The Triangle Game, from Zero

A step-by-step visual guide to a colouring game studied since 1955, a question for which Erdős offered prize money, and the 2026 result that answered it. You do not need a maths background. Each new idea begins with a picture. All 120 connections between sixteen people receiving one of three colours without making a one-colour triangle The animation shows sixteen people with a connection between every pair. That makes 120 connections, and each one receives one of three colours. When the colouring is complete, there is no group of three people whose three mutual connections all have the same colour. Sixteen is the largest group for which this is possible with three colours. Greenwood and Gleason built this example and proved that seventeen is impossible in 1955. Everything in this guide comes from the open source repository at github.com/muchmirul/conjectures: the text, every figure, and the tests behind each number. This guide asks what happens when many more colours are available. More colours let us keep a larger group safe, but how much larger can the group become for each added colour? Erdős offered cash prizes for answering that question. For decades, the best construction and the best limit were so far apart that researchers did not even know whether the long-term rate stopped at a fixed number. A result in 2026 showed that it never stops growing. The guide develops the result in small steps. Most numbered sections also have a page you can play with, where you can change the main choices and watch the picture respond. The code that creates every figure is included in this repository. The tests recalculate the numbers and check the small colourings shown here. When a statement comes from an existing theorem rather than those tests, the text says so. The 2026 work is presented in two documents. One gives the finished proof. The other is a behind-the-scenes guide that explains how the ideas were found, including earlier attempts that failed. Sections 5 to 7 explain why the old methods stalled. Sections 8 to 11 explain the finished construction. A table near the end shows where each idea appears in the two source documents. first, learn the game 1 the rule 2 why six people cannot stay safe 3 what an extra colour changes then, find the question 4 how to compare different colour counts why earlier ideas stalled 5 multiplying safe groups 6 a general upper limit 7 the wide gap between them the finished construction 8 which colours each room omits 9 keeping arrivals on safe teams 10 one fixed referee for every choice 11 stacking the rooms into a tower what the result settles 12 what the result settles Use the controls to change the picture. Open the activity on its own page 1 · The game2 · Six is forced3 · More crayons4 · The question5 · Multiply6 · The ceiling7 · The gap8 · Palettes9 · The trap10 · The referee11 · The tower12 · What it means, and what it does notWhere to find each idea in the sourcesWhat you can check yourselfThe plain words, and the real onesWhere to go next 1 · The game Start with a group of people and draw a connection between every pair. Give every connection a colour. Any choice of three people forms a triangle because each pair among them is connected. You lose if all three connections in one of those triangles have the same colour. We will call this a one-colour triangle. A completed colouring with no one-colour triangle will be called safe. A group together with a safe colouring is a safe table. Two colourings of four people, with a one-colour triangle on the left and a safe result on the right The thick triangle on the left uses one colour on all three sides, so that colouring loses. Four people contain four possible groups of three. On the right, each of those four triangles uses at least two colours, so the colouring is safe. With two colours, a group of five can be kept safe. Apart from renaming the people or swapping the colours, there is only one way to do it. Place the five people around a circle. Use the first colour for neighbouring pairs around the outside, then use the second colour for the five connections that skip across the circle. The ten connections among five people being coloured, followed by a check of all ten triangles The final sweep checks every possible group of three. There are ten of them, and every one uses both colours. Safety is therefore not based on how the picture looks. It comes from checking a short, complete list. The tests in this repository perform the same check. There is also a quick way to understand the pattern. Look at either colour by itself. Its five connections form a ring, and a ring contains no triangle. Since neither colour can make a triangle on its own, the combined colouring is safe. Use the controls to change the picture. Open the activity on its own page 2 · Six is forced Five people can stay safe with two colours. Six people cannot. This is the one theorem proved completely in this guide, and every step can be seen in the animation. Choose one of the six people and look at the five connections leading from that person. There are only two colours available. At least three of those five connections must have the same colour, because two connections of each colour would account for only four. Suppose three of them are red. Three same-coloured connections from one person forcing a red or blue triangle Now focus on the three people at the other ends of those red connections. If any pair among them is joined in red, that connection and the two red connections back to the first person make a red triangle. To avoid that outcome, all three connections among those people would have to be blue. But those three blue connections make a blue triangle instead. A one-colour triangle appears in either case. The first counting step is often called the pigeonhole principle: when more objects are placed into fewer groups, one group must receive several objects. The argument did not depend on how the connections were coloured, so it covers every possible two-colouring of six people. The computer tests check the same claim another way. Six people have fifteen connections, with two choices for each connection, giving 32768 complete colourings. The tests inspect all of them and find that none is safe. All 32768 two-colourings of six people grouped by their number of one-colour triangles The empty bar at zero confirms that every colouring contains a one-colour triangle. The bar at one is empty too. Even the best six-person colourings contain exactly two one-colour triangles, never only one. Goodman recorded this stronger observation in 1959, and the exhaustive test finds it again. The two-colour story is now complete: five people can stay safe, while six force a loss. We will call six the forcing size for two colours, meaning the first group size at which a one-colour triangle is unavoidable. Use the controls to change the picture. Open the activity on its own page 3 · More crayons Adding a third colour makes a much larger safe group possible. The best safe size jumps from five people to sixteen. The opening animation showed the full colouring; the next picture separates its three colours so that each can be checked on its own. The sixteen-person colouring separated into three colour layers, none of which contains a triangle Each panel contains the connections of one colour. Every person has five connections in each panel, but no panel contains a triangle. A one-colour triangle would have to appear entirely inside one of these panels, so the full three-colour pattern is safe. Greenwood and Gleason created this pattern in 1955. This repository rebuilds it from their instructions and checks all 560 groups of three. They also proved that seventeen people cannot stay safe with three colours. A direct computer sweep would require considering three choices on each of 136 connections, which is far beyond the case-by-case checks in this project. The guide therefore quotes their theorem rather than claiming to verify it. After three colours, no exact forcing size is known. The exact answers for one, two and three colours, followed by the open range for four colours For four colours, the forcing size is known to be somewhere between 51 and 62. In other words, a safe colouring of fifty people is known, and every colouring of sixty-two people is known to fail, but the exact point between them remains open. The uncertainty is even wider for five colours. No exact answer is known for any colour count above three. The number of possible colourings grows too quickly for a simple search, so progress depends on general arguments and constructions. Use the controls to change the picture. Open the activity on its own page 4 · The question To compare constructions that use different numbers of colours, we need a fair score. Suppose a safe table uses a certain number of colours. Ask for the number that, when multiplied by itself once for each colour, gives the group size. We will call that score people per colour. This is the usual root of the group size, but the repeated-multiplication picture is all we need here. Five people with two colours score about 2.24, because multiplying 2.24 by itself gives about five. Sixteen people with three colours score about 2.52, because multiplying 2.52 by itself three times gives about sixteen. A larger score means that each colour contributes a larger multiplying effect. Before 2026, the best long-term recipe was based on addition patterns around number circles. As the number of colours grows, its score approaches about 3.28. This does not mean there is a literal five-colour safe group with that score. Such a group would need 380 people, while the simple upper argument in section 6 already forces a triangle by 327 people. The recipe reaches 3.28 only as a long-term rate after its small losses are spread over more and more colours. The people-per-colour scores of known constructions rising toward the old long-term rate The known scores rose