Problem 21 · 1998 AJHSME
Hard
Geometry & Measurement
complementary-counting
A 4 × 4 × 4 cubical box contains 64 identical small cubes that exactly fill the box. How many of these small cubes touch a side or the bottom of the box?
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Answer: B — 52 cubes.
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Hint 1 of 2
Counting the cubes that touch a wall directly is a corner/edge nightmare (you'll double-count). Flip it: count the cubes that touch NOTHING — no wall, no floor — and subtract from 64.
Still stuck? Show hint 2 →
Hint 2 of 2
Read which surfaces matter: 4 side walls and the bottom, but NOT the top (the box is open up there). A 'safe' cube must dodge all four walls and the floor — but it's free to touch the top.
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Approach: complementary counting — subtract the cubes that touch nothing
- Find the cubes touching no side and not the bottom. To miss all four walls, a cube must sit in the inner 2 × 2 of the 4 × 4 footprint (strip one cube off each of the four edges). To miss the bottom, it must be above the floor layer — but it CAN touch the top, since only sides and bottom count. That's the top 3 layers: 2 × 2 × 3 = 12 untouched cubes.
- Everything else touches a side or the bottom: 64 − 12 = 52.
- Why this transfers: 'touches at least one of several surfaces' is far easier counted backwards — count the few that touch NONE, then subtract. The untouched region is a clean little box (here 2×2×3), so it's quick to size up.
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