3 labels
2 labels
1 label
No labels
6x6x6
5x5x5
Results
Formula
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 12, 24, 36, 48…
U3 = 12 = 12x3-24
U4 = 24 = 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
Conclusion
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 6x6x6 cube and seeing if I could find a pattern to write down a formula for the pattern. I now can find out how many small cubes – which make the large cube – have 1 label, 2 labels, 3 labels or no labels on any of its sides for a 17x17x17 and/or 69x69x69 large cube.
In my extension I will use the method I used for the cube on the cuboid and see if I can find a pattern to find a formula for finding how many small cubes – which make the large cuboid – have 1 label, 2 labels, 3 labels, or no labels on any of its sides.
5x3x3
4x3x3
Key
3 labels
2 labels
1 label
No labels
7x3x3
6x3x3
Results
Formula
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
By
Derrick Gachiri 9c
