Zhenodarov R.G.

Eight cells of one diagonal of a chessboard are called a fence. The rook moves around the board, not stepping on the same cell twice and not stepping on the cells of the fence (intermediate cells are not considered visited). What is the maximum number of jumps over the fence that the rook can make?
Evaluation. Note that if the rook jumps over a fence, then either the initial or the final cell of the jump is marked gray on the diagram. Since there are 24 gray cells and through each one a maximum of two jumps can be made, there can be no more than 48 jumps in total. At the same time, if there are exactly 48 jumps, then from each gray cell, two jumps to white cells must be made (in the previous count, a jump from a gray cell to a gray cell would be counted twice!). Then all moves from dark-gray cells will lead to a white square below the diagonal, and from there - only to dark-gray cells (or to other cells in the same square). This means that in this way, the rook will never reach the light-gray cells. Contradiction.

Thus, the number of jumps does not exceed 47.

An example with 47 jumps is shown in the diagram (the numbers in the cells indicate the order in which the rook passes through them).