AMC 10 · 2019 · #6

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 carries an infinite repeating pattern of squares, alternating above and below the line, with a small diagonal segment at one corner of each square. Not counting the identity, four rigid motions are offered: a rotation about a point on the line, a translation along the line, a reflection across the line, and a reflection across a line perpendicular to it. Count how many of them carry the whole figure back onto itself.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

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

Measure the period

Top and bottom are offset by half a period.

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

2STEP 2

Check the translation

Sliding one period matches exactly.

Translation by 4 units → matches

3STEP 3

Check the rotation

A half turn swaps top and bottom and matches.

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

4STEP 4

Check the reflection across the line

Flipping lands them in the wrong places.

Flip across ℓ → above → below at WRONG positions

5STEP 5

Check the perpendicular reflection

The stubs land on the wrong corner.

Vertical flip → diagonal stubs land on wrong corners

6STEP 6

Count the survivors

Exactly 2 survive.

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 12 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.