Problem 18 · 2013 AMC 8
Medium
Geometry & Measurement
volume-differenceshell-counting
Isabella uses one-foot cubical blocks to build a rectangular fort that is 12 feet long, 10 feet wide, and 5 feet high. The floor and the four walls are all one foot thick. How many blocks does the fort contain?
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Answer: B — 280 blocks.
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Hint 1 of 2
Counting the floor and four walls block-by-block is a nightmare of overlaps. Instead, imagine the fort as a solid box, then carve out the empty room inside — subtraction beats addition here.
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Hint 2 of 2
"Solid total minus the hollow" is the go-to for any shell or frame. The only care needed is how much the hollow shrinks: a 1-ft wall on each side removes 2 from a dimension; a floor with no ceiling removes only 1 from the height.
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Approach: solid outer box minus the hollow interior
- Pretend the fort is solid: 12 × 10 × 5 = 600 cubes.
- Now find the empty room. Walls on both sides shave 2 off length (12 → 10) and 2 off width (10 → 8); the floor shaves 1 off height but there's no ceiling (5 → 4). Hollow = 10 × 8 × 4 = 320.
- Blocks actually used = 600 − 320 = 280.
- Watch the trap: the open top is the whole subtlety — height loses only 1 (floor), not 2. Treating it like a closed box would give the wrong hollow.
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