1,496 research outputs found
Building Nim
The game of nim, with its simple rules, its elegant solution and its
historical importance is the quintessence of a combinatorial game, which is why
it led to so many generalizations and modifications. We present a modification
with a new spin: building nim. With given finite numbers of tokens and stacks,
this two-player game is played in two stages (thus belonging to the same family
of games as e.g. nine-men's morris): first building, where players alternate to
put one token on one of the, initially empty, stacks until all tokens have been
used. Then, the players play nim. Of course, because the solution for the game
of nim is known, the goal of the player who starts nim play is a placement of
the tokens so that the Nim-sum of the stack heights at the end of building is
different from 0. This game is trivial if the total number of tokens is odd as
the Nim-sum could never be 0, or if both the number of tokens and the number of
stacks are even, since a simple mimicking strategy results in a Nim-sum of 0
after each of the second player's moves. We present the solution for this game
for some non-trivial cases and state a general conjecture
Compound Node-Kayles on Paths
In his celebrated book "On Number and Games" (Academic Press, New-York,
1976), J.H. Conway introduced twelve versions of compound games. We analyze
these twelve versions for the Node-Kayles game on paths. For usual disjunctive
compound, Node-Kayles has been solved for a long time under normal play, while
it is still unsolved under mis\`ere play. We thus focus on the ten remaining
versions, leaving only one of them unsolved.Comment: Theoretical Computer Science (2009) to appea
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
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