Thanks for the "Indian Clerk" reference -- I already have the Archimedes Codex book (and the SLAC physicist who worked on the project spoke at a lunch meeting at my school last spring!). As for nuclear pennies, I did the activity with my 9-year-old and then with a group of middle school math students at the San Jose Math Circle (http://geometer.org/sjcircle or http://sanjosemathcircle.org ). One thing that my 9-year-old pointed out is that "if you split the first penny, it should really be half a penny on each space, not a whole penny". Then we discussed what happened with further splitting ... and yeah, if you put a quarter penny on each space, that's OK, but it gets hard to keep track when you have 5/8 of a penny somewhere -- how do you represent that? So then we decided that each place should have its value (that's me trying to steer him toward the interesting invariant, instead of having the invariant be that all the penny pieces add up to one whole penny). Starting with the 1 splitting in half, you pretty quickly get a pattern of period 6: 1, 1/2, -1/2, -1, -1/2, 1/2, 1. It's the real part of the complex numbers! So you can prove that translations of a single penny must be multiples of 6 without needing to solve x^2 + 1 = x or dealing with complex numbers. And of course you can start with any two numbers, not just 1 and 1/2. I teased the kids by asking them to assign values to each space that will avoid the fractions ... which they could do ... and then by asking them to avoid the negative numbers ;) --Joshua Zucker