Draw a 9 by 3 rectangle on a square grid. Draw one diagonal. How many squares does the diagonal pass through? Draw some non-similar rectangles with one diagonal. How many squares does the diagonal pass through? Develop a rule to determine the number of squares a diagonal passes through for any rectangle of any size.
I think I'm going to keep it in my back pocket for a day when I need to kill half a period. It might be nice for the first day of school if you like that sort of thing. I don't think I've seen it before. It was sent to me by Øistein Gjøvik - he has a post about it that includes access to a Geogebra file. (One benefit of blogging I would have never predicted: a cool Norwegian sends awesome math problems to my inbox.)
I have been on a bit of a Sketchpad bender since we used it at PCMI, so here's a sketch I made.
I am torn about giving guidance about posting solutions in the comments. I have one way to think about it that works, but I'm sure there are more and I really want to hear them. On the other hand, I don't want to spoil anyone's fun. So maybe if you want to work on it, resist looking at comments?
Another thing I'd like to hear about is, do you see this fitting into a curriculum? Or is it just a nice problem that doesn't have a home in a unit of study?