PotentialBased Strategies for TicTacToe on the Integer Lattice with Numerous Directions We consider a tictactoe game played on the ddimensional integer lattice.  The game that we investigate is a Makerxe2x80x93Breaker version of tictactoe.  In a Makerxe2x80x93Breaker game, the first player, Maker, only tries to occupy a winning line and the second player, Breaker, only tries to stop Maker from occupying a winning line. We consider the bounded number of directions game, in which we designate a finite set of directionvectors cal S subset Bbb Zd which determines the set of winning lines.  We show, by using the Erdxc5x91sxe2x80x93Selfridge theorem and a modification of a theorem by Beck about games played on almostdisjoint hypergraphs, that for the special case when the coordinates of each directionvector are bounded, i.e., when cal S subset  vecv : vecv_infty leq k , Breaker can win this game if the length of each winning line is on the order of d2lg(dk) and d2lg(k), respectively.  In addition, we show that Maker can build winning lines of length up to (1o(1))dlg k  if cal S is the set of  all  directionvectors with coordinates bounded by k.  We also apply these methods to the nconsecutive lattice points game on the Nd board with (essentially) cal S  Bbb Zd, and we show that the phase transition from a win for Maker to a win for Breaker occurs at n (do(1))lg N.

Using Combinatorial Solutions and Atomic Weights to Play Competitive AI Clobber. Clobber is a twoplayer abstractstrategy game with no element of random chance. Introduced by game theorists Michael H. Albert, J. P. Grossman and Richard Nowakowski in 2001, it has subsequently been extensively studied for its combinatorial properties. Since 2005, it has been a major event in the Computer Olympiad.Although the concept of endgame tables has been introduced for Clobber and a use case suggested, usage of such has been scant in the Computer Olympiad or in the further development of other areas of Combinatorial Game Theory. We remedy this gap in this paper and demonstrate how to implement and use combinatorial solutions and atomic weights in an endgame table to perfect the play of Clobber by AI.We find that with the proper leveraging of endgame tables, it is possible to play perfectly on many Clobber positions of up to 70 pieces (on a 10x10 opening position) within time limits of most tournaments, and the solutions of Clobber on some significantly sized boards are likely reachable in the near future.