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 3 of 6: Necessity: exclude side lengths 1,2,51,2,5
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m or n∈{1,2,5}  ⟹  no hook fits without leftover cellsm\text{ or }n\in\{1,2,5\} \implies \text{no hook fits without leftover cells}
Detailed analysis

The hook has a 3×33\times3 bounding box, so it cannot lie in a strip of width 11 or 22. For width 55, look at a corner and the first two rows and columns next to the two boundary sides. A hook meeting the corner has only the finitely many orientations obtained from the displayed cell pattern; checking these orientations, the cells of the hook that lie in the first two boundary layers force one of the three cells in the opposite corner of this 3×33\times3 window to be covered next. That next hook either overlaps the first one or leaves the boundary cell immediately beside the corner uncovered. Thus no tiling can meet a corner of a width-55 rectangle, while every rectangle has four corners. Hence m,n∉{1,2,5}m,n\notin\{1,2,5\} is necessary.