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12 October, 15:44

Suppose a game of 3-D tic tac toe is played on a 6x6x6 cube. How many winning lines of 6-in-a-row are there through the cube?

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  1. 12 October, 17:57
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    On a cube of side k and n dimensions, there are

    [ (k+2) ⁿ - kⁿ] / 2

    winning lines.

    In your case,

    k = 6

    n = 3

    Therefore:

    [ (6+2) ³ - 6³] / 2

    = (8 ³ - 6³) / 2

    = (512 - 216) / 2

    = 148

    Hence, there are 148 6-in-a-row winning lines through the cube.
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