Question 8
A polyomino consists of unit squares joined along whole edges. It has standard size if no smaller similar polyomino exists. Replacing each unit square by an $n\times n$ square gives size $n$. Construct ten pairwise nonsimilar rectangles under these rules:
(i) Use at most ten copies of a chosen polyomino. (ii) It is not rectangular. (iii) Every copy has integer size. (iv) At least one has standard size. (v) At least one has larger size. Rotations and reflections are allowed, without overlap or gaps. The left example is a $9\times4$ rectangle made with six copies and satisfies all rules. The right example is $13\times6$ with four copies of the same polyomino but fails rule (iv).
(i) Use at most ten copies of a chosen polyomino. (ii) It is not rectangular. (iii) Every copy has integer size. (iv) At least one has standard size. (v) At least one has larger size. Rotations and reflections are allowed, without overlap or gaps. The left example is a $9\times4$ rectangle made with six copies and satisfies all rules. The right example is $13\times6$ with four copies of the same polyomino but fails rule (iv).