Puzzle: how many triangles do you see? +-------+-------+-------+-------+-------+-------+-------+ | \ \ \ | / / / / | | \ \ \ | / / / / | | \ \ \ | / / / / | + +-------+-------+-------+-------+-------+ + | \ | \ \ | / / / | | | \ | \ \ | / / / | | | \ | \ \ | / / / | | + + +-------+-------+-------+ + + | | \ | \ | / / | | | | | \ | \ | / / | | | | | \ | \ | / / | | | + + + +-------+-------+-------+-------+ | | / | \ | | \ | | | | | / | \ | | \ | | | | | / | \ | | \ | | | + + + +-------+-------+-------+-------+ | / | \ | / \ \ | \ | | | / | \ | / \ \ | \ | | | / | \ | / \ \ | \ | | + + +-------+-------+-------+ + + | | / \ / | \ \ | \ | | | / \ / | \ \ | \ | | | / \ / | \ \ | \ | + +-------+-------+-------+-------+-------+ + | / / \ | \ \ | | / / \ | \ \ | | / / \ | \ \ | +-------+-------+-------+-------+-------+-------+-------+ Solution: There are 60 triangles in the figure. A simple way of counting is to lock at each vertex and count all triangles having their right angle at this vertex. In the right angles at each point the number of triangles using this angle is noted. This way the numbers are easily checked. +-------+-------+-------+-------+-------+-------+-------+ | \ \ \ 3 | 3 / / / / | | \ \ \ | / / / / | | \ \ \ 1 / / / / | + +-------+-------+-------+-------+-------+ + | \ | \ \ 2 | 2 / / / | | | \ | \ \ | / / / | | | \ | \ \ 2 / / / | | + + +-------+-------+-------+ + + | | \ | \ 1 | 1 / / | | | | | \ | \ | / / | | | | | \ | \ 3 / / 1 | 1 | 1 | + + 2 + +-------+-3-----+-------+-------+ | | / | \ | | \ 1 | 1 | 1 | | | / | \ | | \ | | | | | / | \ | | 1 \ | | | + + 1 + 3 +-------+-------+-------+-------+ | / | \ | / 3 \ \ 1 | \ 2 | 2 | | / | \ | / \ \ | \ | | | / | \ | 1 \ \ | \ | | + + 1 +-------+-------+-------+ + + | | / 2 \ 1 2 \ \ | \ | | | / \ / | \ \ | \ | | | 2 \ 1 / 1 | 1 \ \ | \ | + +-------+-------+-1-----+-------+-------+ + | / / 1 \ 1 | \ \ | | / / \ | \ \ | | 1 / 1 | 1 \ \ | +-------+-------+-------+-------+-------+-------+-------+ References: - BTV4U Postfach 424 D-71604 Ludwigsburg 2003-06-19 www.b-tv.de ->Progamm->Gl\"ucklicht -- http://www.mathematik.uni-bielefeld.de/~sillke/ mailto:Torsten.Sillke@uni-bielefeld.de