A few months ago I posted an article about an interesting little mathematics problem called "The Random Maze", and based on the responses it got it would seem that my readership enjoys such things. And so to keep the enjoyment going, (and as a warm up exercise for the millions of students returning to school next week) I will be proposing another curious puzzle in today's article*.

As with my previous puzzle postings, I suspect that this one has been studied before but the few colleagues that I consulted with have not seen it and as such I do not have a reference for the first appearance. I thought it up myself while watching a science documentary one night, but it is very likely not original. And with the disclaimers out of the way, here is the puzzle:

Suppose you have a piece of graph paper, N x M units in size. Suppose further that you have a piece of stiff wire of whatever length you choose (but larger than NxM units of length). This wire is so stiff that it cannot be bent into a circle, half-circle or quarter circle whose radius is less than 1 unit. 

The question is, can you bend this wire such that it passes through each square of the graph paper once and only once? And if so, what values of N and M allow this to happen?

I contend that it is not possible, but I am curious to see if this claim can be proven or disproven. So go forth and bend wires, whether real or imaginary, and see what you find!

* While I have not posted this puzzle on this blog before, I have posted it on other websites as well as a variation on it called the "Greedy Snake Puzzle".