7. A token is placed on the B1 square of a chessboard. The token can be moved diagonally one square, either up-left or up-right, but it must not leave the board. In how many ways can the token be moved to the C8 square?

![](https://cdn.mathpix.com/cropped/2024_06_03_47603db0a075197c1d6ag-9.jpg?height=716&width=717&top_left_y=1561&top_left_x=681)

The use of a pocket calculator or any reference materials is not allowed.
First solution.

In the image below, all the fields of the chessboard that the token can pass through on its way from field B1 to field C8 are marked in yellow.

The number written in a yellow field indicates how many ways the token can move to that field starting from field B1. The numbers are filled in as follows:

We write the number 1 in field B1. Then we fill in the table from bottom to top. The token can move to a field only from the field that is directly below and to the left or below and to the right. Therefore, the number in each yellow field is obtained by adding the numbers in the yellow fields that are diagonally in the row below that field.

4 POINTS

![](https://cdn.mathpix.com/cropped/2024_05_30_1ae5e90785d026149547g-34.jpg?height=708&width=700&top_left_y=1733&top_left_x=695)

The token can be moved from field B1 to field C8 in 28 ways.

Note 1: If a student makes a calculation error in addition, 1 POINT should be deducted for each error.

Note 2: A similar consideration can be made in reverse, as if we are observing a token returning from field C8 to B1 with moves diagonally downward. The image then looks like this:

![](https://cdn.mathpix.com/cropped/2024_05_30_1ae5e90785d026149547g-35.jpg?height=705&width=696&top_left_y=484&top_left_x=700)