I will be investigating different patterns that can be found in cubes that are constructed from smaller 1cm cubes and are painted on the outside.

Cube Length

No. Of small cubes

No. Of small cubes with 3 painted faces (Y)

2*2*2

8

8

3*3*3

27

8

4*4*4

64

8

5*5*5

25

8

6*6*6

216

8

There fore the formula for this is simply Y=8

The next table will show the formula to find cubes with two sides painted and I how I found it.

Cube Number (X)

Cube dimensions

No. Of small cubes

No. Of small cubes with 2 painted faces (Y)

2

2*2*2

8

0

3

3*3*3

27

2

4

4*4*4

64

24

5

5*5*5

25

36

6

6*6*6

216

48

7

7*7*7

343

60

8

8*8*8

512

72

I first noticed that the number of small cubes (the column in red) was the twelve times table as each number went up in twelve's so I knew I had to base my formula on this, then noticed that it would make sense to minus two from the cube number as for the first two cubes there aren't any cubes with two sides painted. As seen below in blue I get a matching pattern that forms the twelve times table. Simply by knowing the dimensions of a cube I can work out how many small cubes there would be with two sides painted in any cube e.g. the following:

3 -2= 1*12=12

4 -2= 2*12=24

5 -2= 3*12=36

6 -2= 4*12=48

7 -2= 5*12=60

8 -2= 6*12=72

Thus finding and proving the formula Y=(X-2)12

This table shows the number of cubes with only one side painted.

Cube Number (X)

Cube dimensions

No. Of small cubes

No. Of small cubes with 1 painted faces

2

2*2*2

8

0

3

3*3*3

27

6

4

4*4*4

64

24

5

5*5*5

25

54

6

6*6*6

216

96

Again the first thing I noticed was that the number of cubes with one side painted were all multiples of 6.Then I also noticed that the cubes with one side painted increased in squares, so I thought that the formula would also include or have the need to include a squaring. Yet again there were no cubes with one side painted for the first two cubes, so like the previous formula I knew I probably had to include a minus of two in the formula, this was the result.

Y=(X-2)2*6

(2-2)2*6=0

(3-2)2*6=6

(4-2)2*6=24

(5-2)2*6=54

(6-2)2*6=96

This was the proof of the formula, corresponding to the answers found in the above table.

The next table shows the number of cubes with no painted sides.

Cube Number (X)

Cube dimensions

No. Of small cubes

No. Of small cubes with 0 painted faces

2

2*2*2

8

0

3

3*3*3

27

4

4*4*4

64

8

5

5*5*5

25

27

6

6*6*6

216

64

This formula was perhaps the easiest to puzzle out, because of all the clues I have learnt to spot in previous formulas. The first thing I noticed about this is that the number of small cubes with no sides painted mirrored that of the number of total small cubes starting at eight two rows down in red, this led me to believe that a minus of two was needed in the formula. I then noticed that the number of small cubes were obviously all made up of cubed numbers so the formula should also involve this, this is what I came up with:

Y= 3(x-2)

3(2-2)=0

3(3-2)=1

3(4-2)=8

3(5-2)=27

3(6-2)=64

This was the proof of this formula, corresponding to the above tables results.