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Painted Cube Investigation

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Introduction

Investigation Rules

In this investigation I had to investigate the number of faces on a cube, which had had its outer surface painted red. I had to answer the question, 'How many of the small cubes will have no red faces, one red face, two red faces, and three faces?' From this, I hope to find a formula to work out the number of different faces on a cube sized 'n x n x n'.

The Rules and Solving the Problem

To solve this problem, I built different sized cubes (2 x 2 x 2, 3 x 3 x 3, 4 x 4 x 4) and made tables to help find the patterns.image07.png

image08.pngimage10.png

Length of cube

No. of small cubes

No. of small cubes with  painted faces

3

2

1

0

2

8

8

0

0

0

3

27

8

12

6

1

4

64

8

24

24

8

No. of Painted Faces

1 x 1 x 1

2 x 2 x 2

3 x 3 x 3

4 x 4 x 4

n x n x n

0

0

0

1

8

(n – 2)3

1

0

0

6

24

6 (n – 2)2

2

0

0

12

24

12 (n – 2)

3

0

8

8

8

8 unless 1 x 1 x 1

4

0

0

0

0

0

5

0

0

0

0

0

6

1

0

0

0

0 unless 1 x 1 x 1

...read more.

Middle

No. of painted faces

Formula (n = 5)

Prediction

Answer

0

(n – 2)3

27

27

1

6 (n – 2)2

54

54

2

12 (n – 2)

36

36

3

8 unless 1 x 1 x 1

8

8

4

0

0

0

5

0

0

0

6

0 unless 1 x 1 x 1

0

0

image11.png

The table proves that the formulas are correct. All my formulae added together should be equal to the total number of small cubes in the large cube which is n3 . Here are the added formulae: n3 = (n – 2)3 + 6 (n – 2)2 + 12 (n – 2) +8

(n - 2)(n - 2)(n - 2) + 6(n - 2)(n - 2) + 12(n - 2) + 8
(n
2 - 4n + 4)(n - 2) + 6(n2 - 4n + 4) + 12n - 24 + 8
n
3 - 4n2+ 4n - 2n2 + 8n - 8 + 6n2 - 24n + 24 +12n - 24 + 8
n
3 + 6n2 - 6n2 + 24n - 24n + 32 - 32
n
3

Exceptions

None of the formulae will work for the cube 1 x 1 x 1: This pattern does not work because there is more than one vertex on one cube and therefore there is not going to be 8 cubes with 3 painted sides.image12.png

image08.png

...read more.

Conclusion

image08.png

It is impossible for any cube to have 4 or 5 painted faces and for any cube bigger than a 1 x 1 x 1, it is impossible to have a cube with 6 painted faces.

Extending the Investigation

I extended the investigation to cuboids. I used measurements 2 x 3 x 4, 2 x 3 x 5 and 3 x 4 x 5.

image14.pngimage13.pngimage15.png

Length of cuboids

No. of small cubes

Amount of small cubes with painted faces

3

2

1

0

2 x 3 x 4

24

8

12

4

0

2 x 3 x 5

30

8

16

6

0

3 x 4 x 5

60

8

24

22

6

No. of Painted Faces

nth term

0

(x - 2) x (y - 2) x (z - 2)

1

2 x ((x - 2)(y - 2)+(y - 2)(z -2)+(x - 2)(z - 2))

2

4(x+y+z)- 24

3

8

Exceptions to the rule are cuboids with one or two dimensions of 1.image09.png

Check

To check I used a 3 x 4 x 6 cuboid.

No. of Painted faces

Formula

Prediction

Answer

0

(x - 2) x (y - 2) x (z - 2)

8

8

1

2 x ((x - 2)(y - 2)+(y - 2)(z -2)+(x - 2)(z - 2))

28

28

2

4(x+y+z)- 24

28

28

3

8

8

8

...read more.

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