1. **State the problem:** We want to find the number of paths from point X (bottom left black square) to point Y (top row, 4th black square from left) on an 8x8 checkerboard. We have two cases: a) Moving only diagonally forward on black squares. b) Moving only north and east on a grid. 2. **Understanding the board:** - The board is 8 squares across and 8 squares down. - Squares alternate colors, starting with white in the upper left corner. - Black squares form a diagonal pattern. ### a) Moving diagonally forward on black squares: 3. Moves allowed are diagonally forward left or right from one black square to another. Since diagonal forward moves advance one row up and one column left/right, the path moves upward. 4. Position X is bottom row, first black square from the left (row 8, col 1 black square). Position Y is top row, 4th black square from left (row 1, col 7), since black squares appear in alternating columns per row. 5. The movement can be mapped to combinatorics on coordinates: each diagonal move changes position by $(row-1, col\pm 1)$. From X to Y, total upward moves $= 7$ (from row 8 to 1), and the horizontal displacement is from column 1 to 7, which requires moving right 6 columns. 6. Each diagonal move right increases column by 1, diagonal move left decreases column by 1. Total moves = 7 (equal to row difference). If $r$ is number of moves diagonally right, and $l$ is to the left: $$r + l = 7$$ $$r - l = 6$$ Solve system: Adding, $$2r = 13 \implies r = 6.5$$ not integer, so position Y is not reachable via diagonal moves only. **Hence, no valid paths exist moving only diagonally forward black squares from X to Y.** ### b) Moving only north and east on grid: 7. Model the board as a grid with rows numbered 1 to 8 from bottom to top, columns 1 to 8 from left to right. Start at X: (row 1, col 1), target Y at (row 8, col 4) since Y is top row 4th black (considering only grid moves). 8. Number of paths from $(1,1)$ to $(8,4)$ moving only north and east: Number of north moves = $8 - 1 = 7$ Number of east moves = $4 -1=3$ Total moves $= 7 + 3 = 10$ 9. Use combinations to count unique paths: $$\text{Number of paths} = \binom{10}{3} = \frac{10!}{3!7!} = 120$$ **Final answers:** - a) Number of diagonal forward paths on black squares: $0$ - b) Number of paths moving north and east on grid: $120$