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# Find 4 formulae that can work out the number of cubes in a cube that has been painted on the outside with 0 faces painted, 1 face painted, 2 faces painted and 3 faces painted in any sized cubes.

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Introduction

Math’s Painted Cube Investigation

Aim: Our aim is to find 4 formulae that can work out the number of cubes in a cube that has been painted on the outside with 0 faces painted, 1 face painted, 2 faces painted and 3 faces painted in any sized cubes.

Method:

1. We drew out all the different sizes of cubes up to 5x5 like the following:

1x1 cube

2x2 Cube

3x3 Cube

4x4 Cube

5x5 Cube

2. Then counted how many with 1 face painted and imagine the ones that were not shown.

3. The counted how many 2 faces painted and imagine the ones which are not shown.

3.The counted the faces painted and imagined the ones that are not shown.

4. Next we recorded our results in a results table.

4.

Middle

2 Faces Painted

3 Faces Painted

1x1

0

0

0

0

2x2

0

0

0

8

3x3

1

6

12

8

4x4

8

24

24

8

5x5

27

54

36

8

Next we tried to see if there was a pattern like square numbers, cube numbers, triangular numbers, odd numbers or even numbers.

We found out the following:

0 Face: 0,0,1,8,27 (Tasneem with the help of the revision guide found out that these numbers were cube numbers).

1 Face:0, 0,6,24,54 (Asma found out these numbers were from the 6 times table).

2 Face: 0,0,12,24,36 (Alliyah and Aysha.

Conclusion

Before we found out the 3 face painted column was always 8 and found out there is no formula n is always = to 8 so n=8. This is because there are always 4 corners in any sized cube.

## Final Formulae

0 Face Painted: (n-2)3

1 Face Painted: 6n2-24n+24

2 Face Painted: 12n-24

3 Face Painted: Always 8 n=8

## Conclusion

We were able to find all the formulas. We found out that the 0 face painted column were

all cubes numbers The 1 face painted column are in the 6 times table. The 2 faces painted column are in the 12 times table. Also we found out the 3 faces painted never changes.

## Evaluation

To make this investigation you should test more different sized cubes.

This student written piece of work is one of many that can be found in our GCSE Hidden Faces and Cubes section.

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