Patterns in Cube Problems

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geekily

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Jan 24, 2007
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8 small cubes are put together to form one large cube. Suppose that all 6 sides of this larger cube are painted, the paint is allowed to dry, and the cube is taken apart. My math teacher suggested that we all buy boxes of sugar cubes to help us with these kinds of problems, so I did. Built the 8-cube one, no problem:

a. How many of the small cubes will have paint on just one side?: 0
on two sides?:0
on three sides?: 8
on no sides?: 0

b. Answer the questions assuming the large cube is formed by 27 small cubes. I built the 3x3 cube and counted it out, again, no problem:
How many of the small cubes will have paint on just one side?: 6
on two sides?:12
on three sides?: 8
on no sides?: 1

c. Same questions, with 64 cubes. Took forever, but I actually did build it:
How many of the small cubes will have paint on just one side?: 24
on two sides?: 24
on three sides?: 8
on no sides?: 8

d. Here's where the trouble comes in: Answer assuming the large cube is formed with n^3 small cubes. I started to look for patterns, and found that the 3-sides is always 8 small cubes, and that 0 sides is (n-2)^3. However, even after making a nice little chart, I still can't see patterns for the 1 and 2 sides. I even tried building a 125-cube one, but I ran out of cubes - I got most of it, though, so my answers to that were 1 side: 36 / 2 sides: 54 / 3 sides: 8 / 0 sides: 27. Even then, though, I still couldn't see a pattern. It's not a huge deal, so I normally wouldn't ask, but I came across another problem on my homework with another pattern I couldn't find, and I thought they might have something to do with each other:

30 unit cubes are stacked in square layers to form a tower. The bottom is 4 cubes x 4 cubes, next layer is 3 cubes x 3 cubes, next is 2 x 2, top is a single cube. a. Determine the total surface area of the tower. Now, I got 72 u^2, and I think that's right. This is what I did:
area of bottom: 2(4x4) + 4(1x4) = 48, then - (3x3) for the part that's covered = 39
area of 2nd layer: (3x3) + 4(1x3) = 21 - (2x2) = 17
a of 3rd layer: (2x2) + 4(1x2) = 12 - (1x1) = 11
a of top: 5(1x1) = 5
39 + 17 + 11 + 5 = 72

Then it asks the same but with the bottom layer as 8 cubes x8 cubes and going down by 1, and then with 20 x 20. Short of writing out every single one of those, I have no idea how to get the answer, and it seems like there should be some pattern that I'm just not seeing no matter how hard I look at my answers.

I'm sorry that this turned into such a long post! Could someone please give me a hint as to what the patterns might be? I'd really appreciate it.

As always, thank you very much.
 
geekily said:
8 small cubes are put together to form one large cube. Suppose that all 6 sides of this larger cube are painted, the paint is allowed to dry, and the cube is taken apart. My math teacher suggested that we all buy boxes of sugar cubes to help us with these kinds of problems, so I did. Built the 8-cube one, no problem:

a. How many of the small cubes will have paint on just one side?: 0
on two sides?:0
on three sides?: 8
on no sides?: 0

b. Answer the questions assuming the large cube is formed by 27 small cubes. I built the 3x3 cube and counted it out, again, no problem:
How many of the small cubes will have paint on just one side?: 6
on two sides?:12
on three sides?: 8
on no sides?: 1

c. Same questions, with 64 cubes. Took forever, but I actually did build it:
How many of the small cubes will have paint on just one side?: 24
on two sides?: 24
on three sides?: 8
on no sides?: 8

d. Here's where the trouble comes in: Answer assuming the large cube is formed with n^3 small cubes. I started to look for patterns, and found that the 3-sides is always 8 small cubes, and that 0 sides is (n-2)^3. However, even after making a nice little chart, I still can't see patterns for the 1 and 2 sides. I even tried building a 125-cube one, but I ran out of cubes - I got most of it, though, so my answers to that were 1 side: 36 / 2 sides: 54 / 3 sides: 8 / 0 sides: 27. Even then, though, I still couldn't see a pattern. It's not a huge deal, so I normally wouldn't ask, but I came across another problem on my homework with another pattern I couldn't find, and I thought they might have something to do with each other:

30 unit cubes are stacked in square layers to form a tower. The bottom is 4 cubes x 4 cubes, next layer is 3 cubes x 3 cubes, next is 2 x 2, top is a single cube. a. Determine the total surface area of the tower. Now, I got 72 u^2, and I think that's right. This is what I did:
area of bottom: 2(4x4) + 4(1x4) = 48, then - (3x3) for the part that's covered = 39
area of 2nd layer: (3x3) + 4(1x3) = 21 - (2x2) = 17
a of 3rd layer: (2x2) + 4(1x2) = 12 - (1x1) = 11
a of top: 5(1x1) = 5
39 + 17 + 11 + 5 = 72

Then it asks the same but with the bottom layer as 8 cubes x8 cubes and going down by 1, and then with 20 x 20. Short of writing out every single one of those, I have no idea how to get the answer, and it seems like there should be some pattern that I'm just not seeing no matter how hard I look at my answers.

