Contest Math II

Suppose you have a set of 2727 unpainted unit cubes. You want to paint them using three different colors (red, green, and purple). You can color each face of each cube individually (that is, the same unit cube might be multiple colors).

You want the coloring done in such a way that after the painting is done, you can assemble a larger cube using all 2727 unit cubes in three different ways:

1.) such that the entire outside of the larger cube is red.

2.) such that the entire outside of the larger cube is green.

3.) such that the entire outside of the larger cube is purple.

Our goal is to find all the ways we can allocate the initial painting of colors to the unit cubes such that this feat is possible.

if you watched the related TED-Ed video (a portion is shown above and on the next page, with animation by Anton Trofimov) it had a method of reaching one solution, but there are more!

((Note: We are just counting the way of allocating the colors, not different ways of placing the colors on a particular cube. For example, if a cube has 33 red sides, 22 green sides, and 11 purple side, there may be multiple ways of placing those colors on a particular cube, but we're not including that in our count.))

Color Cube Assembly

                     

When the 2727 cubes are assembled, one of the cubes fills the center, leaving 2626 cubes visible on the outside. How many "corner" cubes are there ((with 33 sides visible),), how many are "edge" cubes are there ((with 22 sides visible)) and how many "center" cubes are there ((with 11 side visible)?)?

Color Cube Assembly

                     

Remember, our goal is to be able to reassemble the 2727 cubes in three different ways so in each way only one color is visible.

Given we have 27×6=16227 \times 6 = 162 small cube faces available to paint, how many faces can we leave unpainted while still satisfying the goal of the problem?

Color Cube Assembly

                     

Each cube has 66 sides, so we can think of them as adding numbers that add to 6 where each of the numbers represents a distinct color. For example, 2+2+2 2 + 2 + 2 is possible and represents 22 red sides, 22 green sides, and 22 purple sides.

What other cube types work as part of the solution? (Again, the only colors being used are red, green, and purple.)

Color Cube Assembly

                     

Select one or more

We've established from the previous question the cube types are 3+2+1, 3 + 2 + 1 , 2+2+2, 2 + 2 + 2 , and 3+3. 3 + 3 . (To make it consistent with the other cube types, we will write 3+3+0 3 + 3 + 0 past this point.)

Let's now count how many we'll have of each type, starting with the 3+2+1. 3 + 2 + 1 . Remember this is a cube with 3 side of one color, 2 side of a second color, and 1 side of the third color.

How many 3+2+1 3 + 2 + 1 cubes are there?

Hint: Consider how many "center" cubes you will need of each color.

Color Cube Assembly

                     

Continuing from the previous question, how many 3+3+0 3 + 3 + 0 cubes do we have?

Note they indicate two distinct colors, but the colors can differ; one of them can be red and green while another is green and purple, for instance.

Color Cube Assembly

                     

To summarize, we have 1818 of the 3+2+1 3 + 2 + 1 cubes, and 3 of the 3+3+0 3 + 3 + 0 cubes. How many 2+2+2 2 + 2 + 2 cubes do we have?

Color Cube Assembly

                     

Note of the 2+2+2 2 + 2 + 2 cubes, their color choices are already pre-set since we require all 33 distinct colors and each one will make a 22-face "edge" piece.

There are choices for the other two types of cubes; for example, with 3+2+1, 3 + 2 + 1 , it can vary which color is a corner (matching with "33"), which is an edge (matching with "22"), and which is a center (matching with "11").

We'll start with the 3+3+0, 3 + 3 + 0 , where we must choose two distinct colors for each of our 33 cubes of this type.

Color Cube Assembly

                     

Note that on any particular color, the six 3+2+1 3 + 2 + 1 cubes are used in 6 of the corners, leaving 22 more to be filled by the 3+3+0 3 + 3 + 0 cubes.

Given this fact, which of these is a possible configuration for the 3+3+03+3+0 cubes?

(Interpret the pairs as each giving the two colors on each of the three cubes. For example, "red/green, red/green, red/green" means that all three cubes use just the colors red and green.)

Color Cube Assembly

                     

We've already noted that the six 2+2+2 2 + 2 + 2 cubes have fixed color configurations. The three 3+3+0 3 + 3 + 0 cubes have a fixed configuration as well ((the answer in the previous question is the only way to have 22 of each color).).

So the only flexibility in solving the problem is setting the eighteen 3+2+1 3 + 2 + 1 cubes. Here is a list of every possible way to allocate the colors:

(The letters A through F are added for convenience in referring to a particular cube.)

Suppose there are 22 cubes of type A (the first row). How many cubes of type D are there?

Color Cube Assembly

                     

Our method of finding the problem in the previous question “22” to fill in for cube A, but we could have filled anything from 00 to 6.6. Since the logic after is forced, and the colors choices of the 2+2+2 2 + 2 + 2 and 3+3+0 3 + 3 + 0 cubes are forced, that means there are seven ways to solve the problem, each corresponding to one of the digits from 00 to 66.

If you liked the problem and want to try to extend it, here are two ways:

  1. For this set of questions we just counting the way of allocating the colors, not different ways of placing the colors on a particular cube. If we were to include the physical placement on each cube in counting the number of solutions, would anything change, and if so, what?

  2. What would happen if we wanted to make a 44 by 44 by 44 cube instead, with 6464 small cubes total? How many colors would we be able to make? Would every side still get used in the same way as the 33 by 33 by 33 case?

Color Cube Assembly

                     
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