TOTAL
3 labels = 8
2 labels = 0
1 label = 0
No labels = 0
1st level – Birds Eye View 2nd level – Birds Eye View
3rd level – Birds Eye View
1st level – Birds Eye View 2nd level – Birds Eye View
3rd level – Birds Eye View 4th level – Birds Eye View
1st level – Birds Eye View 2nd level – Birds Eye View
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3rd level – Birds Eye View 4th level –Birds Eye View
5th level – Birds Eye View
So using my formulas I can predict the answers for 6 by 6 by 6 cubes.
3 labels = n=8
= 8
2 labels = 12(n-2)
= 12 x (6-2)
= 12 x 4
= 48
1 label = 6(n-2)²
= 6 x (6-2)²
= 6 x (4²)
= 6 x 16 = 96
No labels = (n-2)³
= (6-2)³
= 4³
= 64
Total = n³
= 6³
= 216
The formula for 3 labels is
n=8
This is because the 3 labeled boxes are only on the corners and there are only 8 corners in any cube, the answer will always be 8.
The formula for 2 labels is
12(n-2)
The formula is this because 2 labeled boxes are only on the edges but not on the edge/corners, thus giving the ‘-2.’ There are 12 edges in any cube, so you must multiply by 12 to get the amount of 2 labeled boxes on the whole cube.
The formula for 1 label is
6(n-2)²
The 1 labeled boxes are in the middle of the faces. The ‘(n-2)²’ comes from the number of boxes in 1 line on the face of the cube minus 2 because the corners are not counted and then cubed to show how many there are in 1 face. Then you must multiply by 6 to work out the total in all 6 faces of the cube.
The formula for no labels is
n-2³
The no labeled boxes are in the centre of the cube and cannot be spotted by looking at the outside of the box. You have to use minus 2 to show that the corners are not included and then cubed because it is the inside.
After looking carefully at my results I have found formulas for how to work out any amount of labels. I have tested these formulas on a 6 by 6 by 6 square. They worked. The edges, corners and faces a cube affected these formulas. Maybe next time I will try to work why this is.
