AMC 10 · 2011 · #2

Grade 4 geometry-2dlogic
Archetype Small-Domain Constrained Search └ Sub-archetype Logic-Deduction / Elimination
spatial-visualizationlogical-deductionif-then-reasoning physical-representationsystematic-enumeration ↑ Prerequisites: spatial-visualization
📏 Medium solution 💡 2 insights 📊 Diagram
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Problem
Overlapping discs are drawn showing only the edge no other disc hides. Find the stacking order.

Pick an answer.

(A)
(C, A, E, D, B)
(B)
(C, A, D, E, B)
(C)
(C, D, E, A, B)
(D)
(C, E, A, D, B)
(E)
(C, E, D, A, B)
How to solve
Strategy Visualize Spatial Relationships

Tool #17 (Visualize Spatial Relationships): the drawing is flat but the question asks about depth, and the only depth information in it is which edge survives at each overlap — so I read every overlap as a statement about which of the two coins is nearer the top. Tool #15 (Organize Information in More Ways): a picture is hard to reason about directly, so I convert it into a plain list of "X is above Y" facts and then merge that list into one order. Tool #3 (Eliminate Possibilities): the easy overlaps between touching neighbours turn out to leave three answer choices alive at once, so I have to find the one further overlap that kills two of them.

1STEP 1

Every pair of coins overlaps

Every pair of discs really does overlap.

two equal coins of radius r overlap ⇔ distance between centres ＜ 2r
2STEP 2

Coin C is drawn whole, so C is on top

One whole outline names the top disc.

arc(C) = 360° → no coin covers C
3STEP 3

Read who covers whom at each overlap

Each broken edge gives one comparison.

C ＞ A, C ＞ E, E ＞ D, D ＞ B, A ＞ B
4STEP 4

Those five facts are not enough

Those comparisons leave three possible orders.

C ＞ E ＞ D ＞ B and C ＞ A ＞ B → three orders survive: (A), (D), (E)
5STEP 5

The A-and-D overlap decides it

One more overlap decides between them.

A's arc stops at 214° = A ∩ D; A ∩ B = {155°, 277°}, A ∩ C = {25°, 263°}, A ∩ E = {250°, 326°}
6STEP 6

Assemble the single order

Assembling gives a single order, choice (E).

C ＞ E ＞ D ＞ A ＞ B → (E) (C, E, D, A, B)
Answer
(C, E, D, A, B)
Run the reasoning backwards: assume the stack is C, E, D, A, B and rebuild the drawing, then compare it with the figure arc by arc. Two of these circles cut each other 61° to either side of the line joining their centres when they are neighbours in the ring, and 38° to either side when they are not. So: C is covered by nothing and stays a full 360° circle. E is covered only by C, hiding 11° through 133°, leaving E drawn from 133° — the figure draws E from 132°. D is covered by C and E, hiding 299° through 74°, leaving D drawn 74° to 299° — the figure draws 75° to 300°. A is covered by C, E and D, leaving it drawn 25° to 214° — the figure agrees exactly. B is covered by all four, leaving it drawn 97° to 227° — the figure draws 96° to 228°. All five arcs match within the figure's rounding to whole degrees, and no other order of the five coins reproduces all five. A second, cruder check: since all ten pairs overlap, the coin lying k-th from the top must be nicked by exactly the k-1 coins above it, and the drawn arcs do shrink steadily down the list — 360°, 238°, 225°, 189°, 132°. Finally, the ten readings contain no loop of the form "X over Y over Z over X", which a drawing could have shown but a real pile of coins never can — so the picture is honest.
💡Key takeaway

The spot where a coin's outline breaks off names the coin lying on top of it — and you have to check the far-apart pairs too, not just the ones sitting side by side.

  • Every pair of coins overlaps
  • Coin C is drawn whole, so C is on top
  • Read who covers whom at each overlap
  • Those five facts are not enough
  • The A-and-D overlap decides it
  • Assemble the single order

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