an old conjecture from the Game of Life solved by amateurs


Configuration of an oscillator that returns identical to itself after 19 generations.

Fifty years after its invention, a mathematical game continues to generate interest and new results. A group of seven enthusiasts, including a woman, published in December his discovery of two “creatures” missing from the catalog of inhabitants of the Game of Life. Imagined by the British mathematician John Conway (1937-2020) in 1970, this “game” is a kind of metaphor for life, despite the simplicity of its rules. On a grid, shapes appear, disappear, multiply, move… However, they are only made up of black or white boxes. The future of a hut, dead (the black ones) or alive (the white ones), is determined at each generation or stage by analyzing the state of its eight neighbors. A dead cell, with exactly three living neighbors, becomes alive. A living cell survives if it has two or three neighbors, otherwise it dies. What John Conway summed up in “birth if three neighbors, survival if two or three neighbors”.

For example, a single living square dies in the next turn. Two boxes that also touch. Three, aligned, on the other hand “come to life”: if they are horizontal, in the next step, the line is vertical. Then becomes horizontal again. This pattern, called “flashing,” belongs to the oscillator family, a pattern that repeats after a fixed number of generations or period (here two).

The question that has just been resolved is whether it is possible to construct oscillators of any period. The “block” is the simplest of oscillators: a square that does not move (period 1)! The 48-cell “pulsar” was proposed by Conway himself and becomes the same every three generations. There spinning top”, every four generations… In 1996, David Buckingham found a method to form oscillators beyond 61. Then, in 2013, Mike Playle did the same for 43. “It’s like hamsters spinning in a bigger and bigger wheel”, describes Tanner Jacobi, one of the authors of the article. The shapes obtained by this process indeed show “objects” appearing to rotate on a circuit.

Engineering, science and creativity

Until 2021, there was no configuration of periods 19, 34, 38 and 41. Challenges 34 and 38 finally “fall” in 2022, the first of which thanks to Mitchell Riley, the only professional mathematician in the group of authors of the last publication. The following year, it was the turn of 19 and 41.

The two discoverers Mitchell Riley and Nico Brown then asked other amateurs they met on the forums dedicated to this game to join them in publishing in an academic style, in the form of a preprint posted on arXiv, a article with a very mathematical title: “Conway’s Game of Life is omniperiodic. » In other words, it is possible to find a configuration which will return equal to itself after any time. Not to be confused with “gliders” or other “ships” which also return identical but having “moved” on the grid.

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