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- Title
GRUNDY NUMBERS OF IMPARTIAL CHOCOLATE BAR GAMES.
- Authors
Ryohei Miyadera; Shunsuke Nakamura; Yushi Nakaya
- Abstract
Chocolate bar games are a variant of CHOMP. A chocolate bar is a rectangular array of squares but with some squares removed. There is a poisoned square at the bottom of the bar (column 0), there are z + 1 columns of squares, and the height of the i-th column is given by a non-decreasing function f(i)+1, with a maximum height of y +1. This is denoted by CB(f, y, z). A move is to break the bar along a horizontal or vertical line and eat that part, not counting the poisoned square. In the case f(i) = [ i/k ] for some even number k, the authors have previously proved that the Sprague–Grundy value of CB(f, y, z) is y ⊕ z. The case of an odd number k is still open. In this paper, the functions f such that the Sprague–Grundy value of CB(f, y, z) is y ⊕ z are characterized, and this result is generalized to the case when the Sprague–Grundy value of CB(f, y, z) is (y ⊕ (z + s)) − s for a natural number s.
- Subjects
ODD numbers; CHOCOLATE; NATURAL numbers; GAMES; SQUARE
- Publication
Integers: Electronic Journal of Combinatorial Number Theory, 2020, Vol 20, p1
- ISSN
1553-1732
- Publication type
Article