Knightmare
The problem
A chess knight moves through a glass cube divided into 8 × 8 × 8 = 512 rooms. Each room has an address (x, y, z), where x, y, and z are each whole numbers from 1 through 8.
In one move, the knight changes exactly two different coordinates: one changes by exactly 2, the other changes by exactly 1, and the third does not change. Each change may add or subtract. A destination counts only if all three of its coordinates remain from 1 through 8. Two moves count as different only when they land in different rooms.
A room’s move number is the number of different rooms the knight can reach from it in one move.
The investigation
Organize your cases and proof on separate paper.
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Determine every move number that occurs somewhere in the cube. Give at least one example room address for each move number you claim is possible.
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Determine how many of the 512 rooms have each possible move number. Which move number is most common?
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Prove that your classification is complete without checking 512 rooms one at a time. Your proof must explain why every room is covered, why no destination is counted twice, and why your room totals add to 512.