Let quasi-square be a figure that consists of an n×n-square with one more 1×1 square attached along one of its sides, so that the unit square shares a side with one of the unit squares of an n×n-square as well as shares a vertex with one of the corner squares of the n×n-square. Thus, the upper two figures on Fig. 3 are quasi-squares, whereas the lower two are not. Determine all possible integer n≥3 for which a plane can be filled with the identical quasi-squares. Quasi-squares can be rotated and reflected but are not allowed to overlap.
Fig. 3
Solution
Fig. 4 shows an example of filling the plane with quasi-squares in which an extra 1×1-square shares a side with an edge square. Fig. 5 shows an example for quasi-squares which have unit square sharing the vertex but not the side with the edge square of an n×n-square.
Fig. 4
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