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?
There need to be 8 corners for each color, occupying 24 faces.
A given piece (1-cm cube) can have one of the following:
A. Two colors arranged around two opposing corners. (3-3)
B. One corner surrounded by three faces of the same color, plus two faces of another color and one face of the remaining color. (3-2-1)
C. All three colors with each face adjacent to the other face of that color. (2-2-2)
Every face that is colored the desired color of the outside must be on the outside, as there are only 9*3=54 faces of each color. Therefore the center cube for any given arrangement must be a 2-color cube (A), of the two colors not involved on the outside.
There must be 8 cubes with three of red, 8 with three of yellow, and 8 with three of blue. As seen above, at least two of each color must be represented thus on a type A cube; ie we must have at least one R-B, one R-Y and one B-Y.
There must be 12 cubes with two R's, 12 with two Y's and 12 with two B's. Since there are only 27 cubes, some of these pairs must be on the same cubes (therefore type C, rather than B).
Posted by Charlie
on 2005-12-12 13:53:02