Rectangle Divided into Incomparable Subrectangles

Theorem

Let $R$ be a rectangle.

Let $R$ be divided into $n$ smaller rectangles which are pairwise incomparable.


Then $n \ge 7$.


The smallest rectangle with integer sides that can be so divided into rectangles with integer sides is $13 \times 22$.



Proof


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Sources

  • 1975 -- 1976: A.C.C. Yao, E.M. Reingold and B. Sands: Tiling with Incomparable Rectangles (Journal of Recreational Mathematics Vol. 8: pp. 112 – 119)
  • 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $7$
  • 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $7$
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