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Ravikumar, B (2009)

A simple multiplication game and its analysis

Accepted for publication in the International Journal of Combinatorial Number Theory.

ISSN/ISBN: Not available at this time. DOI: Not available at this time.



Abstract: ABSTRACT: We study the following two-player game involving players A and B. Let n be a positive integer known to both players. The player A (B) chooses an n-digit integer a (b). They do this without the knowledge of the other player’s choice. Then the numbers are revealed and the product ab is computed. If the leading digit (i.e., the first digit from the left) of ab is 1, 2 or 3, then player A wins, else player B wins. Since six digits favor B and only three favor A, a naive reasoning would suggest a 2 to 1 advantage for player B. The reality is nearly the opposite. More precisely, let pn be the probability that A wins when both players play optimally. We show the somewhat surprising result that limn→∞ pn = log104≈0.6020599913. We also determine the odds in favor of the players when they use some non-optimal strategies. Our analysis of optimal play involves the scale invariance of the well-known Benford distribution


Bibtex:
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Reference Type: Journal Article

Subject Area(s): Applied Mathematics