You can easily make three identical pieces that can be put together to make a cube. Simply take six flat squares, group into three pairs and join the two squares of each pair at right angle. You can then assemble the three right angles into a cube. But they won't stay together. If you just set it down and don't touch it, it might hold together, but jostle it and it will collapse. This cubic trisection is a little peculiar. Make that very peculiar.
This puzzle was invented by Robert Reid. Doesn't seem to be much information about him, possibly since he spent 50 years running a cinema in Peru.