It’s 27 identical blocks, each of which is 4x5x6, to be placed inside a 15x15x15 box. 24<25, and so it should be possible to put all the pieces in the box with room to spare. And indeed it is possible, but it's tough--it took me a couple hours to figure out how to arrange the pieces. I think it's just a beautiful puzzle because all the pieces are identical. Some guy showed me this puzzle once--he pointed out that the 2-D version (with 4 identical rectangles) is trivial and he claimed that the 4-D isn't hard but that he didn't know if the 5-D version had a solution. Anyway, 3-D is hard enough for me. I'm sure someone manufactures it, but I don't know who, so I took a board and sawed it into 27 little pieces one day and made my own version, where it sits in my office inside a little plastic box. If I had one of those digital cameras, and if I were in my office, I'd post a picture of it here. P.S. It's only the second-coolest puzzle because Rubik's cube is the coolest.
What are the numbers on the 2-D version? (I'm thinking four 6-by-9 rectangles in a 15-by-15 box, which would be trivial, but I might be wrong.)
Isabel,
Yes, your example would work.
More generally, the d-dimensional version is d^d identical pieces, each of dimension a_1 x a_2 x … x a_d, in a box of dimension A^d, where A=a_1+a_2+…a_d. The only constraints are that a_1 A (so that you can't cheat by stacking a bunch of pieces together in the thinnest dimension. In the solution, you need to have one of each dimension in each direction, which is like a latin square design in statistics, but there's also the complexity that the pieces have to fit together in d dimensions (which isn't trivial even if d=3).
For practical purposes (e.g., if you make the
pieces out of wood), you don't want the dimensions to be too close to each other. In 3 dimensions, you can't do 3x4x5 (since 3+3+3+3=12, so you could cram 4 pieces in crosswise), so 4x5x6 seemed like reasonable dimensions. Anything smaller (e.g., 7x9x11) and it might be possible to cram 4 pieces in by accident, and anything bigger (e.g., 9x11x13) it might be tricky to feel which are the small, medium, and large dimensions.