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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The first step of the game consists in labeling the midpoints of the square (the blue disks) with the absolute value of the difference of the two numbers at its two closest corners; for instance, in between 7 and 5 there is a 2.

Each successive step continues this process of labeling one square inside another following the same rule into a nested chain of squares. When we reach a stage in which a square gets labeled with four zeros, the game ends.

The number of steps needed to end a game is called its length; for instance, the game is of length 4. By changing the numbers using the sliders, you can experiment to see how the length of a game changes. Can you find numbers that give a game of length 5?

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Contributed by: Jaime Rangel-Mondragon (August 2012)
Suggested by: Sol Lederman
Open content licensed under CC BY-NC-SA


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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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