 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ii/paper-2/33b/b/i/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

Let

I_{0} = [x_{0}, x_{1}], I_{1} = [x_{1}, x_{2}], I_{2} = [x_{2}, x_{3}] .

I_{0} ⟶ I_{1}, I_{1} ⟶ I_{2}, I_{2} ⟶ I_{0}, I_{1}, I_{2} .

A = 001101011 .

The two closed length-three walks

I_{2} \to I_{2} \to I_{2} \to I_{2}, I_{2} \to I_{0} \to I_{1} \to I_{2}

F^{3}

, so

F

must be chaotic. Moreover,

tr A = 1, tr (A^{3}) = 7.

\frac{7 - 1}{3} = 2.

Hence

F

must have at least, and the connect-the-dots realization shows that it need have only,

2

distinct 3-cycles.

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