A217150 Smallest number, two or more, of unequal squares that tile a square in exactly n ways; or 0 if there is no such set of tiles.

21, 25, 28, 24

Offset: 1

Author

Geoffrey H. Morley, Sep 27 2012

Keywords

• nonn
• hard

Comments

The definition precludes the trivial tiling by only one square.

a(8) = 25, a(16) = 26, a(32) = 28, and a(48) = 28. For all other n > 4, a(n) > 29 or a(n) = 0. a(n) > 29 (with a known upper bound) for n = 7, 24, 56, 64, 72, 96, 112, 128 ...

Examples

			See MathWorld link for an explanation of Bouwkamp code used in these examples.
a(2) = 25. The 25 squares of the following perfect square with side 540 can be arranged in one other way by rearranging polygons a-c: (279,261)(98,68,95)(135a,144a)(30,38)(11,84)(55,65,8)(57)(126b,9a)(45c,10)(117c,36c)(116,16)(100)(81c).
a(3) = 28. The 28 squares of the following perfect square with side 408 can be arranged in two other ways by rearranging polygons a-e: (165,102a,141a)(63a,39a)(24a,156b)(99c,61c,92c)(38c,23c)(15c,8c)(7c)(9c,20c,64c)(144c,13c,2c)(11c)(44c)(45d,111d)(87e,21d)(66d). No other set of fewer than 30 unequal squares tiles a square in exactly three ways.
		

Links

• C. J. Bouwkamp, On some new simple perfect squared squares, Discrete Math. 106-107 (1992), 67-75.
• A. J. W. Duijvestijn, Simple perfect squared squares and 2x1 squared rectangles of order 25, Math. Comp. 62 (1994), 325-332.
• Eric Weisstein's World of Mathematics, Perfect Square Dissection

Crossrefs

Cf. A014530, A217151.