AMC 10 · 2011 · #2
Easy mode Grade 4Five round coins lie flat on a table, overlapping each other. They are labeled A through E in the picture. Each coin's edge is drawn only where no other coin covers it. Put the five coins in order, from the coin on top of the pile down to the coin at the bottom.
Pick an answer.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: Five identical round coins lie flat on a table, overlapping one another. The picture draws each coin's outline only along the part of its edge that no other coin hides. From those broken outlines, work out the stacking order of the five coins from top to bottom.
Givens: Five congruent circular coins labelled $A$, $B$, $C$, $D$, $E$, with their centres at the five corners of a regular pentagon; Each coin is wide enough to reach past its neighbours, so the coins overlap; Each coin's outline is drawn only where no coin above it covers that spot; Coin $C$'s outline is a complete unbroken circle; the outlines of $A$, $B$, $D$, $E$ are broken arcs; Answer choices: (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) $(C, E, D, A, B)$
Unknowns: The order of the five coins from top to bottom
Understand
Restated: Five identical round coins lie flat on a table, overlapping one another. The picture draws each coin's outline only along the part of its edge that no other coin hides. From those broken outlines, work out the stacking order of the five coins from top to bottom.
Givens: Five congruent circular coins labelled $A$, $B$, $C$, $D$, $E$, with their centres at the five corners of a regular pentagon; Each coin is wide enough to reach past its neighbours, so the coins overlap; Each coin's outline is drawn only where no coin above it covers that spot; Coin $C$'s outline is a complete unbroken circle; the outlines of $A$, $B$, $D$, $E$ are broken arcs; Answer choices: (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) $(C, E, D, A, B)$
Plan
Primary tool: #17 Visualize Spatial Relationships
Secondary: #15 Organize Information in More Ways, #3 Eliminate Possibilities
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.
Execute — Answer: E
4.G.A.1 Step 1 Every pair of coins overlaps
- Two coins of the same size overlap exactly when their centres are closer together than two radii.
- Here the centres sit at the corners of a regular pentagon and each coin is wide enough that even the two coins across the pentagon from each other still cross.
- Check it on the figure: every one of the ten pairs of circles meets at two points.
- This is worth noticing up front, because it means every pair of coins leaves evidence about which of the two is on top — the picture really does determine the whole order, not just part of it.
💡 If two coins overlap at all, one of them must hide part of the other, and that missing piece is the clue.
K.G.A.1 Step 2 Coin C is drawn whole, so C is on top
- Coin $C$'s outline is a full circle with nothing missing.
- But $C$ overlaps all four other coins, so any coin lying above $C$ would have to cut a gap into $C$'s edge.
- There is no gap anywhere.
- So nothing lies above $C$, and $C$ is the top coin.
- Every answer choice already starts with $C$, so this costs nothing — but it is the one relation the picture states outright, and it fixes the starting point.
💡 A coin whose edge is nowhere interrupted has nothing lying on top of it.
K.G.A.1 Step 3 Read who covers whom at each overlap
- Here is the rule that turns the picture into facts.
- If any drawn piece of coin $X$'s edge runs across the face of coin $Y$, then $X$ is above $Y$ — because that piece of edge would have been erased if $Y$ sat on top of it.
- Apply the rule to the five pairs of coins that sit side by side around the ring.
- $C$'s outline sweeps across the face of $A$ and across the face of $E$.
- $E$'s outline runs across $D$.
- $D$'s outline runs across $B$.
- $A$'s outline runs across $B$.
- That is five over-under facts, read one at a time.
💡 The coin whose edge you can still see crossing the other one is the coin lying on top.
1.MD.A.1 Step 4 Those five facts are not enough
- Chain them together: $C > E > D > B$, and separately $C > A > B$.
- Both chains start at $C$ and finish at $B$, but nowhere do they compare $A$ against $E$ or against $D$.
- So $A$ is loose.
- Exactly three full orders fit all five facts — $(C, A, E, D, B)$, $(C, E, A, D, B)$ and $(C, E, D, A, B)$ — and those are precisely choices (A), (D) and (E).
- Choices (B) and (C) both place $D$ above $E$, which the figure contradicts, so they are already dead.
- Everything now hangs on one question: where does $A$ belong?
💡 Chaining only settles an order once every pair has been compared somewhere along the chain.
4.MD.C.6 Step 5 The A-and-D overlap decides it
- $A$ and $D$ are not side by side in the ring, but Step 1 said their coins still overlap, so the picture compares them too.
