$\underline{\text { Frankin } 5 . \text {. }}$.

A $2 \mathrm{~N} \times 2 \mathrm{~N}$ board is covered with non-overlapping dominoes $1 \times 2$. A lame rook has traversed the board, visiting each cell exactly once (each move of the lame rook is to a cell adjacent by side). We call a move longitudinal if it is a transition from one cell of a domino to the other cell of the same domino. What is

a) the greatest;

b) the least possible number of longitudinal moves?
a) Estimation. The number of longitudinal moves does not exceed the number of $2 \mathrm{~N}^{2}$ dominoes (since each domino contains no more than one longitudinal move).

Example. Take any rook's tour and number the cells in the order of the tour. Let cells $2 \mathrm{k}-1 \mathrm{u} 2 \mathrm{k}$ form a domino for all $\mathrm{k}$ from 1 to $2 \mathrm{~N}^{2}$. Then the number of longitudinal moves is equal to the number of dominoes.

b) The case $\mathrm{N}=1$ is obvious. Let $\mathrm{N}>1$. Estimation. When passing through a corner, one of the two moves will be longitudinal. One corner can be the start of the rook's path, another can be the end, and the remaining corners will have to be passed. Therefore, there will be at least two longitudinal moves.

Example. Place a vertical domino in each of the top corners of the board, and place all the others horizontally.

Let the rook move in a snake-like pattern from the bottom-left corner (see figure). The longitudinal moves will be only in the vertical dominoes.

![](https://cdn.mathpix.com/cropped/2024_05_06_d9b26d382f9c77aca61cg-08.jpg?height=446&width=452&top_left_y=1006&top_left_x=14)

## Answer

a) $2 \mathrm{~N}^{2}$ moves.

b) 1 move for $\mathrm{N}=1 ; 2$ moves for $\mathrm{N}>1$.