A Piece of the Pi: mathematics explained

The 1-2-3 conjecture for graphs

Richard Green — Fri, 26 Jun 2026 00:50:46 GMT

The Petersen graph is a famous graph with 10 vertices and 15 edges. In the picture below, each edge of the graph has been labelled by a (blue) number from the set {1, 2, 3}: the magenta edges are labelled 1, the brown edges are labelled 2, and the cyan edges are labelled 3. Each vertex has been labelled by the (red) number given by the sum of the labels…

Tournaments and dice games

Richard Green — Wed, 06 May 2026 00:40:50 GMT

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The picture above shows a set of seven dice designed by Oskar van Deventer. An arrow between die x and die y indicates that if the dice x and y are thrown randomly, then die x will show a higher number than die y with a probability of 5/9. For example, the yellow die, whose six faces are numbered {6, 6, 8, 8, 19, 19}, will usually beat the pink die, who…

“Lights Out” on a triangular grid

Richard Green — Sun, 22 Mar 2026 13:51:06 GMT

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Lights Out is a game that can be played on an equilateral triangular grid of buttons, each of which may be in a lit or unlit state. A move in the game consists of pressing one of the buttons. The effect of this move is to toggle the state of the button that was pressed, as well as toggling the states of all the buttons touching the button that was press…

How to turn a number into a rooted forest

Richard Green — Sun, 01 Mar 2026 02:55:50 GMT

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The Matula arborification of a number is a way to represent that number as a rooted forest. The picture above shows the arborifications of the numbers 41 up to 49. The connected components of a rooted forest are known as rooted trees. Each vertex in each rooted tree is connected to precisely one vertex lower down, with the unique exception of the root v…

Surfaces and sequence graphs

Richard Green — Sat, 31 Jan 2026 23:02:46 GMT

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The golden ratio, φ=(1+√5)/2, is a famous example of an irrational number, which means that it cannot be expressed exactly as a ratio of two integers. The Kronecker sequence (nφ mod 1) is formed from the numbers (0φ, 1φ, 2φ, 3φ, …) ≈ (0, 1.618, 3.236, 4.854, …) by discarding the part of the number to the left of the decimal point to obtain (0, 0.618, 0.…

Colouring identical cuboids

Richard Green — Tue, 30 Dec 2025 20:18:10 GMT

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How many colours do we need to colour a fixed stack of identical cuboids with integer coordinates in such a way that no two cuboids of the same colour touch each other? The answer depends on the dimensions and orientations of the cuboids.

Modular curve stitching

Richard Green — Wed, 10 Dec 2025 02:08:09 GMT

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Consider the points of a clock face, numbered 1 up to 12, and for each point k, draw a straight line between k and 2k, taking remainders modulo 12. For example, there is a line between 5 and 10 because 10=5×2, and there is a line between 9 and 6, because 18=9×2 and 18 and 6 leave the same remainder when divided by 12.

The poetry of Steiner systems

Richard Green — Fri, 14 Nov 2025 03:18:25 GMT

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A Steiner triple system consists of a set S (such as the set of seven letters {A, E, H, R, T, W, Y}) together with a set of 3-element subsets of S called blocks (such as {YEA, WHY, TRY, HER, RAW, WET, HAT}). These blocks are represented in the diagram above by six straight lines and a circle. The property that makes this a Steiner system is that any pai…

Perfect difference sets and Sidon sets

Richard Green — Thu, 30 Oct 2025 03:10:37 GMT

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Consider a wheel with 21 evenly spaced positions, consecutively numbered 0, 1, 2, up to 20, and highlight the positions 1, 2, 5, 15, and 17 in bold, as in the picture. Can we find two marked positions that are one tick of the circle apart? Yes: 1 and 2. How about two marked positions that are two ticks apart? Yes: 15 and 17. Three ticks apart? Yes: 2 an…

How many cylinders can touch each other?

