Fault-Free Rectangle


A dissection of a rectangle into smaller rectangles such that the original rectangle is not divided into two subrectangles. Rectangle dissections into 3, 4, or 6 pieces cannot be fault-free but, as illustrated above, a dissection into five or more pieces may be fault-free.

More precisely, a complete existence criterion for fault-free rectangles with congruent tiles is given by the following theorem due to Graham (1981, p. 125). A rectangle with integer sides p and q admits a (nontrivial) fault-free tiling by a×b tiles (where a and b are relatively prime integers) if and only if all the following conditions are fulfilled:

1. Each of a and b divides one of p and q.

2. Both the Diophantine equations ax+by=p and ax+by=q have at least two distinct solutions in positive integers.

3. If a=1 and b=2, then p and q are not both equal to 6.

A nowhere-neat dissection is a special case of a fault-free rectangle in which no two squares have a side in common.

See also

Blanche's Dissection, Mrs. Perkins's Quilt, Nowhere-Neat Dissection, Perfect Square Dissection, Rectangle

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Graham, R. L. "Fault-Free Tilings of Rectangles." In The Mathematical Gardner: A Collection in Honor of Martin Gardner (Ed. D. A. Klarner). Belmont, CA: Wadsworth, pp. 120-126, 1981.Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, p. 85, 1999.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 73, 1991.

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Fault-Free Rectangle

Cite this as:

Weisstein, Eric W. "Fault-Free Rectangle." From MathWorld--A Wolfram Web Resource.

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