Chessboard Conundrum

Logic Level 3

Consider a standard $8\times 8$ chessboard as shown. How many 7-move routes, involving black squares only, take you from the black squares on the top edge of the board to the black squares on the bottom edge of the board?

Note : you can start in any of the black squares in the top row and end in any of the black squares in the bottom row.

56 256 296 192

1 solution

Isaac Reid
Dec 5, 2015

The easiest way to count the zig-zag paths from the black squares in the top row to the black squares in the bottom row is to write the number of ways to reach each square, as shown.

The top-row black squares are labelled with 1s, since there is only one way to reach each of them (i.e. by choosing it as the starting square). The number of ways into a square in a subsequent row can then be calculated by summing the values either side of it in the previous row, as illustrated. Continuing this until the final row yields $35, 89, 103$ and $69$ . The total number of ways to reach any black square on the bottom row is the sum of these values: that is, $35+89+103+69=296$ . So the solution is $\boxed{296}$ .

I think you meant 7 move only. I assume the initial position doesn't count as a step. And I think you should mention that you should start from any of the black squares in the first row and end in any of the black squares in the last (eighth) row.

Pi Han Goh - 5 years, 6 months ago

Thank you; that does make the question clearer.

Isaac Reid - 5 years, 6 months ago

Chessboard Conundrum

1 solution

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