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.