Is it possible to cut a square into nine squares and colour one of them white, three of them grey and ve of them black, such that squares of the same colour have the same size and squares of different colours will have different sizes? (3 points)
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1=2
03.09.2010 16:07
Searching for solutions to $a^2+3b^2+5c^2=d^2$, we get the solution $(a,b,c,d)=(2,3,1,6)$, and we exhibit this method of butchering the square:
I can't really draw it here, so here's how to draw it: Draw the big square, cut it into fourths. Color three of those fourths grey. Then with the uncolored fourth, draw a square in one corner of it so that it covers 4/9 of the square. Color that square white. Now you can cut the remaining region into fifths easily using squares that are 1/4 the size of the white square.