- Measure angles at each coin's own centre, with $0^\circ$ pointing right and turning counterclockwise.
- Coin $D$'s outline starts at about $75^\circ$ and keeps going up to about $110^\circ$ before leaving $A$ — that whole stretch of $D$'s edge is drawn lying across $A$'s face, so $D$ is above $A$.
- The other half of the check confirms it: coin $A$'s outline runs from $25^\circ$ around to $214^\circ$ and stops dead.
- The circles of $A$ and $D$ cross at $214^\circ$ and $290^\circ$ on $A$, so $214^\circ$ is exactly where $A$ would have entered $D$ — and everything past it is erased.
- No other coin cuts $A$ there: $B$ cuts $A$ at $155^\circ$ and $277^\circ$, $C$ at $25^\circ$ and $263^\circ$, $E$ at $250^\circ$ and $326^\circ$.
- Only $D$ can have made that break.
💡 The exact spot where a coin's outline stops names the coin that cut it off.
1.MD.A.1 Step 6 Assemble the single order
- Feed $D > A$ back into the chains.
- $C$ is on top.
- From Step 3, $C > E > D$.
- From Step 5, $D > A$.
- From Step 3 again, $A > B$.
- Strung together this reads $C > E > D > A > B$, with every pair now compared.
- Of the three orders still standing after Step 4, only this one puts $D$ above $A$, so the answer is (E).
💡 One extra comparison collapses three possible orders down to one.
4.G.A.1 Two coins of the same size overlap exactly when their centres are closer togethe K.G.A.1 Coin $C$'s outline is a full circle with nothing missing. But $C$ overlaps all f K.G.A.1 Here is the rule that turns the picture into facts. If any drawn piece of coin $ 1.MD.A.1 Chain them together: $C > E > D > B$, and separately $C > A > B$. Both chains st 4.MD.C.6 $A$ and $D$ are not side by side in the ring, but Step 1 said their coins still 1.MD.A.1 Feed $D > A$ back into the chains. $C$ is on top. From Step 3, $C > E > D$. From Review
Reasonableness: 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 $61^\circ$ to either side of the line joining their centres when they are neighbours in the ring, and $38^\circ$ to either side when they are not. So: $C$ is covered by nothing and stays a full $360^\circ$ circle. $E$ is covered only by $C$, hiding $11^\circ$ through $133^\circ$, leaving $E$ drawn from $133^\circ$ — the figure draws $E$ from $132^\circ$. $D$ is covered by $C$ and $E$, hiding $299^\circ$ through $74^\circ$, leaving $D$ drawn $74^\circ$ to $299^\circ$ — the figure draws $75^\circ$ to $300^\circ$. $A$ is covered by $C$, $E$ and $D$, leaving it drawn $25^\circ$ to $214^\circ$ — the figure agrees exactly. $B$ is covered by all four, leaving it drawn $97^\circ$ to $227^\circ$ — the figure draws $96^\circ$ to $228^\circ$. 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 — $360^\circ$, $238^\circ$, $225^\circ$, $189^\circ$, $132^\circ$. 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.
Alternative: Work from the answers instead of from the figure. Take each listed order in turn, assume that stack, and predict where each outline should break; keep only the order that reproduces all five arcs. Choices (B) and (C) put $D$ above $E$, which would leave a gap in $E$'s outline on the side facing $D$ — there is none, so both fail immediately. Choice (A) puts $A$ directly under $C$ alone, which would leave $A$'s outline running all the way to $263^\circ$; choice (D) puts $A$ under $C$ and $E$, which would leave it running to $250^\circ$. The figure stops $A$ at $214^\circ$, so both fail. Only $(C, E, D, A, B)$ survives. It is the same single measurement doing the work, reached from the answer end rather than the figure end.
CCSS standards used (min grade 4)
K.G.A.1Describe positions of objects using above, below, beside, in front of (Reading each overlap in the picture as the statement that one named coin lies above another.)1.MD.A.1Order three objects by length and compare lengths indirectly (Chaining the separate over-under pairs into one top-to-bottom order, and spotting that A is never compared with D or E by the easy overlaps.)4.G.A.1Draw points, lines, line segments, rays, angles, and identify in figures (Identifying the arcs, the points where two circles cross, and the fact that all ten pairs of circles cross.)4.MD.C.6Measure angles in whole-number degrees using a protractor (Locating in degrees the exact place where coin A's outline stops, and matching it against the crossing points of each other circle to name the coin that cut it.)
⭐ 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.
⭐ 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.
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