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
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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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