with more points, a heart-shaped outline called a cardioid becomes visible.
Write non-negative whole numbers in a ring. Replace every entry at once by its absolute difference from the clockwise neighbour, using only the old ring to calculate the new one. Four entries always reach all zeroes, but three entries can repeat forever: the ring 0, 1, 1 cycles through 1, 0, 1 and 1, 1, 0 before returning.
Number people from 1 around a circle, count person 1 as the first, and remove every second person while continuing around the ring. With five people, the removals are 2, 4, 1 and 5, leaving person 3. For this counting rule, moving the leading 1 in the group size’s binary notation to the end gives the survivor’s position.
Start with a non-negative whole number written in base 10 and multiply its digits together, repeating until only one digit remains. For 39, the products are 27, 14 and 4, so it takes three rounds. Every start finishes, but it remains unknown whether any base-10 start needs twelve or more rounds.
Begin with the digit 1 and describe each consecutive group by its count followed by its digit. One 1 becomes 11, two 1s become 21, and one 2 followed by one 1 becomes 1211. From this start only 1, 2 and 3 ever appear, and Conway proved that the length ratio approaches a constant near 1.303577. Repeating the description makes longer strings whose growth this page tracks.
Start with a positive whole number, add its positive divisors other than itself, and repeat with the total. For 12, those divisors are 1, 2, 3, 4 and 6, giving 16. Some chains reach 0 and stop, while others repeat; for starts such as 276, their eventual behaviour remains unknown.
Langton’s ant follows a tiny rule invented in 1986: on an all-white square grid it turns right on white and left on black, then flips the square’s colour and moves forward one square. From this start, a repeating diagonal path appears after about 10,000 steps; this finite wrapping grid lets you inspect its formation, not prove its infinite continuation.
In 1742 Goldbach and Euler exchanged letters about sums of primes, and Euler’s reply gave the form still open today: every even whole number greater than 2 is a sum of two primes. Choose such an even number and look for two primes whose sum equals it. For example, 10 is both 3 + 7 and 5 + 5, so the primes may be equal. Extensive searches have found no counterexample, but a proof remains unknown.
Mark distinct points on a circle, join every pair with a straight line segment and count the regions inside the circle. When no three segments cross at the same interior point, one through five points give 1, 2, 4, 8 and 16 regions. Six points give 31, not 32, which is the break Moser made famous: the early counts alone do not establish a doubling rule.
A prime is a whole number greater than 1 divisible only by 1 and itself. Rearrange all the digits of a prime and ask whether every ordering is still prime: 13 passes because 31 is prime too, while 19 fails because 91 = 7 × 13. This page checks those rearrangements and explains which digits can survive.
In the 1970s Penrose found two diamond-shaped tiles, one wide and one narrow, with rules for which edges may meet. The laboratory divides triangular half-tiles into smaller pieces: a narrow half becomes one narrow and one wide half. Repeating this builds a finite patch. Valid infinite tilings repeat local arrangements but have no repeating grid; removing the matching rules allows ordinary repeating patterns.
In 1963 Ulam wrote the whole numbers in an outward spiral on a square grid, starting with 1 at the centre, then 2, 3, 4 and so on. Mark the primes, the whole numbers greater than 1 divisible only by 1 and themselves. Diagonal streaks appear among the marks; the question is which number formulas create them and whether those formulas produce infinitely many primes.
Archimedes trapped π, a circle’s circumference divided by its diameter, between polygons drawn inside and outside the circle. For a circle of diameter 1, an inscribed regular hexagon has perimeter 3, so π must be greater than 3. Add sides to polygons inside and outside the circle to narrow guaranteed bounds, then compare them with a statistical estimate from randomly dropped needles and their line crossings.
A power repeats multiplication: 3 ↑ 3 means 3 × 3 × 3 = 27. A second arrow repeats powers, so 3 ↑↑ 3 means 3 raised to the power 27. Continuing this hierarchy leads to the 64-stage construction of Graham’s number, which appeared as an upper bound in a problem about colouring the connections of a high-dimensional cube. Its full decimal expansion is too large to display, but its last ten digits can be computed exactly.
In the 1950s Stein showed that estimating several means at once can beat the obvious guess. Suppose three measurements have independent, zero-centred normal errors, each with variance 1. The James–Stein formula adjusts them together. For measurements 4, 3 and 0, square and add them to get 25, then multiply each measurement by 1 − 1/25 = 24/25, giving 3.84, 2.88 and 0. This lowers the average total squared error over repeated measurements, whatever the true values are, but it can make an individual trial worse. The theorem applies in three or more dimensions.
