You have a standard 8x8 chessboard with 64 squares. Two diagonally opposite corners of the chessboard are removed, leaving 62 squares. You are given 31 dominoes, each of which covers exactly two squares. The puzzle asks whether it is possible to cover the entire chessboard with these 31 dominoes, given that each domino must cover exactly two adjacent squares.

Objective:
Determine whether it is possible to cover the modified chessboard completely with the 31 dominoes, and if not, explain why.

Solution

To solve this puzzle, we need to consider the nature of the chessboard and the way dominoes cover the squares. The key to the solution lies in analyzing the color pattern of the chessboard.

Step-by-Step Solution for the Chessboard Puzzle:

Step 1: Analyze the Chessboard Colors

A standard 8x8 chessboard has an alternating color pattern with 32 black squares and 32 white squares.

• When two diagonally opposite corners are removed, they are of the same color (both black or both white), because the chessboard is symmetrical.
• After removing these two squares, the modified chessboard now has 30 squares of one color (e.g., 30 black squares) and 32 squares of the other color (e.g., 32 white squares).

Step 2: Consider How Dominoes Cover the Board

Each domino covers exactly two squares, and when placed on the chessboard, it will always cover one black square and one white square (since they are adjacent).

• On a normal 8x8 chessboard, with 32 black and 32 white squares, 31 dominoes can perfectly cover the board because they can pair all the black and white squares evenly.

Step 3: Evaluate the Modified Chessboard

After removing the two diagonally opposite corners:

• The modified chessboard has an imbalance: 30 squares of one color and 32 squares of the other.
• Since each domino must cover one black and one white square, it’s impossible to cover an uneven number of squares (30 black and 32 white) with the dominoes.

Step 4: Conclusion

The imbalance in the number of black and white squares means that there will always be two squares of the same color that cannot be covered by a domino, no matter how you arrange the dominoes. Therefore, it is impossible to cover the entire modified chessboard with the 31 dominoes.

Final Answer

It is not possible to cover the entire modified chessboard with the 31 dominoes because the removal of two diagonally opposite corners creates an imbalance in the number of black and white squares, making it impossible for the dominoes to pair all the squares.