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 4 of 4: Conclusion
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θ=180∘n,n≥2 an integer\theta=\frac{180^\circ}{n},\qquad n\ge2\ \text{an integer}
Detailed analysis

Steps 2 and 3 together show that Mulan can guarantee victory in finitely many steps if and only if θ=180∘/n\theta=180^\circ/n for some integer n≥2n\ge2.