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Interview prep / Probability

Number of winning lines in 4x4x4 noughts and crosses

Question

Ordinary 3×33\times3 noughts and crosses has 88 winning lines (three rows, three columns, two diagonals). Now play on a 4×4×44\times4\times4 cube, where a win is four of your marks in any straight line, including vertical lines and all planar and space diagonals. How many winning lines are there?

Short answer

7676. Surround the cube with a one-cell shell to get a 6×6×66\times6\times6 cube. Each winning line, extended both ways, exits into exactly two shell cells, and every shell cell is hit by exactly one line, so the count is (63−43)/2=152/2=76(6^3-4^3)/2=152/2=76.

Adapted from QuantQuestions.io; question and answer rewritten by SelfMadeQuant.com.

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