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Problem of the Week 1023

What's the Real Diff

Place four real numbers 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.

True or False: Any starting sequence of real numbers 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.

Correction: In fact, this problem has a very long history. The problem is known as the Ducci problem and the discovery of 4 reals for which the process does not terminate is due to M. Lotan in 1949. A recent paper on the subject that discusses the history and more recent directions of research is due to Marc Chamberland (Grinnell College): Unbounded Ducci Sequences. The paper appeared in Journal of Difference Equations and Applications 9 (2003) 887-895.

Note the difference with Problem of the Week 1022.

© Copyright 2005 Stan Wagon. Reproduced with permission.

20 January 2005