AMC 10 · 2004 · #7

Grade 4 geometry-3d
Archetype Arithmetic Expression Decomposition └ Sub-archetype Multi-Segment Path or Count Sum
spatial-visualizationmulti-digit-arithmetic pattern-recognitionidentify-subproblems ↑ Prerequisites: multi-digit-arithmetic
📏 Medium solution 💡 1 insight
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Problem
Oranges are stacked so the bottom layer is a 5-by-8 rectangle. Each orange in a higher layer sits in the pocket made by four oranges below it. The stack ends with a single straight row of oranges. Find the total number of oranges in the whole stack.

Pick an answer.

(A)
96
(B)
98
(C)
100
(D)
101
(E)
134

AMC 10 2004 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Visualize Spatial Relationships

The whole problem hinges on seeing the geometry (Tool #17): an orange resting in a pocket of four sits over the gap where four oranges meet, so it lands between rows and between columns of the layer below. That means each new layer has one fewer orange along each side than the layer under it. Once you see that, the sizes form a clear shrinking pattern (Tool #5): 5×8, then 4×7, then 3×6, and so on. Finally the stack splits into simple subproblems (Tool #7): count each rectangular layer by multiplying its two side lengths, then add the layers. The trap answer is to forget how the stack ends or to miscount the last layer.

1STEP 1

See why each layer is one smaller on each side

Each orange drops into the gap where four below meet, so an m by n layer supports one that is (m-1) by (n-1).

m × n layer → (m-1) × (n-1) layer above
2STEP 2

List the size of every layer

Shrink each side by 1 from the 5-by-8 base: 5×8, 4×7, 3×6, 2×5, 1×4 — five layers, and 1×4 is the finishing row.

5{×}8, 4{×}7, 3{×}6, 2{×}5, 1{×}4
3STEP 3

Count the oranges in each layer

Each layer is a rectangle, so multiply its two sides: 40, 28, 18, 10, 4 oranges from bottom to top.

40, 28, 18, 10, 4
4STEP 4

Add the layers for the total

Add the five counts: 40+28=68, 68+18=86, 86+10=96, 96+4=100 oranges, choice (C).

40+28+18+10+4 = 100 → (C)
Answer
100
The base alone is 5×8=40 oranges, and each higher layer is smaller, so the total must be more than 40 but not by a huge amount. Five shrinking layers landing at 100 fits: it is more than double the base but far below 134, which would require far more or larger layers. It is also just above 96 (the base plus a rough guess), and the careful layer-by-layer sum lands cleanly on 100, so (C) is sensible while (E) is clearly too big.
💡Key takeaway

Each orange sits in the gap between four below it, so every layer is one smaller on each side; multiply each layer's sides and add: 40+28+18+10+4 = 100.

  • See why each layer is one smaller on each side
  • List the size of every layer
  • Count the oranges in each layer
  • Add the layers for the total

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