Here's the basic shape:
Xavi was upset a bit when I started experimenting to get the angles and shapes. "No extra folds!"
First: In half, then each half in half to get four parallel quarters. Then with outside edges folded in, fold the short edge to the long edge.
Second: open sheet, then fold in the corner of the right quarter, and the corner of the left half.
Third: fold in the right quarter, with the corner tucked in.
Fourth: repeat with the opposite corner. This picture shows the first corner fold. Remember to tuck in the small corner dogear.
Fifth: Once both corners are done, fold them in to the center square. Xavi had a good test for if the pieces would fit together - see how well they nest.
Sixth: Assemble into a cube.
We had no instructions for that. We built one cube, but there were gaps at the edges, which was not the case with the one he had brought home. The teacher put together the cube for anyone who was having troubles with it in class. When we noticed that each edge had a square 'covering' it, that gave us the clue to get it put together. Every tab goes in a slot turned out to be an important characteristic.
But where's the math? There was some in recognizing and naming the shapes, and some problem-solving in figuring out the cube. But then...?
I was interested in how it fit together. So I decorated our plain white cube by tracing those edge-squares that turned out to be a good clue. I added the dots because I thought it would help make the cube into a puzzle, and it would be interesting to help study how it fit together. I just asked Xavi what he thought the pieces would look like and he took off with it. Great work, and with the barest of nudges, he wrote it down.
This was a great context for talking about conjectures and proof. He repeatedly talked about how fun this was, and was very keen on sharing the results. If only he knew someone with a math blog...
While you might think the child of a mathematician is naturally interested in school mathematics, but that's not the case. At some point, teachers started to have more authority than the father did about how things could be. But maybe that's a post for another day.