Richard Green — Mon, 13 Oct 2025 21:52:22 GMT

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In 1968, J.E. Littlewood asked whether it is possible to arrange seven identical infinite cylinders in three-dimensional space in such a way that each of them touches each of the others. The answer to Littlewood’s question is “yes”, as evidenced by the arrangements of seven cylinders shown above and below, which were constructed by Sándor Bozóki, Tsung-…

The golden ratio as a number base

Richard Green — Sat, 27 Sep 2025 22:15:54 GMT

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The Fibonacci numbers (1, 1, 2, 3, 5, 8, 13, 21, 34, 55, …) are one of the most famous sequences of integers. Every number in the sequence after the first two is the sum of the two previous numbers: 1+1=2, 1+2=3, 2+3=5, and so on. It turns out that every positive integer can be written as a sum of distinct Fibonacci numbers, possibly in several differen…

Matroid bingo

Richard Green — Mon, 15 Sep 2025 02:38:53 GMT

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Matroid bingo is a simplified version of the game of bingo. Each player receives a card containing a subset of the numbers 1 up to n. The caller has a jar of balls numbered 1 up to n, selects a ball at random from the jar and calls out the number, and each player whose card contains that number marks that number off. The winner is the first player to ma…

Transparent rectangle visibility graphs

Richard Green — Mon, 01 Sep 2025 21:17:44 GMT

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A graph consists of a set of vertices, some of which are joined in pairs by edges. It is possible to reconstruct the graph shown above and to the left from the arrangement of rectangles shown on the right. Each of the 20 rectangles corresponds to one of the 20 vertices in the graph, and there is an edge between two vertices if and only if there is a dir…

Balanced Steinhaus triangles

Richard Green — Sat, 16 Aug 2025 16:41:49 GMT

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A familiar fact from the game of pool is that 15 objects can be arranged in an equilateral triangle of side length 5. This is because 15=1+2+3+4+5 is the fifth triangular number, T(5), where T(n)= n(n+1)/2. If we have balls of five different colours, it is possible to create an arrangement like this where there are equally many balls of each colour, for…

How to make hyperbolic wallpaper

Richard Green — Wed, 06 Aug 2025 21:02:21 GMT

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Given a repeating wallpaper pattern in the Euclidean plane, is there a good algorithm to turn it into a wallpaper pattern in the hyperbolic plane, as in the Circle Limit series of pictures by M.C. Escher? According to the recent paper The Smooth Power of the "Neandertal Method" by Aaron Montag, Tim Reinhardt, and Jürgen Richter-Gebert, the answer is “ye…

Spiral Sudoku

Richard Green — Wed, 23 Jul 2025 22:23:13 GMT

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Suppose we have a 5×5 grid with 15 circled positions in a spiral shape as shown above. Is it possible to fill the grid with the digits 1 up to 5 in such a way that (a) each row and each column contains each digit precisely once, and (b) the circled digits include precisely one 1, two 2s, three 3s, four 4s, and five 5s? Although the short answer is “yes”…

The binary sphere packing problem

Richard Green — Sat, 12 Jul 2025 00:22:13 GMT

What is the densest way to fill space with spheres of the same size? Is there a way to do it that is denser than the obvious regular way to stack cannonballs on a square base, as shown in the picture above? According to the Kepler conjecture, posed by the mathematician and astronomer Johannes Kepler in 1611, the answer is “no”. The Kepler conjecture was…

Tiling the plane with congruent polygons

Richard Green — Sun, 29 Jun 2025 16:46:15 GMT

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It is impossible to tile the plane with regular pentagons of the same size, because the internal angle of a pentagon is 108°, and 360 is not a multiple of 108. However, it is possible to tile the plane with copies of a pentagon with angles of 60°, 80°, 100°, 140°, and 160°, as shown in the picture above. It is necessary to flip some of the tiles over in…

Games in projective space

Richard Green — Mon, 16 Jun 2025 19:11:24 GMT

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Projective geometry is a version of geometry in which there are points, lines, and planes, but in which there are no such things as distance or parallel lines. There is also a version of projective geometry that uses finite sets, and this can be used to design games like the card game Dobble (also known as Spot It!). The deck of Dobble cards is designed…

Intersections of chords of a circle

Richard Green — Wed, 04 Jun 2025 03:08:07 GMT

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The picture above shows a regular polygon with 15 vertices, including all the diagonals between the 15 vertices, and all the intersections between the diagonals. These points of intersection, shown in red, are not distributed evenly throughout the shape, and they tend to be located away from the centre. In 1962, Jovan Karamata proved a formula that pred…