## TGIF Puzzle March 3, 2006

### The Puzzle

You are in a rowboat in the middle of a circular lake. A goblin who wants to eat you is on the shore. The goblin can run K times as fast as you can row, but you can outrun the goblin on dry land. There is a threshhold value for K below which you can row to shore and escape being eaten by the goblin and above which he can invariably run you down and enjoy his dinner. What is that threshhold value of K?

### The Simulation Applet

The applet on this page displays a simulation of this puzzle. The knob is preset to the threshhold value of K.

### The Optimum Strategies

Your optimum strategy has two parts: You start by rowing directly away from the goblin. As he pursues you, you turn so as to always keep the center of the lake between you and the goblin. You can maintain this strategy until you are 1/K times the radius of the lake (R) from its center and the path you row will be a semicircle. The second part of your flight is to continue in a straight line tangent to the circle in which you were rowing.

The optimum strategy for the goblin (if you also adopt the optimum strategy) is to run as fast as he can in one direction around the lake. If you abandon your optimum strategy before you reach the the R/K circle, the goblin simply reverses direction as necessary to minimize your advantage. Regardless, the goblin can do no better than being as far away as possible when you cross the R/K circle. After you cross the R/K circle and are following the tangent to the shore, if the goblin reverses direction, you simply turn onto the chord centered on your position and head away from the goblin; you will reach the shore with an even greater margin.

### The Answer

The threshhold value of K is 4.60333885 (to 8 decimal places). This is the solution to:

sqrt(1-1/k^2) = (pi + acos(1/k))/k.

The left side of the equation is the time you need to travel along the tangent line (for R = 1, and speed = 1) and the right side is the time for the goblin to arrive at the same point (speed = K).

### Other Observations

The time required for the first phase of your flight allows the goblin to travel one quarter of the way around the lake (pi/2). Your flight path requires the goblin to travel nearly all the way around the lake.