AMC 10 · 2006 · #14

Grade 4 geometry-2dpattern

Archetype Arithmetic Expression Decomposition └ Sub-archetype Multi-Segment Path or Count Sum

sequences-arithmeticspatial-visualizationpattern-recognition identify-subproblems ↑ Prerequisites: sequences-arithmetic

📏 Medium solution 💡 2 insights 📊 Diagram

Problem

Rings hang linked on a peg, one below the next, and every ring is 1 cm thick. Going down, each ring's outside diameter is 1 cm less than that of the ring above it, starting from a top ring 20 cm across the outside and ending at a bottom ring 3 cm across the outside. Find the distance from the top of the top ring to the bottom of the bottom ring.

Pick an answer.

(A)

171

(B)

173

(C)

182

(D)

188

(E)

210

AMC 10 2006 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

Adding up the outside diameters (20 + 19 + … + 3) overcounts, because linked rings overlap where one hangs inside the other. Tool #7 (Identify Subproblems) says: cut the total height into pieces that do not overlap. Tool #1 (Draw a Diagram) shows what the pieces are — each ring's top bar tucks inside the hole of the ring above it, so what truly stacks up are the inside diameters, bracketed by just the top bar of the first ring and the bottom bar of the last. Tool #5 (Look for a Pattern) then adds the inside diameters fast: they are exactly the whole numbers 1 through 18, which pair up neatly. The whole trick is measuring inside holes, not outside edges.

1STEP 1

Switch from outside to inside diameters

The 1 cm metal eats 1 cm off each side of the hole, so inside = outside - 2. The outsides 20, 19, …, 3 become the holes 18, 17, …, 1.

inside = outside - 2; 20→18, 19→17, …, 3→1

2STEP 2

See how linked rings overlap

Each lower ring's top bar tucks inside the hole above, so only the holes stack — plus a 1 cm bar at the very top and one at the bottom.

total = 1_top bar + (18 + 17 + … + 1) + 1_bottom bar

3STEP 3

Add the inside diameters by pairing

Pair the ends of 1 + 2 + … + 18: 1 + 18 = 19, 2 + 17 = 19, … — that is 9 pairs of 19, so the sum is 171.

1 + 2 + … + 18 = 9 × 19 = 171

4STEP 4

Add the two end bars

Add the leftover metal, 1 cm on top and 1 cm on the bottom: 171 + 1 + 1 = 173 cm, which is choice (B).

171 + 1 + 1 = 173 → (B)

Answer

173

Sanity-check the two traps. Adding the inside diameters alone gives 171, which is choice (A) — that forgets the top and bottom bars, so it must be a bit small. Adding the outside diameters 3 + 4 + … + 20 = 207 is far too big, because it counts every overlap twice; note 207 is not even a choice, warning you overlaps must be removed. Our answer 173 sits just above 171 by the 2 cm of end bars, which is exactly the small correction the picture demands. Also check the count: outside diameters 3 through 20 is 18 rings, matching 18 inside diameters, so nothing was dropped.

💡Key takeaway

Linked rings overlap, so measure the inside holes that truly stack — here the holes are 1 through 18 (sum 171), plus one centimeter of metal at the very top and very bottom, giving 173.

Switch from outside to inside diameters
See how linked rings overlap
Add the inside diameters by pairing
Add the two end bars

Switch from outside to inside diameters

See how linked rings overlap

Add the inside diameters by pairing

Add the two end bars

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