# Cutting a 27×27 square into incomparable rectangles

How can I cut a 27×27 square into 8 incomparable rectangles?

A rectangle with width w and height x is incomparable with a rectangle of width y and height z iff $$wz$$ or $$w>y\land x. (Assume $$\text{width}\le\text{height}$$.)

All rectangles must have integer side lengths.

The book Fractal Music, Hypercards and more... by Martin Gardner says

but eight rectangles can tile a square of side 27

• "(Assume width <= height)" Does this mean 4×7 and 8×5 rectangles are comparable? Commented Aug 15 at 1:53
• @AxiomaticSystem Yes, they are. Commented Aug 15 at 1:54
• The layman's definition for incomparable rectangles is just "neither will fit inside the other", which seems somewhat easier to grasp than the given notation.
– Bass
Commented Aug 15 at 2:37
• @Bass It might be possible to rotate one rectangle (by an angle that is not 0 or 90 degrees or something equivalent) and fit it inside the other. For example, a 1x6 can fit inside a 5x5 (with rotation). Commented Aug 15 at 2:39
• @Bass's statement implies parallel edges, which would be explicitly stated in a proper description. Commented Aug 15 at 2:42