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 1 of 6: Fix notation for the hook
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hook=6 cells={(0,0),(0,1),(0,2),(1,2),(2,1),(2,2)} in a 3×3 box\text{hook}=6\text{ cells}=\{(0,0),(0,1),(0,2),(1,2),(2,1),(2,2)\}\text{ in a }3\times3\text{ box}
Detailed analysis

A hook fits in a 3×33\times3 bounding box, using 66 of its 99 unit cells and leaving an L-tromino of 33 cells empty; equivalently it is a column of 33 cells and a row of 33 cells sharing a corner, with one extra cell attached beyond the far end of the row. Two hooks placed so that one exactly fills the L-tromino hole of the other tile a 3×43\times4 rectangle.