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Composite number For composite numbers n, an n X n squares square will always have at least two n-cell patterns. This is due to the fact that composite numbers have at least 2 factor pairs: a 1, f1f2, X n pattern and an f1, X f2. ( pattern, where 1 < f1 and <= f2 are not necessarily distinct < n and f1f2 f1*f2 = n.) n-cell patterns For example, a 14 X 14 square can be made that tessellate the tiled using fourteen 1 X 14 rectangles or fourteen 2 X 7 rectangles; a 15 X 15 square by creating n can be tiled using fifteen 1 X 15 rectangles that are 1 or fifteen 3 X f1f2. Or, the n 5 rectangles; a 9 X n 9 square can be tessellated tiled using n nine 1 X 9 rectangles that are f1 or nine 3 X f23 squares (as in Sudoku!).

For example, for prime numbers p, a 14 p X 14 p square, its factor pairs are 1,14 and 2,7. It can always be tiled using fourteen 1 X 14 tessellated with p rectangles or fourteen 2 that are 1 X 7 rectanglesp.

For a 15 X 15 square, its factor pairs are 1,15 and 3,5. It can be tiled using fifteen 1 X 15 rectangles or fifteen 3 X 5 rectangles.

For a 9 x 9 square, its factor pairs are 1,9 and 3,3. It can be tiled using nine 1 X 9 rectangles or nine 3 X 3 squares (Sudoku!).

Prime number p X p squares can always be tessellated with p rectangles that are 1 X p.

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Composite number n X n squares will always have at least two n-cell patterns. This is due to the fact that composite numbers have at least four factors 2 factor pairs: 1, f1f2, and f1, f2. (f1, and f2, are not necessarily distinct and f1f2 = n. ) n-cell patterns can be made that tessellate the square by creating n rectangles that are 1 X f1f2. Or, the n X n square can be tessellated using n rectangles that are f1 X f2.

For example, for a 14 X 14 square, its factor pairs are 1,14 and 2,7. It can be tiled using fourteen 1 X 14 rectangles or fourteen 2 X 7 rectangles.

For a 15 X 15 square, its factor pairs are 1,15 and 3,5. It can be tiled using fifteen 1 X 15 rectangles or fifteen 3 X 5 rectangles.

For a 9 x 9 square, its factor pairs are 1,9 and 3,3. It can be tiled using nine 1 X 9 rectangles or nine 3 X 3 squares (Sudoku!).

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Composite number n X n squares will always have at least two n-cell patterns. This is due to the fact that composite numbers have at least two prime four factors p1 1, f1, f2, and p2f1f2. n-cell patterns can be made that tessellate the square by creating n rectangles that are 1 X p1p2f1f2. Or, the n X n square can be tessellated using n rectangles that are p1 f1 X p2f2.

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proposed

Composite number n X n squares will always have at least two n-cell patterns. This is due to the fact that composite numbers have at least two prime factors p1 and p2. Nn-cell patterns can be made that tessellate the square by creating n rectangles that are 1 X p1p2. Or, the n X n square can be tessellated using n rectangles that are p1 X p2.

Prime number pXp p X p squares can always be tessellated with p rectangles that are 1 X p.

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