Unsolved problem: Is there a 5x5 board with just four hits or just four misses in which it is possible to deduce the positions of all five ships? I suspect the answer is no, but I'm not sure. Trick-tac-toe
1. Betty must put her O in the center square to avoid losing. If both players play as well as possible, the outcome is a draw.
2. Alfie must also place his O in the center to avoid losing. Alfie wins by playing his second O adjacent to his first move. This tic-tac-toe variation was invented by mathematician Frank Harary.
3. The first player can easily win by placing an X in any but the four corner squares. Then wherever O moves -
-, X takes another inside square diagonal to the O -
-. O can block only one end of the row -
-. X wins by completing the other end of the row -
4. Here are the four possible draws on a 4x4 board.
Puzzle solver Wei-Hwa Huang writes: "The trick to solving problem 4 is to consider the four center squares. No player can take three of the four center squares without forcing a three-in-a-row, so in a draw, these center squares must split two-two. If it's a "parallel" split [in which one symbol occupies both upper squares and the other symbol both lower squares], you get the first solution. If it's a "cross" split [in which identical symbols are diagonal to each other], then the corners are forced. Putting the identical symbol next to a corner symbol forces five more positions, leading to the second and third solutions. When each corner symbol is surrounded by its opposite, you get the last solution." Blocked In
1. Four chains of three, four, five, and six boxes each remain to be claimed. If both players play as well as possible, then Betty will start by drawing a line that cedes the shortest chain (three boxes long) to Alfie. Alfie then draws a line that gives the chain of four boxes to Betty, and so on. Betty wins, 13 to 12.
2. Alfie is tempted to claim the two boxes in the bottom left corner, but then he would be forced to draw a line that would yield the rest of the board to Betty. So instead Alfie draws the line shown above in red, creating a "domino" of two boxes. Now Betty must draw a line to claim the two boxes of this domino, then draw a line that gives the rest of the board back to Alfie. (If she ignores the domino, she loses by an even bigger margin.) This is a "double-dealing" move.
3. Alfie wins by repeatedly using double-dealing moves, closing off the last two boxes of each chain as a domino. Although this sacrifices two boxes to Betty, it forces her to return control to Alfie on the next chain. Alfie claims all of the final chain, winning the game 13 to 12. For more about dots and boxes strategy, see Elwyn Berlekamp's book The Dots and Boxes Game,
published by AK Peters. You can play dots and boxes on-line at www.puzzles.com
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