Talk:Black Path Game

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"the second player can instead win by using a domino tiling including every square but the one containing the first player's first move." -

is a domino tiling including every square but the one ("X") difinitely exists?

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[ ] [X] [ ]

Maksim-e (talk) 18:17, 10 September 2008 (UTC)[reply]

A proper domino tiling only exists if the removed field is at a position (row, col) such that row + col is even. In the example: 3 + 2 = 5, which is odd. This is similar to the Mutilated chessboard problem.

The trick to solving the Black Path Game in these cases is that we are also allowed to pair up nonadjacent fields, if they are connected by an existing path on the board.

In your example, if the tile at position X is of the third kind (cross), so that the initial path extends upwards to the middle field, then you can create a tiling where the fields labeled `a` are horizontally connected:

 bcd
 bcd
 a+a

Similarly, if the tile at X is of the first kind (curve from bottom to right):

 dcc
 dab
 a/b

These two cases are the main two variations that can be extended to boards of different sizes.

MaksVerver (talk) 20:43, 24 June 2021 (UTC)[reply]