In the 2022 AMC 10A competition, Problem #18 investigates composite transformations in the Cartesian plane using rotations and reflections:

Contest Problem Statement:
"Let $T_k$ be the transformation of the coordinate plane that first rotates the plane $k$ degrees counterclockwise around the origin and then reflects the plane across the $y$-axis. What is the least positive integer $n$ such that performing the sequence of transformations $T_1, T_2, T_3, \dots, T_n$ returns the point $(1, 0)$ back to itself?"

Understanding the Transformation $T_k$

Let a point have polar angle $\theta$. What does $T_k$ do?

1. Counterclockwise rotation by $k^\circ$: Adds $k^\circ$ to the angle: $$\theta \mapsto \theta + k^\circ$$
2. Reflection across the $y$-axis: A reflection across the vertical $y$-axis replaces $x$ with $-x$. In terms of polar angle, $(x, y) = (\cos\phi, \sin\phi)$ becomes $(-\cos\phi, \sin\phi) = (\cos(180^\circ - \phi), \sin(180^\circ - \phi))$: $$\phi \mapsto 180^\circ - \phi$$

Combining both operations, $T_k$ transforms any angle $\theta$ into: $$T_k(\theta) = 180^\circ - (\theta + k^\circ) = 180^\circ - k^\circ - \theta$$

Tracking the Angles Step by Step

Starting at $(1, 0)$ with initial angle $\theta_0 = 0^\circ$:

• Step 1 ($k=1$): $\theta_1 = 180^\circ - 1^\circ - 0^\circ = 179^\circ$.
• Step 2 ($k=2$): $\theta_2 = 180^\circ - 2^\circ - 179^\circ = -1^\circ$ (or $359^\circ$).
• Step 3 ($k=3$): $\theta_3 = 180^\circ - 3^\circ - (-1^\circ) = 178^\circ$.
• Step 4 ($k=4$): $\theta_4 = 180^\circ - 4^\circ - 178^\circ = -2^\circ$.

Do you see the alternating pattern?

• For an even number of steps $n = 2m$: $$\theta_{2m} = -m^\circ$$ For $\theta_{2m} \equiv 0^\circ \pmod{360^\circ}$, we need $m = 360 \implies n = 720$.
• For an odd number of steps $n = 2m + 1$: $$\theta_{2m + 1} = 180^\circ - (m + 1)^\circ$$ For $\theta_{2m + 1} \equiv 0^\circ \pmod{360^\circ}$, we need $m + 1 = 180 \implies m = 179$!
  Thus $n = 2(179) + 1 = 359$.

Because $359 < 720$, the least positive integer $n$ is 359!

In this lesson, we write a simulation in Lua using dcos() and dsin() to verify this coordinate orbit.

Move the mouse over a dotted box for more information.

The simulation demonstrates that rotation combined with reflection creates an alternating parity pattern, returning $(1, 0)$ to itself at step 359.