Perfect Squared Squares
A "perfect" squared square is a square such that each of the smaller squares has a different size.
It is first recorded as being studied by R. L. Brooks, C. A. B. Smith, A. H. Stone and W. T. Tutte at Cambridge University. They transformed the square tiling into an equivalent electrical circuit — they called it "Smith diagram" — by considering the squares as resistors that connected to their neighbors at their top and bottom edges, and then applied Kirchhoff's circuit laws and circuit decomposition techniques to that circuit.
The first perfect squared square was found by Roland Sprague in 1939.
If we take such a tiling and enlarge it so that the formerly smallest tile now has the size of the square S we started out from, then we see that we obtain from this a tiling of the plane with integral squares, each having a different size.
Martin Gardner has published an extensive article written by W. T. Tutte about the early history of squaring the square.
Read more about this topic: Squaring The Square
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