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Exposed cube sides

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In this investigation I will be finding out how many labels are on an exposed side of a 3x3x3 cube when 27 small cubes are put together to make the large 3x3x3 cube.


I will first start to count the number of small cubes out of the 27; have no labels, 1 label, 2 labels and 3 labels. Once I have done this I will the do the same for a 4x4x4 cube, 5x5x5 cube and finally a 6x6x6 cube and see if I can find a pattern. When I have found a pattern I

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= 12x4-24

U5 = 36 = 12x5-24

Un = 12(n-2)

The pattern for 1 label is 6, 24, 54, 96

U3= 6 = 6x1 = 6x1² = 6(3-2) ²

U4= 24 = 6x4 = 6x2² = 6(4-2) ²

U5= 54 = 6x9 = 6x3² = 6(5-2) ²

Un = 6(n-2)²

The pattern for no labels is 1, 8, 27, 64

U3= 1 = 1³= (3-2) ³

U4= 8 = 2³ = (4-2) ³

U5= 27 =3³= (5-2) ³

Un= (n-2) ³

3 labels= 0n+8

2 labels= 12(n-2)

1 label= 6(n-2) ²

No labels= (n-2) ³


In this investigation I have found a formula for finding how many labels will be on the exposed side of any cube using my successful method of counting the number of labels on an exposed side of a 3x3x3, 4x4x4, 5x5x5 and finally 6

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2 labels





1 labels





No labels






The pattern for 3 labels is 8, 8,8

U3 = 8 = 0x3+8          

U4 = 8 = 0x4+8

U5 = 8 = 0x5+8

Un= 0n+8

The pattern for 2 labels is 16, 20, 24, 28

U4 = 16 = 4x4

U5 = 20 = 4x5

U6 = 24 = 4x6

Un = 4n

The pattern for 1 label is 10, 14, 18, 22

U4= 10 = 4x4-6

U5= 14 = 4x5-6

U6= 18 = 4x6-6

Un = 4n-6

The pattern for no labels is 1, 2, 3, 4

U4= 1 = 1x4-3

U5= 2 = 1x5-3

U6= 3 =1x6-3

Un= n-3

3 labels= 0n+8

2 labels= 4n

1 label= 4n-6

No labels= n-3


Derrick Gachiri 9c

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