# The Four Numbers Game

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The Four Numbers Game is played starting with four non-negative integers (not necessarily different) placed at the vertices of a square (shown on orange disks above).

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Contributed by: Jaime Rangel-Mondragon (August 2012)

Suggested by: Sol Lederman

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

It is easy to see that the length of a game does not change if the initial configuration of numbers is rotated or reflected (eight possibilities in total); the length of the game is invariant under the symmetries of the square. One may wonder whether every game ends. The answer is affirmative; more precisely, if is the largest of the four non-negative integers we start with and is the least integer such that , then the length of the game is at most . For example, if then and the game lasts at most 44 steps.

For every positive integer , there is a game of length . Most games are of length 4 or 6; is of length 9. If we play with non-negative real numbers, some games have finite length; for instance, has finite length.

Reference

[1] J. D. Sally and P. J. Sally Jr.,* Roots to Research: A Vertical Development of Mathematical Problems*, Providence, RI: American Mathematical Society, 2007.

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