24 research outputs found
A restriction of Euclid
Euclid is a well known two-player impartial combinatorial game. A position in
Euclid is a pair of positive integers and the players move alternately by
subtracting a positive integer multiple of one of the integers from the other
integer without making the result negative. The player who makes the last move
wins. There is a variation of Euclid due to Grossman in which the game stops
when the two entrees are equal. We examine a further variation that we called
M-Euclid in which the game stops when one of the entrees is a positive integer
multiple of the other. We solve the Sprague-Grundy function for M-Euclid and
compare the Sprague-Grundy functions of the three games
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