Minimally Squared Rectangles

Divide a rectangle of size into the minimal number of squares. For a rectangle of size 17×19, the solution (with nine squares) is nontrivial to find. This Demonstration gives precalculated solutions for rectangles up to size 380×380.

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Let represent the minimal number of squares for an rectangle. Conjecture: for all and .
These solutions were compiled in the mistaken belief that a counterexample would be found. So far, the conjecture holds true for . Solutions are found with a method involving Young tableaux; a single solution can require hours to find [1].
Some rectangles can be divided into squares of different sizes. These are known as perfect rectangles, and are much easier to find [2].
References
[1] B. Felgenhauer. "Filling Rectangles with Integer-Sided Squares." (Mar 17, 2013) int-e.eu/~bf3/squares.
[2] S. Anderson. "Squaring.Net 2013." (Mar 26, 2013) www.squaring.net.
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