You want to make up a set of 27 1-cm cubes with their faces variously colored red, yellow and blue. You want to do this in such a way that you can form any of three 3-cm cubes: one that is all red on the outside, one that is all yellow on the outside or one that is all blue on the outside. You can't repaint the original 27 cubes again--the same set of colorations for the 1-cm cubes must work regardless of whether you want the outside to be red, yellow or blue. How must you color the faces of the 27 1-cm cubes?
Then consider the same problem with 64 1-cm cubes using four colors this time, fitting together to make any one of four 4-cm solid-color-on-the-outside cubes.
Is there a method that will work for n^3 unit cubes with n colors?
Consider the cude all red:
1) the center cube must be 3 blue side and 3 yellow sides.
2) The eight corners are made of 1 3R3B, 1 3R3Y, 3 3R2B1Y, and 3 3R1B2Y.
3) the six in the center of each face are 3 1R2B3Y and 3 1R3B2Y.
4) the 12 with 2 red sides (non coner/non center face) are 3 2R3B1Y, 3 2R1B3Y, and 6 2R2B2Y.
This fits to have all three colors where there are the 54 faces of each color and eight corner pieces, 6 center faces, 12 two sided, and one piece in the center of the other two colors.
Posted by Patrick
on 2005-12-12 15:55:09