15 research outputs found
Finite subtraction games in more than one dimension
We study two-player impartial vector subtraction games (on tuples of
nonnegative integers) with finite rulesets, and solve all two-move games.
Through multiple computer visualizations of outcomes of two-dimensional
rulesets, we observe that they tend to partition the game board into periodic
mosaics on very few regions/segments, which can depend on the number of moves
in a ruleset. For example, we have found a five-move ruleset with an outcome
segmentation into six semi-infinite slices. We prove that games in two
dimensions are row/column eventually periodic. Several regularity conjectures
are provided. Through visualizations of some rulesets, we pose open problems on
the generic hardness of games in two dimensions.Comment: 38 page
When are translations of P-positions of Wythoff's game P-positions?
We study the problem whether there exist variants of {\sc Wythoff}'s game
whose -positions, except for a finite number, are obtained from those of
{\sc Wythoff}'s game by adding a constant to each -position. We solve
this question by introducing a class \{\W_k\}_{k \geq 0} of variants of {\sc
Wythoff}'s game in which, for any fixed , the -positions of
\W_k form the set , where is the golden ratio.
We then analyze a class \{\T_k\}_{k \geq 0} of variants of {\sc Wythoff}'s
game whose members share the same -positions set . We establish
several results for the Sprague-Grundy function of these two families. On the
way we exhibit a family of games with different rule sets that share the same
set of -positions
How far can Nim in disguise be stretched?
A move in the game of nim consists of taking any positive number of tokens
from a single pile. Suppose we add the class of moves of taking a nonnegative
number of tokens jointly from all the piles. We give a complete answer to the
question which moves in the class can be adjoined without changing the winning
strategy of nim. The results apply to other combinatorial games with unbounded
Sprague-Grundy function values. We formulate two weakened conditions of the
notion of nim-sum 0 for proving the results.Comment: To appear in J. Combinatorial Theory (A