Competition · AMC preparation · step 4 of 4
AMC 8 · 2013 · #18
Grade 5 geometry-3darithmetic
Pick an answer.
AMC 8 2013 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Counting wall blocks face-by-face is messy because the corner columns belong to two walls at once and the floor blocks meet the walls along the bottom edge — easy to double-count. Tool #16 (Complement) flips the question: instead of "how many blocks are filled?" ask "how many blocks would fill the whole 12 × 10 × 5 box?" and "how many blocks fit in the empty room inside?" Subtract the second from the first. Tool #17 (Visualize Spatial Relationships) is needed to see that the inner room loses 1 ft on each of the two long sides, 1 ft on each of the two short sides, and 1 ft from the bottom only (no ceiling). Tool #7 (Identify Subproblems) then keeps the three computations — outer volume, inner volume, subtraction — neatly separated.
Count the solid fort blocks
If the fort were solid, its block count is just the outer volume: 12 × 10 × 5 = 600.
Filling a rectangular box with unit cubes and multiplying the three side lengths is the Grade 5 volume-as-multiplication idea.
5.MD.C.5Change Focus Count The ComplementMeasure the inner room
From above, the walls trim 1 ft off each end: inner length 12 − 2 = 10 ft, inner width 10 − 2 = 8 ft.
Picturing the floor plan and trimming 1 ft off each wall is spatial reasoning about a unit-cube layout — Grade 5 volume thinking.
5.MD.C.3Visualize Spatial RelationshipsHandle the inner height
No ceiling, so subtract only the floor from the height: inner height 5 − 1 = 4 ft.
Noticing the asymmetry (floor only, no ceiling) keeps you from over-subtracting — this is the careful spatial step in any "hollow box" problem.
5.MD.C.3Visualize Spatial RelationshipsCount the empty inner blocks
The empty inner room is a box too, so the unused blocks = 10 × 8 × 4 = 320.
The inner room is itself a rectangular prism, so the same length × width × height rule applies.
5.MD.C.5Identify SubproblemsSubtract to get the blocks used
Blocks used = solid − empty room = 600 − 320 = 280, which is (B).
Counting the complement (empty cubes) and subtracting is faster and safer than tallying corner-sharing walls one by one.
The number of blocks Isabella actually used equals the count that would fill the whole solid box minus the count that fills the empty room inside.
▸ Why?
The whole solid box splits into exactly two parts with no gaps and no overlaps — the cells she filled with blocks and the empty cells of the inner room — so the used count plus the empty count is the whole box's count.
▸ Why?
Since the used count plus the empty count equals the whole count, taking the empty count away from the whole count leaves exactly the used count behind.
This AMC 8 problem only needs Grade 5 volume = length × width × height — count what the fort would be if it were solid, subtract the empty room inside, and you're done!
- Count the solid fort blocks
- Measure the inner room
- Handle the inner height
- Count the empty inner blocks
- Subtract to get the blocks used
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