MathLabs

Problem 3

Define a hook to be a figure made up of six unit squares consisting of a row of three squares and a column of four squares sharing a corner square, together with one further square attached at the far end of the row (or any figure obtained from this one by rotations and reflections). Determine all m×nm\times n rectangles that can be covered without gaps and without overlaps by such hooks, with no part of a hook covering area outside the rectangle.
Step 4 of 6: Pair up hooks by their holes
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pair each hook withthe one filling its L-hole  ⟹  hooks come in pairs forming one of two 12-cell tiles\text{pair each hook \quad\text{with}\quad the one filling its L-hole} \implies \text{hooks come in pairs forming one of two }12\text{-cell tiles}
Detailed analysis

Call the unique concave cell of a hook its inside cell. Fix a hook H and place coordinates so that its six cells are H={(0,0),(0,1),(0,2),(1,2),(2,1),(2,2)}H=\{(0,0),(0,1),(0,2),(1,2),(2,1),(2,2)\}; its inside cell is the missing central cell of the 3×33\times3 box. The hook covering that cell must be a translated rotation or reflection of H and must be disjoint from H. Listing the eight orientations and translating them until they contain the inside cell leaves exactly the two reciprocal configurations (a configuration and its reflection); in each, the inside cell of the second hook is one of the six cells of H. Therefore the hook covering H's inside cell is paired back with H, so every hook belongs to exactly one pair. The two possible unions are the two 12-cell tile types in the source diagram; the coordinate check also shows that no third orientation can occur.