Question:

You are in a boat in the exact center of a perfectly circular lake. There is a goblin on the shore of the lake. The goblin wants to do bad things to you. The goblin can’t swim and doesn’t have a boat. Provided you can make it to the shore — and the goblin isn’t there, waiting to grab you—you can always outrun him or land and get away. The problem is this: The goblin can run four times as fast as the maximum speed of your boat. He has perfect eyesight, never sleeps, and is extremely logical. He will do everything in his power to catch you. How would you escape the goblin?

Answer: Just so you understand the problem: The obvious plan is to make a beeline for the shore at the point farthest from where the goblin is right now. This gives you a substantial distance advantage. You have only to travel a radius (r) of the circular lake. The no swimming goblin has to run in a semicircular arc along the shore, amounting to half the lake’s circumference. This comes to Jtr. The goblin thus has to cover π times the distance you do. Pi is a little more than three. Were the goblin only three times as fast as your boat, you could narrowly beat him to the far shofe. That’s why the puzzle says the goblin is four times as fast. No matter where you choose to land, the goblin will be there to grab you. Like many puzzles, this one asks you to sort out some significant ambiguities. Is the goblin just a mindless “magnet” sliding along the shore at the nearest point to you—or is he a thinking being? That business about the goblin being “extremely logical” implies the latter. It would seem you’ve got to fake out the goblin. But the scope of any “faking out” is restricted. There is no place to hide in the middle of a lake. An extremely logical goblin must carefully consider all possible strategies on your part and cannot truly be taken by surprise. For the moment, pretend the goblin is a mindless magnet tracking your every motion and trying to stay as dose to you as possible. Here’s one way to rattle his cage: Make a tiny circle about the center of the lake. This will drive the goblin nuts. He will want to circle the entire lake (while your boat is making a circle of a few feet). The goblin won’t be able to keep up with your boat since his circle is so much larger than yours. This means you can, by circling, put yourself beyond the halfway point of a line drawn from the goblin through the lake’s center to the far shore. That suggests a solution. Ask yourself “What is the largest circle, concentric with the lake, I can travel on, such that the goblin can just keep up with me?

“It must be a circle where you cover only 1/4 the distance of the four-times-faster goblin. It’s a circle with radius r/4.Travel clockwise on this circle, and the goblin will be forced to run at top speed clockwise, just to stay at the point closest to you onshore. Travel counterclockwise, and the goblin has to run counterclockwise. Here’s the clever part. Should you travel on a path of radius just under r/4, the goblin will be unable to keep up with you. He will lag gradually behind.

That means that you can manage to put yourself 11/4 radii from the goblin. One way to do it is to spiral out from the center of the lake, approaching yet/not quite attaining the circle of radius r/4. As long as you stay within that charmed circle, the goblin will be unable to keep up with you. You can keep playing him out until the goblin falls a full 180 degrees behind you. That puts your boat opposite the goblin and nearly 5/8 of the way across the lake from him (the lake’s center is halfway across the water from the goblin, and you are almost but not quite a 1/4 radius or 1/8 diameter beyond that). This geometry allows you to escape. You abruptly stop your circling and make a beeline to the far shore. The distance you need to cover is just over 3/4 times r. The goblin has to cover πr. That is 4π/3 times as great, and since the goblin is four times faster, he covers it in π/3 the time you do. Pi over three is slightly bigger than one(1.047 …). Do everything according to plan, and you will have already landed and started running by the time the goblin gets there.

Does this really solve the puzzle? What if the goblin is smart and has already heard about this plan? He doesn’t have to follow you around like a lapdog, not when he realizes what you’re up to.

Yes, but even when the goblin knows exactly what you’re up to, he can’t do any better. You can pick up a bullhorn and announce “Hey, Goblin! Here’s exactly what I’m going to do. I’m running around in this little circle just under one-quarter the radius of the lake. You do the math!

The instant I’m one hundred eighty degrees from you, I bolt for the shore, and we both know I’ll beat you there. Now we can do this the easy way, the hard way, or the stupid way. The easy way is for you to realize you’re defeated. Stay put, and let me swing around to the other side and make my escape. The hard way is for you to chase me. That’s more work for both of us. The outcome will be exactly the same. Finally, there’s the stupid way. Should you try and pull a ‘contrary’ strategy —such as chasing at less than your top speed, chasing in the wrong direction, running back and forth, or even running away from the water — any of those things will just make it easier for me to get one hundred eighty degrees away from you, at which point I’m out of here!”

At various companies, the anecdote takes other forms. Sometimes you’re in the middle of a circular field, surrounded by barbed wire, with a killer dog on the outside trying to get you. Another version has a fox attempting to get a duck in the middle of a circular lake (though it’s hard to imagine a duck knowing the geometry).