AMC 10 · 2015 · #7

Grade 8 geometry-2d

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

line-symmetryrotation-isometryequal-spacing spatial-visualizationidentify-subproblems ↑ Prerequisites: line-symmetryrotation-isometry

📏 Long solution 💡 2 insights

Problem

A regular many-sided figure has fold lines and a smallest turn that both leave it unchanged. Add the two counts.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

How to solve

Strategy Visualize Spatial Relationships

Both L and R ask for an exact count, and an exact count is really two claims: these ones work, and nothing else does. Finding fold lines and turns that work is easy to see by picturing the 15 corners spaced evenly on a circle. Ruling out every other fold line and every smaller turn is the part that carries the weight, so the plan is to first pin down where a symmetry can possibly send the center and the corners, then let that squeeze the count from above.

1STEP 1

Split into two exact counts

The question is really two counts.

L = #{fold lines}, R = min{α ＞ 0 : turn by α works}

2STEP 2

Every symmetry keeps the center still

Every symmetry keeps the centre still.

symmetry(O) = O

3STEP 3

A crease holds exactly one corner

An odd count puts one corner on each crease.

180°/24° = 7.5 ∉ Z

4STEP 4

So L = 15, no more and no fewer

So there are exactly 15 creases.

L = 15

5STEP 5

Corners sit 24° apart

The corners sit 24 degrees apart.

360°/15 = 24°

6STEP 6

So R = 24, and nothing smaller

Nothing smaller can work.

α = 24° k, k ∈ Z ⟹ R = 24°

7STEP 7

Add the two results

Adding gives 39, choice (C).

L + R = 15 + 24 = 39

Answer

Test the method on shapes you can already picture. An equilateral triangle gives L = 3 and R = 120°; a square gives L = 4 and R = 90°; a regular pentagon gives L = 5 and R = 72°. Each matches L = n and R = 360°/n, so L = 15 and R = 24° fit the pattern. A useful sanity note: R = 24° is also the exterior angle of the 15-gon, which makes sense because the turn that carries one side onto the next side turns by exactly that much. The wrong answers show what happens if a half of the argument is skipped. Choice (A) 24 is what you get by forgetting L entirely. Choice (E) 54 is 30 + 24, and 30 is the total number of symmetries of the figure, 15 turns plus 15 folds, or equivalently what you get by counting each crease twice as two half-lines from the center. Only the "exactly one corner per crease" argument tells you 15 rather than 30, so that step is what the problem really turns on.

💡Key takeaway

A regular 15-gon has one crease per corner and turns by 24° at a time, and the real work is showing there is nothing else.

Split into two exact counts
Every symmetry keeps the center still
A crease holds exactly one corner
So L = 15, no more and no fewer
Corners sit 24° apart
So R = 24, and nothing smaller
Add the two results