Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.7

Chocolate Numbers

Caleb Ji
The Bear Creek School
Redmond, WA 98053

Tanya Khovanova
Department of Mathematics
Cambridge, MA 02139

Robin Park
Thomas Jefferson High School for Science and Technology
Alexandria, VA 22312

Angela Song
Phillips Exeter Academy
Exeter, NH 03833


In this paper, we consider a game played on a rectangular m × n gridded chocolate bar. Each move, a player breaks the bar along a grid line. Each move after that consists of taking any piece of chocolate and breaking it again along existing grid lines, until just mn individual squares remain. This paper enumerates the number of ways to break an m × n bar, which we call chocolate numbers, and introduces four new sequences related to chocolate numbers. Using various techniques, we prove interesting divisibility results regarding chocolate number sequences.

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(Concerned with sequences A000142 A045468 A257281 A261746 A261747 A261964.)

Received September 21 2015; revised versions received November 29 2015; December 26 2015. Published in Journal of Integer Sequences, December 29 2015.

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