The large circle, radius R, is the lake. The smaller circle, radius r, contains the escape starting point for the boat. The ogre's speed multiple is f > 1, so R = f.r, using the period to mean multiplication. The rower has thus achieved her greatest possible separation from the ogre for which she can maintain the lake's center directly between them. We now compare two escape strategies.

**[1] Radial escape.**

The three black dots are, from right to left, the Ogre's and the rower's starting points, and the point on the shore that each heads for as the chase begins. The black circular arc is the ogre's path, and the black leftward arrow is the rower's path. The equation of simultaneous arrival is simply

f.x = R.pi = f.r.pi

x = r.pi

Noting that x = R-r = r(f-1) we obtain

f = 1 + pi = **4.14159...**

as the maximum ogre speed factor advantage that can be overcome using radial escape.

She must be able to row** 0.966 mph.**

**[2] Tangential escape.**

The red dot now is the point on the shore that each is headed for. The sum of the black and red circular arcs is the ogre's new path, and the red upward arrow is the rower's new path. The equation of simultaneous arrival is a little more complicated, but not much.

First, we need to find the extra path length for the ogre. It's simply a.R [the red arc] where a is the angle from horizontal to the new destination point. Thus

f.y = R(pi + a) = f.r(pi + a)

y = r(pi + a)

Noting that tan(a) = y/r

tan(a) = pi + a which gives a = 77.45 degrees.

Noting that cos(a) = y/R = 1/f, we obtain f = 1/cos(77.45)

f = **4.6033**

This is an 11% increase in the ogre's speed that can be overcome using tangential escape compared to radial escape.

Thus the rower can row 11% slower and still escape.

She need only row **0.869 mph.**

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Two comments.

First, this is also only a sufficient condition. This analysis does not rule out a more efficient escape strategy. Spirals are good next steps to investigate. Piecewise linear escape paths would be easier to investigate.

Second, one could ask whether the ogre, once he sees the direction taken by the rower, might not help himself by switching directions and taking the shorter route to the red dot. The answer is no. If he does, the rower fires up photoshop, constructs a new, larger inner circle, and heads tangentially for shore in the other direction, the result of which is actually to the rower's net advantage. The ogre's best strategy is to stay the course [clockwise in this case] for the red dot.

The drawing is for a value of f in the general range we are considering. That is, a is not exact, but close to 77.45 degrees. Thus it is valid to perceive the advantage of tangential escape by noting that y is not significantly larger than x, while a + pi is significantly larger than pi. Tangential escape makes the ogre's task more difficult.