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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.   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

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 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.  Conclusion 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.   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. 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

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