Cube & Dice
🔒 Log in to trackmedium importance~0-1 Q in Tier 16 formulas⚡ 7 shortcuts3 subtopics
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Dice: finding opposite faces from positions
Standard die
1+6 = 2+5 = 3+4 = 7
Opposite faces of a standard die add to 7.
Hidden faces of a standard die
\text{hidden} = 21 - \text{shown}
All six faces add to 21.
Painted cube cut into small cubes
Two faces painted
12(n-2)
Cubes on the edges, not the corners.
One face painted
6(n-2)^2
Cubes in the middle of each face.
No face painted
(n-2)^3
The hidden inner block.
Total check
8 + 12(n-2) + 6(n-2)^2 + (n-2)^3 = n^3
The four counts add up to all the small cubes.