AMC 10 · 2008 · #11
Grade 6 geometry-3dlogic
Pick an answer.
Chasing 13 visible numbers is messy; chasing the 5 hidden ones is short, so flip to the complement and subtract from the fixed grand total. But the complement only helps once we know what the hidden faces are allowed to be, and that is a folding question: on a stacked cube the covered top and bottom are opposite faces, so we must fold the net to learn which numbers sit opposite which. That single fact is what stops us from simply hiding the smallest numbers. After the folding gives a lower bound on the hidden sum, we still have to build an actual stack that reaches it, since a bound nobody can hit is not an answer.
Locate the five hidden faces
Exactly five faces end up hidden, in a two-two-one pattern.
A cube in the middle of a tower is blocked above and below, while the cube on top is blocked only underneath.
K.G.A.1Visualize Spatial RelationshipsFold the net to find the opposite pairs
Folding the net fixes which numbers sit opposite.
Walking around a cube you return to the start after four faces, so squares two apart in a strip land back-to-back.
Walking around a cube returns to the start after four faces, so squares two apart in a strip land back to back.
▸ Why?
The four side faces form a closed ring, so stepping four times brings you back where you began.
▸ Why?
Each face has exactly one face directly opposite it, so the pairing is complete with none left over.
Rule out hiding the two smallest
The pairings rule out hiding the two smallest numbers together.
Hiding a face forces its partner into hiding too, so cheap numbers with expensive partners are not cheap to hide.
6.G.A.4Eliminate PossibilitiesSwap the question for its complement
Maximizing the visible is minimizing the hidden.
The pile of numbers is fixed, so pushing the seen part up and the covered part down are the same move.
4.OA.A.3Change Focus Count The ComplementPush the hidden total to its floor
The hidden total cannot drop below 25.
Three separate choices with no strings between them let you take the best option in each one at the same time.
4.NBT.A.2Extreme PrincipleBuild a stack that reaches the floor
A real stack reaches that floor, giving 164, choice (E).
Once you can point at a real tower that hides only 25, the lower bound stops being a guess and becomes the answer.
4.NBT.B.4Create A Physical RepresentationCovering the top and bottom of a cube always hides two opposite faces, so you can only hide a number together with the one across from it.
- Locate the five hidden faces
- Fold the net to find the opposite pairs
- Rule out hiding the two smallest
- Swap the question for its complement
- Push the hidden total to its floor
- Build a stack that reaches the floor
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