In 1952 W. E. Roth published two theorems, one of which has come to be known as Roth’s removal rule and (slightly generalised) goes as follows. [Recall that square matrices M, N are similar when there is an invertible matrix R such that RMR-1 = N. The matrix entries can be elements from any field, although for simplicity we shall call them ‘numbers’.]
A selection process used by some compilers of big-prize competitions is to present a triangle (usually equilateral) divided into (say) nine congruent sub triangles by the lines joining corresponding trisection points of the sides, and to ask how many triangles there are in the figure. It is amusing to obtain the general result when each side of the original triangle A is divided into n equal parts and pairs of dividing points are joined by parallels to the sides in all possible ways.
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