MathLabs

Problem 4

Shan-Yu and Mulan are playing a game. Let θ\theta be an angle with 0∘<θ<180∘0^\circ<\theta<180^\circ known to both players. Initially, Shan-Yu makes a paper triangle TT with measurements of his choice. Then, they repeatedly perform the following steps: if TT has at least one angle measuring exactly θ\theta, then the game stops and Mulan wins; otherwise, Mulan chooses a point PP on the perimeter of TT, different from its three vertices, and makes a straight cut from PP to the opposite vertex of TT, splitting it into two triangles; Shan-Yu discards one of the two triangles, and the remaining triangle becomes the new TT. For which real values of θ\theta can Mulan guarantee her victory in finitely many steps, no matter how Shan-Yu plays?
Step 1 of 4: Any angle equal to kθ is a winning position for Mulan
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angle kθ (k≥1) ⟹cut into θ+(k−1)θ Mulan wins in at most k−1 cuts\text{angle }k\theta\ (k\ge1)\ \stackrel{\text{cut into }\theta+(k-1)\theta}{\Longrightarrow}\ \text{Mulan wins in at most }k-1\text{ cuts}
Detailed analysis

We prove by induction on k≥1k\ge1 that Mulan can win from any triangle having an angle equal to kθk\theta. For k=1k=1 the game ends immediately. For k≥2k\ge2, Mulan cuts from the vertex of angle kθk\theta to the opposite side so as to split that angle into θ\theta and (k−1)θ(k-1)\theta: if Shan-Yu keeps the piece with angle θ\theta, Mulan wins at once; if he keeps the piece with angle (k−1)θ(k-1)\theta, Mulan wins in at most k−2k-2 further cuts by the induction hypothesis.