Painting Cubes

Cubes

The challenge is to figure out the most efficient way to paint the cubes for the puzzles.

Puzzles ‘R’ Us is a small company that makes three-dimensional puzzles and sells them over the Internet. One of its most popular puzzles is a cube builder. The cube builder consists of a set of small red and white cubes that have to be assembled into a larger cube. Some of the cubes are painted red on just one face, some on two adjacent faces, some on three adjacent faces. Some cubes are all white.

The puzzle is to assemble all the small cubes into a larger cube, called a 4-Cube, so that the outside of the large cube is all red and all the small cubes that don’t touch an edge are all white. The 4-Cube gets its name because it is four small cubes long in each direction.

Problem 1

Your job is to help Puzzles ‘R’ Us streamline its process of making puzzles by telling its employees exactly how many cubes to paint in different ways to make a 4-Cube puzzle. Answer the following questions:

How many small cubes will be needed to construct a 4-Cube puzzle?
How many small cubes should be painted red on three adjacent sides?
How many small cubes should be painted red on two adjacent sides?
How many small cubes should be painted red on just one side?
How many small cubes should be painted all white?

Problem 2

Puzzles ‘R’ Us wants to expand its line of cube puzzles to larger and smaller cubes--all made from the same basic pieces. Your job is to help the company by answering the same five questions for an N-Cube; that is, a cube with N small cubes along each edge.

Hints:

Start by thinking about the basic properties of a cube: How many faces (surfaces) does it have? How many vertices (corners)? How many edges? These properties are the same for any cube, regardless of its size.
Try to build your knowledge by starting to answer the five questions for a 2-Cube, and a 3-Cube. If you can do those, a 4-Cube should be fairly easy. Then if you look for patterns in your data, you can answer the questions for an N-Cube.
Keep track of your results in a table like this one:

Name of Cube	Total Number of Cubes Needed
		Number of Cubes Painted with:
		3 red sides	2 red sides	1 red side	all white sides
2-Cube
3-Cube
4-Cube
5-Cube

N-Cube

Course:

Algebra [1]
Geometry [2]

Result/Solution(s)

Solution: Painting Cubes Math Puzzle

There are a number of different ways that students have solved this puzzle. Here are three of them.

Najib’s Solution

I got confused when I first tried to figure out Problem 1 just using paper and pencil. I made some drawings, but I couldn’t figure out what was on the back and inside the 4-Cube. So I went and got a whole bunch of sugar cubes and built a 4-Cube out of them. I glued them together with rubber cement because I wanted to take the 4-Cube apart later. Then I spray painted the top and 4 sides of my 4-Cube. When the paint was dry, I carefully turned the 4-Cube over and painted the bottom. When the bottom paint was dry, I took the whole thing apart and counted all the cubes to see how they were painted.

I found the following results:

I used 64 sugar cubes all together
8 were painted red on 3 sides
24 were painted red on 2 sides
24 were painted red on 1 side
8 were all white (not painted)

I had fun putting the puzzle back together. It wasn’t as easy as I thought, especially the bottom, but I got it.

Then I decided not to work on Problem 2 because I had used up almost all my sugar cubes.

Janifa’s Solution

I decided to follow the hints and think about a 2-Cube and a 3-Cube first. The 2-Cube was easy. Look at the picture. It has 8 small cubes, and all of them are painted on 3 sides. It doesn’t need any other cubes.

The 3-Cube was a little harder. Once I drew a 3-Cube, I could see that it had three layers of 9 cubes each, or 27 cubes total. Then I tried to figure out how many corners it had. Each corner is painted red on 3 sides, and I could see that there are 8 of them, just like the 2-Cube.

Then I put 2 dots on every cube I could see that had 2 sides painted red, and 1 dot on every cube that had just 1 side painted red.

Cube

I could see a 1-dot cube in the middle of 3 visible sides, and I knew there were 3 sides I couldn’t see. So there must be 6 1-dot cubes, or 6 cubes painted on 1 side.

I could see that there were 4 2-dot cubes on the top of the 3-Cube. So there must be 4 more on the bottom. Then I went around the cubes on the middle layer. I could see 3 2-dot cubes, and I knew there was 1 more along the far edge that I couldn’t see. So there are 4 2-dot cubes in the middle layer. That makes 12 2-dot cubes altogether, or 12 painted on 2 sides.

So far I have

8 cubes painted on 3 sides
6 cubes painted on 1 side
12 cubes painted on 2 sides

That makes 26 cubes all together, so there must be 1 more cube in the middle that is all white.