I'm sorry that this turned into such a long post! Could someone please give me a hint as to what the patterns might be? I'd really appreciate it.

As always, thank you very much.
Click to expand...

The corner cubes are painted on three sides - so that number is always 8 (except for n=1)

The cubes that are on the edges will be painted on two sides (except for the 8 corners) - there are 12 edges with 'n' cubes [but count only (n-2) - subtracting the corners] and the number of cubes painted on two sides now are 12*(n-2) (except for n= 1 & 2)

The that have paints on side only would be the cubes on the faces (except for the edges and the corners. There are 6 faces - so # of cubes painted on one side would be 6*(n-2)^2 (except for n= ??)

Now find the # of interior cubes (no paint).
 
geekily said:
8 small cubes are put together to form one large cube. Suppose that all 6 sides of this larger cube are painted, the paint is allowed to dry, and the cube is taken apart. My math teacher suggested that we all buy boxes of sugar cubes to help us with these kinds of problems, so I did. Built the 8-cube one, no problem:

a. How many of the small cubes will have paint on just one side?: 0
on two sides?:0
on three sides?: 8
on no sides?: 0

b. Answer the questions assuming the large cube is formed by 27 small cubes. I built the 3x3 cube and counted it out, again, no problem:
How many of the small cubes will have paint on just one side?: 6
on two sides?:12
on three sides?: 8
on no sides?: 1

c. Same questions, with 64 cubes. Took forever, but I actually did build it:
How many of the small cubes will have paint on just one side?: 24
on two sides?: 24
on three sides?: 8
on no sides?: 8

d. Here's where the trouble comes in: Answer assuming the large cube is formed with n^3 small cubes. I started to look for patterns, and found that the 3-sides is always 8 small cubes, and that 0 sides is (n-2)^3. However, even after making a nice little chart, I still can't see patterns for the 1 and 2 sides. I even tried building a 125-cube one, but I ran out of cubes - I got most of it, though, so my answers to that were 1 side: 36 / 2 sides: 54 / 3 sides: 8 / 0 sides: 27. Even then, though, I still couldn't see a pattern. It's not a huge deal, so I normally wouldn't ask, but I came across another problem on my homework with another pattern I couldn't find, and I thought they might have something to do with each other:

30 unit cubes are stacked in square layers to form a tower. The bottom is 4 cubes x 4 cubes, next layer is 3 cubes x 3 cubes, next is 2 x 2, top is a single cube. a. Determine the total surface area of the tower. Now, I got 72 u^2, and I think that's right. This is what I did:
area of bottom: 2(4x4) + 4(1x4) = 48, then - (3x3) for the part that's covered = 39
area of 2nd layer: (3x3) + 4(1x3) = 21 - (2x2) = 17
a of 3rd layer: (2x2) + 4(1x2) = 12 - (1x1) = 11
a of top: 5(1x1) = 5
39 + 17 + 11 + 5 = 72

Then it asks the same but with the bottom layer as 8 cubes x8 cubes and going down by 1, and then with 20 x 20. Short of writing out every single one of those, I have no idea how to get the answer, and it seems like there should be some pattern that I'm just not seeing no matter how hard I look at my answers.

I'm sorry that this turned into such a long post! Could someone please give me a hint as to what the patterns might be? I'd really appreciate it.

As always, thank you very much.

The best way to solve any problem of this sort is to examine more than one case and determine a pattern or relationship between the givens and the results. For your problem, if you make a table of the numbers of painted surfaces for different size cubes, by inspection, you will be able to define a set of expressions that will give you the appropriate numbers for any size cube. Lets see what we get. PS stands for Painted Surfaces.
Cube size No. of cubes 4 PS 3 PS 2 PS 1 PS 0 PS
2x2 8 0 8 0 0 0
3x3 27 0 8 12 6 1
4x4 64 0 8 24 24 8
5x5 125 0 8 36 54 27
6x6 216 0 8 48 96 64

A little study will quickly lead you to the general expressions for an "n" sided cube. There can be no more than 8 cubes painted on three surfaces, namely the 8 corners of the basic large cube. The number of painted surfaces within a cube n x n x n can now be determined from the following.

3 faces N(3) = 8
2 faces N(2) = 12(n - 2)
1 face N(1) = 6(n - 2)^2
0 face N(0) = (n - 2)^3
Click to expand...
 
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