6. In how many different ways can a chess king move from square $e 1$ to square $h 5$, if it is allowed to move only one square to the right, up, or diagonally to the right-up?
Answer: 129 ways.

Solution: Sequentially (starting from e1) find the number of ways to reach each cell. Each number (except 1) is obtained by summing the neighbors below, to the left, and diagonally left-below.

![](https://cdn.mathpix.com/cropped/2024_05_06_0823b63ccaab99760f27g-2.jpg?height=659&width=642&top_left_y=1121&top_left_x=290)