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Problem of the Week 1022

What's the Diff

Place four integers at the four corners of a square and then, at the center of each edge of the square, place the difference between the two numbers at the ends of that square, always taking absolute values so that all entries are nonnegative. Call this the "diffy" operation.

If we repeat the diffy operation starting with the sequence (3, 7, 1022, 2005) we get the following.

(3, 7, 1022, 2005)
(4, 1015, 983, 2002)
(1011, 32, 1019, 1998)
(979, 987, 979, 987)
(8, 8, 8, 8)
(0, 0, 0, 0)

True or False: Any starting sequence of integers leads to the zero vector in finitely many steps.

Source: A paper about to appear in the Amer Math Monthly by A. Behn, C. Kribs-Zaleta, V. Ponomarenko.

Stan provides Mathematica code to simplify things. The FixedPointList command repeats the function on the list (edit as you wish) until a fixed point appears.

diffy[A_] := Abs[RotateLeft[A] - A]
FixedPointList[diffy, {0, Sqrt[2], Sqrt[3], [Pi]}]

© Copyright 2005 Stan Wagon. Reproduced with permission.

20 January 2005