6,345 research outputs found
11 x 11 Domineering is Solved: The first player wins
We have developed a program called MUDoS (Maastricht University Domineering
Solver) that solves Domineering positions in a very efficient way. This enables
the solution of known positions so far (up to the 10 x 10 board) much quicker
(measured in number of investigated nodes).
More importantly, it enables the solution of the 11 x 11 Domineering board, a
board up till now far out of reach of previous Domineering solvers. The
solution needed the investigation of 259,689,994,008 nodes, using almost half a
year of computation time on a single simple desktop computer. The results show
that under optimal play the first player wins the 11 x 11 Domineering game,
irrespective if Vertical or Horizontal starts the game.
In addition, several other boards hitherto unsolved were solved. Using the
convention that Vertical starts, the 8 x 15, 11 x 9, 12 x 8, 12 x 15, 14 x 8,
and 17 x 6 boards are all won by Vertical, whereas the 6 x 17, 8 x 12, 9 x 11,
and 11 x 10 boards are all won by Horizontal
Hitting time results for Maker-Breaker games
We study Maker-Breaker games played on the edge set of a random graph.
Specifically, we consider the random graph process and analyze the first time
in a typical random graph process that Maker starts having a winning strategy
for his final graph to admit some property \mP. We focus on three natural
properties for Maker's graph, namely being -vertex-connected, admitting a
perfect matching, and being Hamiltonian. We prove the following optimal hitting
time results: with high probability Maker wins the -vertex connectivity game
exactly at the time the random graph process first reaches minimum degree ;
with high probability Maker wins the perfect matching game exactly at the time
the random graph process first reaches minimum degree ; with high
probability Maker wins the Hamiltonicity game exactly at the time the random
graph process first reaches minimum degree . The latter two statements
settle conjectures of Stojakovi\'{c} and Szab\'{o}.Comment: 24 page
Move-minimizing puzzles, diamond-colored modular and distributive lattices, and poset models for Weyl group symmetric functions
The move-minimizing puzzles presented here are certain types of one-player
combinatorial games that are shown to have explicit solutions whenever they can
be encoded in a certain way as diamond-colored modular and distributive
lattices. Such lattices can also arise naturally as models for certain
algebraic objects, namely Weyl group symmetric functions and their companion
semisimple Lie algebra representations. The motivation for this paper is
therefore both diversional and algebraic: To show how some recreational
move-minimizing puzzles can be solved explicitly within an order-theoretic
context and also to realize some such puzzles as combinatorial models for
symmetric functions associated with certain fundamental representations of the
symplectic and odd orthogonal Lie algebras
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