Over Thanksgiving dinner we were discussing a calculation that had come up, and my wife (born and raised in Japan) commented that it was a "tsuru-kame (crane and turtle)" situation. My son and I, both puzzled, asked for an explanation of what a crane and turtle situation was. My wife was surprised I had never heard of it, and proceeded to try to explain the calculation algorithm. Unfortunately, too many years had intervened between her middle school, where she learned the technique, and the dinner, and she couldn't quite explain.

So we all looked it up later, and I, in the process, learned some interesting cultural tidbits, and also learned another algorithm for solving what I call coin problems. You've got some number of things with one value associated, and some number with another value, know the total number of things, and know the total value. Find the number of each item.

This is a typical lesson in New York's Algebra curriculum. It appeared in Integrated Algebra, Math A before that, and, I assume, in whatever name they gave Algebra before Math A. I assume it's in CCLS Algebra, as well. We teach to solve using the technique of simultaneous equations and substitution.

The Tsuru-Kame method (鶴亀) takes a different approach that seems more like using area to solve.

Let's pretend we have a bunch of cranes and turtles, totaling 10 animals. Let's also say that there are 34 legs all together in this menagerie. Cranes each have two legs, and turtles have four. So the difference between them is two legs. If we were to assume all the animals are turtles, we'd come up with 40 legs. This is six extra legs than we actually have. Divide that six by the difference between crane and turtle leg count (2) and we get three. So that is the number of cranes. Which means there must have been seven turtles. (Check: 3 cranes @ 2 legs is 6 legs, 7 turtles @ 4 legs is 28 legs, for a total of 34 legs. It works.)

This method avoids algebra, but is pretty intriguing to me. I am almost always happy to learn a new algorithm.