Problem

Mulder is imprisoned in a circular field of radius \( R \), and an alien—4 times faster—is constrained to the perimeter. If Mulder reaches an unguarded point on the fence, he can instantly escape. Can he do it?

Variables

  • Let \( v \): Mulder’s speed
  • Alien’s speed: \( 4v \)
  • \( C \): center of the circle
  • \( A \): alien’s initial position
  • \( M \): Mulder’s current position inside the circle
  • \( x \): the fraction of radius \( R \) that Mulder is from center toward the escape point \( P \)

Strategy Overview

  1. Initial Phase: Mulder spirals or moves inside the circle to deceive the alien, maintaining a central position where the alien must guess his trajectory.
  2. Escape Preparation: Mulder moves to a point \( M \) that is closer to the perimeter, precisely at distance \( xR \) from center \( C \), along the line connecting \( A \) and the center.
  3. Straight Run: Once \( M, C, A \) are collinear with \( C \) between \( M \) and \( A \), Mulder sprints toward the point \( P \) on the perimeter opposite the alien’s position.

Timing Condition

Mulder’s run from \( M \) to \( P \):
\[ \text{Time}_M = \frac{(1 - x)R}{v} \]

Alien’s run from \( A \) to \( P \) along the perimeter:
\[ \text{Time}_A = \frac{\pi R}{4v} \]

Mulder escapes if: \[ \frac{(1 - x)R}{v} < \frac{\pi R}{4v} \Rightarrow x > 1 - \frac{\pi}{4} \]

This gives: \[ x > 1 - \frac{\pi}{4} \approx 0.2146 \]

So, if Mulder is at least ~21.5% of the way from the center to the edge, the alien cannot intercept him in time.

Angular Speed Constraint

Mulder must reach this point \( xR \) without being intercepted. To do that, he orbits around the circle of radius \( xR \). The alien tries to rotate along the outer circle, but Mulder’s angular speed exceeds that of the alien when:

\[ \frac{v}{xR} > \frac{4v}{\pi R} \Rightarrow x < \frac{1}{4} \]

So Mulder can safely orbit at any radius less than \( \frac{R}{4} \) to position himself.

Final Strategy

  • Mulder sets \( x = 1 - \frac{\pi}{4} + \epsilon \), e.g., \( \epsilon = 0.01 \).
  • He orbits at radius \( xR \) until perfectly aligned opposite the alien.
  • He then sprints straight for the fence.

Conclusion

\[ \boxed{\text{Yes, Mulder can escape by running smart—not just fast.}} \]

This problem elegantly combines geometry, relative motion, and clever timing.

Reference