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Colored Blocks (Posted on 2005-12-12) Difficulty: 2 of 5
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?

  Submitted by Brian Smith    
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Solution: (Hide)
Compiled from comments by Cory Taylor and Steve Herman

For a 2x2 cube or even a 3x3 cube, the combinatorics of the painting requirements are not prohibitive and can be worked out relatively quickly. Beyond a 3x3 cube the color selections get very complex and a more general solution would be required. As the general solution for this is relatively straightforward it will be explained and it can then be applied to any desired size of cube.

For any sized large block, made up of "n" blocks per side, there are n blocks total. Each block has 6 faces so the total number of faces available for painting is 6*n.

Each face of the larger block contains n smaller block faces, meaning that the block as a whole has 6*n faces painted. If n colors are used, the total number of required painted faces is 6*n.

This means that, for any sized block, no face of any block can remain unpainted. At this point we have not shown that the solution is possible or impossible, simply that it needs to be 100% efficient. If a re-arrangement algoritm can be derived whereby through several (n) repetitions (of the same algorithm) each face of each smaller block is exposed exactly once, then this algorithm can be used to paint the faces in accordance with the problem requirements.

The following alogorithm (in three steps) successfully re-arranges the cubes so that successive paintings after each completion of the algorithm can achieve the desired final result. Moreover, after completion of the painting (painting the "n'th" side), the cube can be cycled through the paintings by repeating the algorithm.

Move the leftmost plane of cubes to the right extreme of the cube.
Then, move the frontmost plane of cubes to the back extreme of the cube.
Then, move the bottommost plane of cubes to the top extreme of the cube.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
3x3 solutionPatrick2005-12-12 15:55:09
Solutionre: ideas solution of 3x3x3Charlie2005-12-12 14:31:17
Some ThoughtsideasCharlie2005-12-12 13:53:02
SolutionSpoiler: General Solution and ConstructionSteve Herman2005-12-12 13:34:59
Some Thoughtsno real informationCory Taylor2005-12-12 10:21:21
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