A famous claim sets 1 + 2 + 3 + 4 + ... equal to −1/12. A short argument derives that number from 1 − 1 + 1 − 1 and from 1 − 2 + 3 − 4, but the same shifting also proves 0 = 1. Ordinary addition tells another story: the running totals are 1, 3, 6, 10, the total after k terms is k(k+1)/2, and those totals are growing without bound. They do not approach −1/12. That value comes from a different procedure called regularisation, such as smoothing, or from the Riemann zeta function at −1.
In the 1930s Gödel showed how arithmetic can talk about its own statements by giving formulas whole-number codes that can be decoded again. In this laboratory, the formula 0 = 0 has code 5, so arithmetic with 5 can refer to that expression as data. Substituting codes into formulas builds a route to self-reference. The laboratory demonstrates numbering and substitution used in that work, not a proof system or the incompleteness theorem itself.
Write three decimal digits, such as 532, reverse them to get 235 and subtract the smaller arrangement from the larger. Keep the difference in three places, reverse it and add the two: 297 + 792 = 1089. This always gives 1089 when the original outside digits differ, including cases that need a leading zero.
A classic probability puzzle asks how crowded a room must be before two people are likely to share a birthday. Add people to a room and compare their birthdays, looking for any two with the same date. If each person independently receives one of 365 equally likely dates, 23 people give a little over a 50% chance of a matching pair. The model probability describes repeated groups, not a guarantee that a particular room contains a match.
Begin with 0 and make a copy in which every 0 becomes 1 and every 1 becomes 0, then append that changed copy. Repeating this gives 0, then 01, then 0110, then 01101001. This is the Thue-Morse sequence: its infinite continuation repeats many smaller blocks but never settles into one endlessly repeated block.
Begin with 1 unit and assign a nominal growth rate of 100% over one unit of time. Adding that growth once gives 2; splitting it into two 50% increases gives 1 × 1.5 × 1.5 = 2.25. Splitting it into n equal increases gives (1 + 1/n)^n, which approaches Euler’s number e, about 2.71828. Follow that limit, then see how powers of e describe growth, decay and rotation.
In the puzzle named for game-show host Monty Hall, three doors hide one car and two goats. You point at a door, and the host, who knows where the car is, opens one of the other two to show a goat and offers you the door still closed. Sticking wins in 1 of the 3 equally likely arrangements and switching wins in the other 2, so switching wins 2/3 of the time. That advantage is made of the host’s knowledge: a host opening a door at random who happened to miss the car would leave both choices worth one half.
Chebyshev noticed that primes leaving remainder 3 on division by 4 seem to lead those leaving remainder 1. Divide each prime by 4 and it leaves a remainder of 1 or 3, except for 2 itself. Counting the primes as they come, the remainder-3 team is ahead at 3, at 7, at 11 and at almost every stopping point anyone would try by hand. It is not ahead forever: at the prime 26,861 the remainder-1 team is in front for the first time, by one.
Write a sequence of 1s and 2s, split it into runs of equal terms, and count each run: 1, 2, 2, 1, 1, 2 splits into 1, then 22, then 11, then 2, so its run lengths are 1, 2, 2, 1. The Kolakoski sequence is the one whose list of run lengths is the sequence itself, beginning 1, 2, 2, 1, 1, 2, 1, 2, 2, 1. Whether 1s and 2s end up equally common is unproved.
Take 2, square it, square the result, and keep squaring: 2, 4, 16, 256, 65,536. Adding one to each gives the Fermat numbers 3, 5, 17, 257 and 65,537, and every one of those five is prime. Fermat conjectured that each number built this way is prime. The next one, F5 = 4,294,967,297, is 641 × 6,700,417, so the conjecture is false, and whether any Fermat prime beyond those five exists is still unknown.
Barnsley’s chaos game marks three corners of a triangle and starts on one of them. Pick a corner at random, move half the way there, and leave a dot; repeat from where you landed. Starting at (0, 0) in the triangle (0, 0), (1, 0), (0, 1), a draw of (1, 0) puts the dot at (0.5, 0). The dots never fill the triangle: they land on the Sierpinski gasket, the holes Pascal’s triangle shows when each entry is reduced modulo 2. The random draws decide the order the dots appear in, not the shape they appear on.
