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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