Aaron has learned that Erin always plays a fixed distance — call it *e* — along the correct angle. Erin knows that Aaron knows this, and that Aaron will pick a distance *a* (and an angle at random, presumably) to maximize *P*(Aaron wins). So Erin should pick a value for *e* that minimizes this probability.

So we need to determine *P*(Aaron wins) for given values of *e* and *r*. If *r* is less than *e*/2, Aaron should choose to stay at the center, thus guaranteeing a win. Otherwise, Aaron should pick the distance *a* that maximizes the *fraction* of the circle of radius *a* located within a distance of |*r*−*e*| of the flag.

Maximizing the fraction of the circle of radius *a* inside the radius-|*r*−*e*| circle is equivalent to maximizing the subtended angle determined by the circles’ intersection points. This gives rise to the arcsin integral above. (The *r* multiplier inside both integrals and 2 multiplier generated by the bottom integral is to account for polar coordinates — e.g. if the flag has an *ε* probability of being placed in a tiny annulus of radius ≈*δ*, it has a 2*ε* probability of being placed in a tiny annulus of radius ≈2*δ*.)

Numerically solving for the *e* that minimizes the probability above gives *e* ≈ 0.501306994212753; plugging that into the integral above gives *P*(Aaron wins) ≈ 0.166186486474004.

**Congrats to this month’s solvers!**