AMC 10 · 2019 · #8

Grade 8 arithmetic

Archetype Small-Domain Constrained Search

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

📏 Short solution 💡 2 insights 📊 Diagram

Problem

The figure below shows line $\ell$ with a regular, infinite, recurring pattern of squares and line segments.

How many of the following four kinds of rigid motion transformations of the plane in which this figure is drawn, other than the identity transformation, will transform this figure into itself?

some rotation around a point of line $\ell$
some translation in the direction parallel to line $\ell$
the reflection across line $\ell$
some reflection across a line perpendicular to line $\ell$

$\textbf{(A) } 0 \qquad\textbf{(B) } 1 \qquad\textbf{(C) } 2 \qquad\textbf{(D) } 3 \qquad\textbf{(E) } 4$

Pick an answer.

(A)

(B)

(C)

(D)

(E)

View mode:

Toolkit + CCSS Solution

Understand

Restated: A line $\ell$ 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 $\ell$, (2) some translation along $\ell$, (3) reflection across $\ell$, (4) some reflection across a line perpendicular to $\ell$?

Givens: Pattern repeats every $4$ units along $\ell$; Squares alternate above and below the line; Each square has a small diagonal segment at one corner that points outward; Four candidate motions; identity is excluded; Choices: (A) $0$, (B) $1$, (C) $2$, (D) $3$, (E) $4$

Unknowns: How many of the four motions are symmetries of this frieze pattern

Understand

Plan

Primary tool: #10 Create a Physical Representation

Secondary: #17 Visualize Spatial Relationships, #1 Draw a Diagram, #3 Eliminate Possibilities

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.

Execute — Answer: C

#1 Draw a Diagram 4.G.A.3 Step 1

Sketch the pattern: along $\ell$, the squares alternate above ($\square$ up) and below ($\square$ down).
Above-squares sit at positions $0, 4, 8, \ldots$ and below-squares sit shifted by $2$ to be at $2, 6, 10, \ldots$.
Each square also has a small diagonal stub at one corner pointing away from $\ell$.

$$\text{Period along }\ell: 4 \text{ units} \quad\text{Above and below offset by } 2$$

💡 Picture the alternating $\square$-above / $\square$-below stripe.

#17 Visualize Spatial Relationships 8.G.A.1 Step 2

Test motion (2) Translation along $\ell$.
Slide the entire figure $4$ units to the right: an above-square at $0$ maps to the above-square at $4$, the below-square at $2$ maps to the below-square at $6$, diagonals shift along too.
The figure is infinite and the period is exactly $4$, so EVERY part lands on a matching part.
YES.

$$\text{Translation by } 4 \text{ units} \Rightarrow \text{matches}$$

💡 Period $4$ means shifting by $4$ leaves the figure unchanged.

#10 Create a Physical Representation 8.G.A.1 Step 3

Test motion (1) Rotation about a point on $\ell$.
Pick the midpoint of $\ell$ halfway between an above-square (at $x = 0$) and the next below-square (at $x = 2$) — i.e., the point $(1, 0)$.
Rotate the whole figure $180^\circ$ about it.
The above-square swaps with the below-square (above and below flip, left and right flip), and the diagonal stub on the above-square corner now lies exactly where the diagonal stub on the below-square corner was.
The pattern is invariant.
YES.

$$180^\circ \text{ rotation about } (1, 0) \Rightarrow \text{above } \leftrightarrow \text{ below, matches}$$

💡 Rotating $180^\circ$ about the midpoint between an up- and down-square swaps them perfectly.

#17 Visualize Spatial Relationships 8.G.A.1 Step 4

Test motion (3) Reflection across $\ell$.
Flip the figure across the horizontal line $\ell$.
Every above-square now points down; every below-square now points up.
But the ORIGINAL pattern has above-squares at $x = 0, 4, 8, \ldots$ and below-squares at $x = 2, 6, 10, \ldots$ — the reflected pattern has below-squares at $x = 0, 4, \ldots$ and above-squares at $x = 2, 6, \ldots$.
These don't match.
NO.

$$\text{Flip across }\ell \Rightarrow \text{above } \to \text{ below at WRONG positions}$$

💡 Flipping across the line moves above-squares to wrong below-square slots.

#17 Visualize Spatial Relationships 8.G.A.1 Step 5

Test motion (4) Reflection across a line perpendicular to $\ell$.
A vertical flip line.
Squares (which are left-right symmetric on their own) would map okay positionally — but each square has a small diagonal stub at one specific corner.
Reflecting across a vertical line moves the stub from (say) the upper-right corner of a square to its upper-left corner.
The original pattern only has stubs at one side, not the mirror side.
NO.

$$\text{Vertical flip} \Rightarrow \text{diagonal stubs land on wrong corners}$$

💡 The diagonal stub is at one specific corner — vertical mirror lands it on the wrong corner.

#3 Eliminate Possibilities K.MD.B.3 Step 6

Tally the YESes: translation YES, rotation YES, reflection across $\ell$ NO, reflection across perpendicular NO.
Count $= 2$.

$$\text{YES count} = 2 \Rightarrow \textbf{(C)}$$

💡 Tally the motions that work.

[1] #1 4.G.A.3 Sketch the pattern: along $\ell$, the squares alternate above ($\square$ up) and

[2] #17 8.G.A.1 Test motion (2) Translation along $\ell$. Slide the entire figure $4$ units to t

[3] #10 8.G.A.1 Test motion (1) Rotation about a point on $\ell$. Pick the midpoint of $\ell$ ha

[4] #17 8.G.A.1 Test motion (3) Reflection across $\ell$. Flip the figure across the horizontal

[5] #17 8.G.A.1 Test motion (4) Reflection across a line perpendicular to $\ell$. A vertical fli

[6] #3 K.MD.B.3 Tally the YESes: translation YES, rotation YES, reflection across $\ell$ NO, ref

Review

Reasonableness: 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 + $180^\circ$ 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).

Alternative: Tool #10 (Physical): cut out a strip showing one period of the figure on transparent paper, then physically slide / rotate / flip the strip over the original. Translation by $4$ and $180^\circ$ turn about a midpoint both make the transparent strip coincide with the original; the two flips clearly don't. Same count: $2$.

CCSS standards used (min grade 8)

8.G.A.1 Verify experimentally the properties of rotations, reflections, and translations (Testing each of the four rigid motions (translation, rotation, horizontal reflection, vertical reflection) against the frieze pattern and deciding which preserve the whole figure.)
4.G.A.3 Recognize a line of symmetry for a two-dimensional figure (Identifying that the figure has no horizontal or vertical mirror lines because of the alternating up/down squares and the diagonal stubs.)
K.MD.B.3 Classify objects into given categories and count the numbers in each (Tallying how many of the four motions came out YES.)

⭐ 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 $4$); $180^\circ$ 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: $2$.

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: C

Review

CCSS standards used (min grade 8)

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