Competition · AMC preparation · step 4 of 4

AMC 10 · 2019A · #8

Grade 8 geometry-2d

Archetype Small-Domain Constrained Search └ Sub-archetype Sum-Target Search

spatial-visualizationreflection-symmetryrotation-isometrytransformations-compositionline-symmetry caseworkphysical-representation ↑ Prerequisites: reflection-symmetryrotation-isometry

📏 Short solution 💡 2 insights 📊 Diagram

Problem

A line ℓ has an infinite, repeating pattern of squares (alternating above and below the line) with small diagonal segments at one corner of each square. How many of these four rigid motions (other than identity) carry the whole figure back onto itself: (1) some rotation about a point on ℓ, (2) some translation along ℓ, (3) reflection across ℓ, (4) some reflection across a line perpendicular to ℓ?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 10 2019 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Create a Physical Representation

Frieze symmetry questions are best answered by physically tracing or sliding/flipping a copy of the picture (Tool #10). Tool #17 lets us mentally rotate or reflect after the physical step. Tool #1: sketch the pattern, mark a candidate center / axis, and check each piece. Tool #3 sweeps the four motions one by one — yes/no per motion, then total the yeses.

1STEP 1

Sketch the repeating pattern

Sketch it: up-squares at x = 0, 4, 8…, down-squares offset by 2, each with a diagonal stub facing away from ℓ — a strip of period 4.

Period along ℓ: 4 units Above and below offset by 2

2STEP 2

Test the translation

Motion (2) translation: slide the figure right 4 units — up→up, down→down, stubs come along. Period is exactly 4, so all match. YES.

Translation by 4 units → matches

3STEP 3

Test the rotation

Motion (1) rotation: a 180° half-turn about (1, 0), the midpoint between an up- and down-square, swaps them stub-onto-stub. Invariant. YES.

180° rotation about (1, 0) → above ⇔ below, matches

4STEP 4

Test the reflection across the line

Motion (3) reflect across ℓ: ups become downs, but they land at x = 0, 4… where ups belong — positions clash. NO.

Flip across ℓ → above → below at WRONG positions

5STEP 5

Test the perpendicular reflection

Motion (4) reflect across a vertical line: squares fit, but the corner stub jumps to the mirror corner where nothing was. NO.

Vertical flip → diagonal stubs land on wrong corners

6STEP 6

Count the motions that work

Tally: translation YES, rotation YES, both reflections NO. Count = 2.

YES count = 2 → (C)

Answer

Frieze patterns are classified into 7 groups by symmetry, and each group has a specific combination of translation, glide-reflection, rotation, horizontal-reflection, and vertical-reflection. Our pattern has translation + 180° half-turn rotation but NO horizontal or vertical mirror (the diagonal stubs and the above/below alternation both break the mirrors). That matches the frieze group p2 — exactly 2 non-identity symmetries from the listed four. Consistent with answer (C).

💡Key takeaway

This AMC 10 problem only needs Grade 8 rigid-motion thinking you already know: test each of the four motions by sliding or flipping a copy onto the pattern. Translation along the line works (period 4); 180° rotation about a midpoint between an up- and down-square works; the two flips both fail because the diagonal stubs land on the wrong corners. Count: 2.

Sketch the repeating pattern
Test the translation
Test the rotation
Test the reflection across the line
Test the perpendicular reflection
Count the motions that work

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