Hello Math-Fun, Write an integer somewhere on a grid, one digit per square: +---+---+---+---+---+---+ | | | | | | | +---+---+---+---+---+---+ | | | | | | | +---+---+---+---+---+---+ | | 1 | 3 | 4 | 0 | | +---+---+---+---+---+---+ | | | | | | | +---+---+---+---+---+---+ | | | | | | | +---+---+---+---+---+---+ | | | | | | | +---+---+---+---+---+---+ The digits will now move vertically: - odd digit j will climb j squares - even digit k will sink k squares - zero stays were it is In our example, the digits of 1340 move like this: +---+---+---+---+ | | 3 | | | +---+---+---+---+ | | | | | +---+---+---+---+ | 1 | | | | +---+---+---+---+ | . | . | . | 0 | +---+---+---+---+ | | | | | +---+---+---+---+ | | | | | +---+---+---+---+ | | | | | +---+---+---+---+ | | | 4 | | +---+---+---+---+ Encircle now those new digits with an elastic band. The resulting shape is not a right triangle - which is a pity because we are precisely looking for integers moving into right triangles! The additional constraint being that ALL the new digits must be ON the elastic band. [you can see this more clearly there: http://www.cetteadressecomportecinquantesignes.com/RightTriangleNumbers.htm ] Here are a few good examples: Integer 130 (the right angle is on 1): +---+---+---+---+ | | 3 | | | +---+---+---+---+ | | | | | +---+---+---+---+ | 1 | | | | +---+---+---+---+ | . | . | 0 | | +---+---+---+---+ Integer 313333 (right angle on 1): +---+---+---+---+---+---+ | | | | | | | +---+---+---+---+---+---+ | | | | | | | +---+---+---+---+---+---+ | 3 | | 3 | 3 | 3 | | +---+---+---+---+---+---+ | | | | | | | +---+---+---+---+---+---+ | | 1 | | | | | +---+---+---+---+---+---+ | . | . | . | . | . | | +---+---+---+---+---+---+ Integer 2201022 (right angle on 1): +---+---+---+---+---+---+---+ | | | | | | | | +---+---+---+---+---+---+---+ | | | | 1 | | | | +---+---+---+---+---+---+---+ | . | . | 0 | . | 0 | . | . | +---+---+---+---+---+---+---+ | | | | | | | | +---+---+---+---+---+---+---+ | 2 | 2 | | | | 2 | 2 | +---+---+---+---+---+---+---+ | | | | | | | | +---+---+---+---+---+---+---+ Here is a bad example: Integer 21000 (yes we have a triangle, yes there is a right angle --on 1--, but the band doesn't pass through ALL digits): +---+---+---+---+---+---+---+ | | | | | | | | +---+---+---+---+---+---+---+ | | | 1 | | | | | +---+---+---+---+---+---+---+ | | . | . | 0 | 0 | 0 | | +---+---+---+---+---+---+---+ | | | | | | | | +---+---+---+---+---+---+---+ | | 2 | | | | | | +---+---+---+---+---+---+---+ | | | | | | | | +---+---+---+---+---+---+---+ Questions: a) Find the 40 first integers producing a right triangle b) Find the biggest integer producing a right triangle c) Find the smallest integer producing a rectangle. [Answers here in December 2010] Best, É.