By Hukum Singh
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S4 (the gap is 1) in 5 moves. s S5 and S6 (the gap is always 1)? 21 Illustration S5 : S6 : We challenge the reader to spend some time with these concrete goal sets S5 and S6 before reading the proof of the general theorem below. Example 2: Let the goal set be the 4 vertices of the “unit square” S = S4 “Tic-Tac-Toe set” “unit square” S4 S9 A simple pairing strategy shows that Maker cannot build a “unit square” S4 on the infinite grid ZZ2 , but he can easily do it on the whole plain. The trick is to get a trap pairing strategy on 2 trap We challenge the reader to show that Maker can always build a congruent copy of the “unit square” S4 in the plane in his 6th move (or before).
3 below. 1 Ordinary 32 Tic-Tac-Toe is a draw but not a Strong Draw. ) case study. However, it seems ridiculous to write a whole book about games such as Tic-TacToe, and not to solve Tic-Tac-Toe itself. To emphasize: by including this case study an exception was made; the book is not about case studies. Examples: Tic-Tac-Toe games 47 We have already proved that the first player can force a draw in the 32 game; in fact, by a pairing strategy. It remains to give the second player’s drawing strategy.
The game that we have been studying in Sections 1–2 (the “S-building game in the plane,” where S is a given finite point set) was a Maker–Breaker game. One player – called Maker – wanted to build a goal set (namely, a congruent copy of S), and the other player – called Breaker – simply wanted to stop Maker. Tic-Tac-Toe and its variants are very different: they are not Maker–Breaker games, they are games where both players want to build, and the player declared the winner is the player who occupies a whole goal set first.