The catalogue
Each one begins with something surprising you can manipulate yourself, then follows it as far as the mathematics honestly goes - to an explanation, a proof, or the edge of what anyone knows.
28 explorations
Write 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. Repeating this description makes longer strings; from this starting digit, only 1, 2 and 3 ever appear, and the length ratio approaches about 1.303577.
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.
Place an ant on an all-white square grid. On white it turns right; on black it turns left. Each turn is 90 degrees, after which it 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.
Choose an even whole number greater than 2 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. Goldbach’s conjecture says every such even number has a pair; 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, showing why the first few counts 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.
Penrose tilings use 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.
Write 1 at the centre of a square grid and continue with 2, 3, 4 and so on in an outward spiral. 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.
The number π is a circle’s circumference divided by its diameter. 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, used in a problem about colouring cube connections. Its full decimal expansion is too large to display, but its last ten digits can be computed exactly.
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.
Add the positive whole numbers in order and the running totals are 1, 3, 6, 10 and so on, growing without bound. They do not approach −1/12. That value comes from a different procedure called regularisation, which changes how the infinite expression is treated; this page shows the ordinary sums alongside the precise procedures that assign −1/12.
Assign mathematical 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 Gödel’s 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.
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.
Often more inviting than “number theory” or “combinatorics”
Listed honestly, not scaffolded as empty pages
Each of these needs a working kernel, an exhaustive test suite, sourced claims and a representation that shows something a paragraph cannot. They appear here when that is true of them, and not before. Read the method →