I used the same reasoning and method for a 4-Cube. It has 4 layers of 16 cubes each, so it must have 4 x 16, or 64 cubes total. Just like before, I could see there are exactly 8 corners with 3 sides painted red. The number of corners doesn’t change. When I drew my 4-Cube, I put dots on the top side to see if I could figure everything out that way. I put 2 dots for cubes with 2 red sides and 1 dot for cubes with 1 red side.

Cube

There are 4 1-dot cubes on the top side of the 4-Cube. Since there are 6 sides all together, there must be 6 x 4, or 24, cubes painted on 1 side.

There are 8 2-dot cubes on the top layer. There must be 8 more on the bottom layer. Then looked at the 2 middle layers. I decided to put dots on them to be sure.

Cube

Now I could see 6 cubes with 2 dots, and I know there is one more on the hidden edge. So there are 8 2-dot cubes in the middle 2 layers. Adding in the 8 from the top and 8 from the bottom, there are 24 cubes painted red on two sides.

So far I have

8 cubes painted red on 3 sides
24 cubes painted red on 2 sides
24 cubes painted red on 1 side

That makes 56 cubes. So there must be 8 all-white cubes (a little 2-Cube) in the middle to make 64 cubes all together.

This is the answer to Problem 1.

For Problem 2, I put my data into a table. I used the patterns in the table and what I’ve already done to figure out the values for a 5-Cube and an N-Cube.

Name of Cube	Total Number of Cubes Needed
		Number of Cubes Painted with:
		3 red sides	2 red sides	1 red side	all white sides
2-Cube	8	8	0	0	0
3-Cube	27	8	12 = (12 x 1)	6 = (6 x 1)	1
4-Cube	64	8	24 = (12 x 2)	24 = (6 x 2 x 2)	8
5-Cube	125	8	36 = (12 x 3)	54 (6 x 3 x 3)	27

N-Cube	N x N x N	8	12 x (N - 2)	6 x (N - 2) x (N - 2)	(N - 2) x (N - 2) x (N - 2)

The last line is the answer to Problem 2.

Ming’s Solution

I used algebra to solve Problem 2 first. Then once I knew the formulas for an N-Cube, I just plugged in numbers for a 4-Cube to solve Problem 1.

I started by exploring the properties of a cube. I observed that every cube has 6 sides, 8 corners, and 12 edges. This allowed me to solve the problem using algebra.

Cube

First, to find the total number of small cubes needed for an N-Cube, I multiplied N by itself 3 times N x N x N, or N³.

Next I imagined what the corners would be painted. Each corner cube has to be painted red on three sides. Since there are 8 corners, every N-Cube has exactly 8 small cubes painted red on 3 sides.

Cube

Next I sketched a cube with 1 edge filled in with cubes painted red on 2 sides only. Not including the corners, each edge on an N-Cube has N - 2 cubes painted red on two sides. Since every cube has 12 edges, the number of cubes painted red on 2 sides is 12 x (N - 2).

Cube

Next I needed to consider the faces of the cube, which are painted red on 1 side only. Each face is a square made up of (N - 2) x (N - 2) cubes. Since there are 6 faces, the total number of cubes painted on one side only is 6 x (N - 2)².

Cube

Finally, inside the large cube, there is a smaller cube, an N-2-Cube, made up of all white cubes. This requires (N - 2)³ small white cubes.

Cube

I put my results into a table to make it easier to calculate. By setting N = 4, I can find the results for a 4-Cube, to solve Problem 1.

Name of Cube	Total Number of Cubes Needed
		Number of Cubes Painted with:
		3 red sides	2 red sides	1 red side	all white sides
N-Cube	N³	8	12 x (N - 2)	6 x (N - 2)²	(N - 2)³
4-Cube	64	8	24 = (12 x 2)	24 = (6 x 4)	8 = 2³

I always like to check my work. One way to do this is to add up all four kinds of cubes and check that they equal the total.

For the 4 cube I get

64 = 8 + 24 + 24 + 8 = 64v

For the N-cube I get

N3 = 8 + 12(N - 2) + 6(N - 2)² + (N - 2)³

When I expand the right hand side, I get

8 + 12N – 24 + 6N² – 24N + 24 + N³– 6N² + 12N – 8

Then collecting like terms I get

[8 - 24 + 24 – 8] + [12N – 24N + 12N] + [6N² – 6N2 ] + N³ = N³v

So N³ = N³ and my formulas check!