Newcomb noticed, and Benford later popularised, that many lists lead with 1 far more often than 9. Take positive whole numbers, write each one in base 10, and record only its first digit. Powers of two give 1, 2, 4, 8, 1, 3, 6, 1, 2, 5; across the first thousand powers, digit 1 appears 301 times where an even split would give about 111. The predicted share of digit d is log(1 + 1/d) in the written base. It is a property of particular processes: integers from 1 to 999,999 drawn uniformly lead with each digit one ninth of the time.
The integral of sin(x)/x from 0 to infinity is exactly π/2. Multiply in sin(x/3)/(x/3) and it is π/2 again. Keep multiplying in the next odd denominator, sin(x/5)/(x/5), sin(x/7)/(x/7) and so on, and the exact answer stays π/2 for a long run of factors, then at one factor it falls short by an amount no plot can show. These are the Borwein integrals; the page computes each one exactly as a fraction of π.
Three six-sided dice are thrown against each other and the higher face wins. Die A has faces 2, 2, 4, 4, 9, 9, die B has 1, 1, 6, 6, 8, 8 and die C has 3, 3, 5, 5, 7, 7. Counting all 36 pairings, A beats B 20 to 16, B beats C 20 to 16, and C beats A 20 to 16, so there is no best die. Throw each die twice and add the faces, and the order changes: A no longer beats B.
Game A wins with probability 0.495. Game B wins with probability 0.095 when your capital is a multiple of 3, and 0.745 otherwise. Each loses on average, by 0.0100 and 0.0087 per round, because a 0.005 edge is taken off every winning chance. Yet choosing between them with a fair coin each round gains on average. The paradox is only apparent, since mixing in A changes how often B meets its bad odds. A larger edge, or a B that ignores your capital, removes the gain.
Apart from 2 and 5, every prime ends in 1, 3, 7 or 9, and each ending turns up about a quarter of the time. Take a prime and look at the last digit of the next prime: 7 is followed by 11, which ends in 1, and 11 is followed by 13, which ends in 3. If the next ending were chosen independently, a prime ending in 1 would be followed by another ending in 1 one time in four. Counting every prime below ten million, a repeated ending turns up far less often than that.
You start with 10 units and the casino with 10. Each round you stake one unit on a coin that wins with probability 1/2 or loses, and you play until one of you holds all 20. The chance you end ruined is exactly 50%. Keep the bet fair but give the casino 90 units instead, and the chance you are ruined rises to 90%, about as high as for a bet that wins only 45% of rounds against an equal bankroll, which is 88%. On average a fair bet loses nothing, yet who ends with everything depends on how the bankrolls compare. The average hides how the ending splits.
The harmonic series 1 + 1/2 + 1/3 + 1/4 + … grows without limit, though it needs 12,367 terms to pass 10. Now delete every term whose denominator contains a 9: 1/9, 1/19, 1/90 and the rest. Only 19 of the first hundred denominators are deleted, yet the sum that is left stops growing and approaches about 22.92067, a value computed by Baillie in 1979. The page counts why: in long numbers, most whole numbers contain a 9.
Two players each bet on a pattern of three coin flips, such as HHT. The first player chooses, then the second. One fair coin is then flipped over and over, both players watching the same flips, and whoever’s pattern turns up first in the run wins. Whatever the first player bets, the second can reply so as to win at least 2 games in 3: the opposite of the first player’s middle call, then the first two calls. Against HHT the reply is THH, which wins 3 games in 4, although both take 8 flips to appear on average. There is no best pattern.
No. 03
You bet on a pattern of three flips, I pick mine after you, and with one fair coin I win at least two games in three: whatever you pick, there is a pattern that beats it. A positive result from a 99% accurate test for a rare illness is right only about 1 time in 11, and a kidney-stone treatment can do better for small stones and for large ones yet worse overall: each time, we counted the wrong thing.
9:13 · 23 October 2026Episode notes →
Different paths into the same field.
Curated paths
The interesting part of a mathematical pattern is often the control that breaks it. So you can change the input, alter the rule, sweep a whole range, and look for the case where it stops working. On some explorations you are asked to commit to a guess before the figure answers, and on every exploration you can keep a trail of what you tried and share it as a link.
Every exploration runs from a plain description through the intuition and the key invariant to a formal statement, then says exactly where the argument stops being complete.
Every claim carries its status, its exact scope and the date it was checked. A million successful runs are not a proof of an infinite statement, and nothing here will imply otherwise. How we handle claims →