# Simple Perfects by Difference (width - height)

In the search for squared squares, Brooks, Smith, Stone and Tutte constructed squared rectangles by drawing plausible diagrams and solving algebraic equations describing how the squares had to fit together. Tutte writes  "The first of the new perfect rectangles was found by Stone. It was the 11th order, that is, it was dissected into 11 unequal squares. As with all subsequent perfect rectangles the side-lengths of the squares were commensurable. We could therefore take them to be all integers, and we could divide by their common factor to get them to lowest terms. The sides of Stone's rectangle were then 176 and 177. Stone remarked sardonically, that it was 'nearly square' and there was not much further to go. Our other examples were mostly of the 9th, 10th and 11th orders. Higher orders required more difficult calculations. Alas, no perfect square made its way into our catalogue."

The perfect rectangle found by Stone has a width - height difference of 1. The squared square was not found using this method. Eric Friedman has called rectangles with a difference of 1 'Almost squares'. An almost square is a rectangle with sides which are consecutive integers. Eric Friedman has shown almost squares 1 x 2, 2 x 3, 3 x 4, . . . n x (n+1) can be tiled inside another almost square when n = 1, 3, 8, and 20.

SPSRs from orders 9 to 17 sorted by difference (width - height). o9-17spsr-w-h.csv

Format; (width+height) (width-height) (widthxheight) (width/height) order width height e1 e2 e3 ... en (elements in bouwkamp code order);
for example 65 1 1056 1.03 9 33 32 9 10 14 8 1 7 4 18 15

### References

1. W.T. Tutte, "Graph Theory As I Have known it", Oxford Science Publications, Oxford, 1998, p3
2. Eric Friedman Almost Squares