Problem 21 · 1995 AJHSME
Hard
Geometry & Measurement
spatial-reasoning
A plastic snap-together cube has a protruding snap on one side and receptacle holes on the other five sides. What is the smallest number of these cubes that can be snapped together so that only receptacle holes are showing?
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Answer: B — 4.
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Hint 1 of 2
The real requirement: every snap must DISAPPEAR — it has to plug into some other cube's hole. So count the snaps that need hiding, and ask how to hide them all.
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Hint 2 of 2
Picture each snap as an arrow that must point into a neighbor. Try to make the arrows form a closed loop so each cube's snap feeds the next cube.
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Approach: every snap must be absorbed — chain them into a closed loop
- Reframe the goal: 'only holes showing' means every snap is buried inside another cube's hole. With n cubes there are n snaps, and each needs its own neighbor's hole to vanish into.
- Why not fewer? Each cube's snap must hide in some OTHER cube, so the snaps form a chain 'cube → cube → …' with no loose end — meaning the chain must close into a loop. With cubes you can only turn a 90° corner per step, so the smallest loop that actually returns to its start is a square ring of 4 (a straight line or an L always leaves an end snap exposed).
- Four cubes in a square, each snap pointing into the neighbor ahead, hide all four snaps; only holes face outward. So the answer is 4.
- Why this transfers: 'each thing must point into another, with none left over' is a closed-loop (cycle) idea — the minimum is the smallest loop that has no loose end. You meet it again in handshake, gift-exchange, and arrow-chasing puzzles.
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