57 research outputs found
The switch operators and push-the-button games: a sequential compound over rulesets
We study operators that combine combinatorial games. This field was initiated
by Sprague-Grundy (1930s), Milnor (1950s) and Berlekamp-Conway-Guy (1970-80s)
via the now classical disjunctive sum operator on (abstract) games. The new
class consists in operators for rulesets, dubbed the switch-operators. The
ordered pair of rulesets (R 1 , R 2) is compatible if, given any position in R
1 , there is a description of how to move in R 2. Given compatible (R 1 , R 2),
we build the push-the-button game R 1 R 2 , where players start by playing
according to the rules R 1 , but at some point during play, one of the players
must switch the rules to R 2 , by pushing the button ". Thus, the game ends
according to the terminal condition of ruleset R 2. We study the pairwise
combinations of the classical rulesets Nim, Wythoff and Euclid. In addition, we
prove that standard periodicity results for Subtraction games transfer to this
setting, and we give partial results for a variation of Domineering, where R 1
is the game where the players put the domino tiles horizontally and R 2 the
game where they play vertically (thus generalizing the octal game 0.07).Comment: Journal of Theoretical Computer Science (TCS), Elsevier, A
Para{\^i}tr
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
Restrictions of -Wythoff Nim and -complementary Beatty Sequences
Fix a positive integer . The game of \emph{-Wythoff Nim} (A.S.
Fraenkel, 1982) is a well-known extension of \emph{Wythoff Nim}, a.k.a 'Corner
the Queen'. Its set of -positions may be represented by a pair of increasing
sequences of non-negative integers. It is well-known that these sequences are
so-called \emph{complementary homogeneous}
\emph{Beatty sequences}, that is they satisfy Beatty's theorem. For a
positive integer , we generalize the solution of -Wythoff Nim to a pair
of \emph{-complementary}---each positive integer occurs exactly
times---homogeneous Beatty sequences a = (a_n)_{n\in \M} and b = (b_n)_{n\in
\M}, which, for all , satisfies . By the latter property,
we show that and are unique among \emph{all} pairs of non-decreasing
-complementary sequences. We prove that such pairs can be partitioned into
pairs of complementary Beatty sequences. Our main results are that
\{\{a_n,b_n\}\mid n\in \M\} represents the solution to three new
'-restrictions' of -Wythoff Nim---of which one has a \emph{blocking
maneuver} on the \emph{rook-type} options. C. Kimberling has shown that the
solution of Wythoff Nim satisfies the \emph{complementary equation}
. We generalize this formula to a certain '-complementary
equation' satisfied by our pair and . We also show that one may obtain
our new pair of sequences by three so-called \emph{Minimal EXclusive}
algorithms. We conclude with an Appendix by Aviezri Fraenkel.Comment: 22 pages, 2 figures, Games of No Chance 4, Appendix by Aviezri
Fraenke
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