No Slippage

When we place \(1 \times 2 \) dominoes on a board that isn't completely covered, sometimes we are able to move a domino horizontally or vertically. For example, in this \( 3 \times 2 \) board, the yellow domino can move horizontally while the blue domino is stuck.

Consider an \( 8 \times 8 \) board tiled with several \( 1 \times 2 \) dominos. What is the minimum number of dominos for which there exists an arrangement in which no domino can move (horizontally or vertically) into another position?

Details and Assumptions:

  • The dominos are not allowed to stick out of the board or overlap.
  • We get to choose the placement of the dominos.
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