Bogdanov I.I.
A chess piece can move 8 or 9 squares horizontally or vertically. It is forbidden to move to the same square twice.
What is the maximum number of squares this piece can visit on a $15 \times 15$ board? (The tour can start from any square.)
The central cell cannot be reached from any other cell. Cells in the central vertical (or horizontal) can only be reached from cells of this vertical. Conversely, starting from such a cell, the figure will not leave the central vertical, meaning it will cover no more than 14 cells.

The remaining $225-29=196$ cells can be covered. Note that in 13 moves vertically, all cells of the vertical (except the central one) can be covered if starting from one of the cells adjacent to the central cell and ending on the other. To cover 196 cells, we start from a cell diagonally adjacent to the central cell, cover the vertical in 13 moves, make a horizontal move, cover a new vertical in 13 moves, and so on. It is not difficult to see that in 13 horizontal moves, the figure will visit all verticals (which alternate in the same way as the cells), except the central one, and each time it will leave and return to the vertical through a cell adjacent to the central one.

## Answer

196 cells.