The Grasshopper Problem

This is my newest problem, to appear in the October 2010 issue of La Voz.

What is this picture asking? Ben suggested that instead of bold black squares, I should have indicated the squares with something especially significant to grasshoppers. Something tasty or something dangerous?

Where did this come from? This problem is related to a derivation of the formula for Primitive Pythagorean Triples: Find all relatively prime natural number triples (a,b,c) that satisfy the equation

The first step in my favorite elementary number theory proof is to show that not both a and b are odd. This would imply that c is one of the shaded squares above: 2, 6, 10, 14, …

Does the grasshopper land on a (boxed) square?


  1. Nice graphic! Having come here roundaboutly, I had the picture without the problem statement, hence the metaproblem "What is this problem?" which was rather nice to solve.

  2. Thanks, Allen! I'm very happy with it (but might tweak it a bit with Ben's suggestion in mind).
    I've taken up problem posing for my monthly column in La Voz (, and am slowly developing my rudimentary computer graphics skills.
    How's your new job treating you?

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