A king starts at the lower left corner of a standard 8 × 8 chessboard. How many ways can the king move to the upper right corner using exactly eight moves? The king can move one square horizontally, vertically, or diagonally on each move.

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We have a chess king starting at the lower left corner of an 8 by 8 chessboard, at position 1,1. The goal is to reach the upper right corner at position 8,8 using exactly 8 moves. The king can move one square horizontally, vertically, or diagonally. This is a combinatorial problem where we need to find all possible paths with exactly 8 moves. To analyze this problem, we need to understand the required displacement. From position 1,1 to position 8,8, we need a net displacement of plus 7 in both x and y directions. Each diagonal move contributes plus 1 to both coordinates, while horizontal moves contribute plus 1 only to x, and vertical moves contribute plus 1 only to y. If we use d diagonal moves, h horizontal moves, and v vertical moves, we get the system: d plus h equals 7, d plus v equals 7, and d plus h plus v equals 8. Let's solve the system of equations step by step. We have d plus h equals 7, d plus v equals 7, and d plus h plus v equals 8. From the first two equations, we get h equals 7 minus d and v equals 7 minus d. Substituting into the third equation: d plus 7 minus d plus 7 minus d equals 8, which simplifies to 14 minus d equals 8, so d equals 6. Therefore, we need 6 diagonal moves, 1 horizontal move, and 1 vertical move. Now we need to count the arrangements. We have 8 total moves: 1 diagonal, 6 horizontal, and 1 vertical. This is a multinomial coefficient problem. The number of ways to arrange these moves is 8 factorial divided by 1 factorial times 6 factorial times 1 factorial, which equals 8. We can list all 8 arrangements by considering where to place the single diagonal move and single vertical move among the 8 positions. In conclusion, the king can reach the upper right corner in exactly 8 moves in precisely 8 different ways. The key insight is that the constraints force exactly 1 diagonal move, 6 horizontal moves, and 1 vertical move. The multinomial coefficient gives us the final count of 8 arrangements. This systematic approach guarantees we found all possible 8-move paths from position 1,1 to position 8,8.