I think what you’re looking for is in *Sixth Book of Mathematical Games from Scientific American*.

There’s an article in there about using these graphs to solve “vessel problems”, you know, the kind of problems like: You have a 7 quart vessel and a 5 quart vessel. How do you measure 3 quarts?

This type of problem is closely tied to greatest common divisors; given two vessels, one holding a units, the other holding b units, the greatest common divisor of a and b is the smallest amount that can be measured. For example, if you have a vessel holding 12 oz., a vessel holding 30 oz., then 6 oz. is the smallest amount you can use those to measure.

Anyway, what’s described in the article is a method of solving these types of problems using a parallelogram and having a ball, beam of light, whatever, bounce around.

It’s really hard to describe without pictures, but I’ll try. Picture a parallelogram–top and bottom sides horizontal, each 7 units long; left and right sides 5 units long each, and making an angle of 60[sup]o[/sup] to the horizontal. Mark off unit intervals on the sides, like you would in a standard xy-plane (except the y axis is tilted).

Now we put a ball inside the parallelogram. At any moment, the position of the ball tells how many quarts are in either vessel (just like in standard coordinate axes).

Start the ball in the bottom left corner ((0,0)–corresponding to the state of nothing being in either vessel). Roll the ball to the right along the bottom side (filling up the 7 quart vessel). The ball hits the corner ((7,0)–vessel full), then bounces off and hits the top side at the point (2,5)–the 7 quart vessel has been used to fill the 5 quart vessel. And so on.

The smallest point the ball hits on the x or y-axis corresponds to the greatest common divisor of the two numbers–in this example, eventually the ball will hit both (1,0) and (0,1), since 1 is the gcd of 7 and 5.