3,232 research outputs found
Heap games, numeration systems and sequences
We propose and analyse a 2-parameter family of 2-player games on two heaps of
tokens, and present a strategy based on a class of sequences. The strategy
looks easy, but is actually hard. A class of exotic numeration systems is then
used, which enables us to decide whether the family has an efficient strategy
or not. We introduce yet another class of sequences, and demonstrate its
equivalence with the class of sequences defined for the strategy of our games.Comment: To appear in Annals of Combinatoric
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
Deciding the Winner of an Arbitrary Finite Poset Game is PSPACE-Complete
A poset game is a two-player game played over a partially ordered set (poset)
in which the players alternate choosing an element of the poset, removing it
and all elements greater than it. The first player unable to select an element
of the poset loses. Polynomial time algorithms exist for certain restricted
classes of poset games, such as the game of Nim. However, until recently the
complexity of arbitrary finite poset games was only known to exist somewhere
between NC^1 and PSPACE. We resolve this discrepancy by showing that deciding
the winner of an arbitrary finite poset game is PSPACE-complete. To this end,
we give an explicit reduction from Node Kayles, a PSPACE-complete game in which
players vie to chose an independent set in a graph
Invariant and dual subtraction games resolving the Duch\^e-Rigo conjecture
We prove a recent conjecture of Duch\^ene and Rigo, stating that every
complementary pair of homogeneous Beatty sequences represents the solution to
an \emph{invariant} impartial game. Here invariance means that each available
move in a game can be played anywhere inside the game-board. In fact, we
establish such a result for a wider class of pairs of complementary sequences,
and in the process generalize the notion of a \emph{subtraction game}. Given a
pair of complementary sequences and of positive integers, we
define a game by setting as invariant moves. We then
introduce the invariant game , whose moves are all non-zero
-positions of . Provided the set of non-zero -positions of
equals , this \emph{is} the desired invariant game. We give
sufficient conditions on the initial pair of sequences for this 'duality' to
hold.Comment: 11 pages, 2 figure
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
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