Philip Henderson

Monte Carlo Tree Search in Hex

Hex, the classic board game invented by Piet Hein in 1942 and independently by John Nash in 1948, has been a domain of AI research since Claude Shannons seminal work in the 1950s. Until the Monte Carlo Go revolution a few years ago, the best computer Hex players used knowledge-intensive alpha-beta search. Since that time, strong Monte Carlo Hex players have appeared that are on par with the best alpha-beta Hex players. In this paper, we describe MoHex, the Monte Carlo tree search Hex player that won gold at the 2009 Computer Olympiad. Our main contributions to Monte Carlo tree search include using inferior cell analysis and connection strategy computation to prune the search tree. In particular, we run our random game simulations not on the actual game position, but on a reduced equivalent board.

Solving hex: beyond humans

For the first time, automated Hex solvers have surpassed humans in their ability to solve Hex positions: they can now solve many 9×9 Hex openings. We summarize the methods that attained this milestone, and examine the future of Hex solvers.

WOLVE 2008 wins Hex Tournament

SIX, the gold medallist in each Hex competition since 2003, uses a two-ply truncated-width alpha-beta search and a Shannon-style electric-circuit evaluation function in which cell adjacencies are augmented by virtual connections. The virtual connection engine (VCE) uses Anshelevich and/or closure operations. From a list of moves to consider, SIX prunes dead cells with low degree; it also prunes cells outside of the virtual connection mustplay region, which are provably losing.

Probing the 4-3-2 Edge Template in Hex

For the game of Hex, we find conditions under which moves into a 4-3-2 edge template are provably inferior.

Hex, braids, the crossing rule, and XH-search

We present XH-search, a Hex connection finding algorithm. XH-search extends Anshelevichs H-search by incorporating a new Crossing Rule to find braids, connections built from overlapping subconnections.

Stronger Virtual Connections in Hex

For connection games such as Hex or Y or Havannah, finding guaranteed cell-to-cell connection strategies can be a computational bottleneck. In automated players and solvers, sets of such virtual connections are often found with Anshelevichs H-search algorithm: initialize trivial connections, and then repeatedly apply an AND-rule (for combining connections in series) and an OR-rule (for combining connections in parallel). We present FastVC Search, a new algorithm for finding such connections. FastVC Search is more effective than H-search when finding a representative set of connections quickly is more important than finding a larger set of connections slowly. We tested FastVC Search in an alpha-beta player Wolve, a Monte Carlo tree search player MoHex, and a proof number search implementation called Solver. It does not strengthen Wolve, but it significantly strengthens MoHex and Solver.

How to play Reverse Hex

We present new results on how to play Reverse Hex, also known as Rex, or Misere Hex, on n i? n boards. We give new proofs-and strengthened versions-of Lagarias and Sleators theorem (for n i? n boards, each player can prolong the game until the board is full, so the first/second player can always win if n is even/odd) and Evanss theorem (for even n , the acute corner is a winning opening move for the first player). Also, for even n ? 4 , we find another first-player winning opening (adjacent to the acute corner, on the first players side), and for odd n ? 3 , and for each first-player opening, we find second-player winning replies. Finally, in response to comments by Martin Gardner, for each n ? 5 , we give a simple winning strategy for the n i? n board. Highlights? We extend a classic result of Lagarias and Sleator, and also one of Evans. ? For n -by- n boards, we give new strengthened proofs that each player can prolong the game until the board is full. ? For 2 k -by- 2 k boards with k at least 2, we find a new first-player winning opening. ? In response to a comment of Martin Gardner, we give simple winning strategies on small boards.

Automatic strategy verification for Hex

We present a concise and/or-tree notation for describing Hex strategies together with an easily implemented algorithm for verifying strategy correctness. To illustrate our algorithm, we use it to verify Jing Yangs 7×7 centre-opening strategy.