Hi, and welcome back from the summer to a new series of mathematical riddles! I’ve decided to include only one riddle per post this semester, which will hopefully mean that I’ll be able to keep a more consistent posting schedule. As always, if you have a nice mathematical riddle that you would like to share, then let us know and we’ll see if we can post it. Today’s riddle is a nice puzzle about swimming and monsters; check out the post below to read more!

Imagine that you are at the center of a circular pool of unit radius. At the very edge of the pool, there is a monster who is trying to catch you. The monster cannot swim, so it can only run around the edge of the pool. You know that you can run faster than it while on the ground, so if you can manage to reach any point on the edge of the pool before the monster can get there then you will escape. However, the monster runs four times faster than you can swim. Is it possible for you to escape from the pool and the monster?

In order to clarify the riddle, suppose that the monster instead ran three times faster than you could swim. Then the obvious thing to do would be to swim in a straight line to the edge of the pool directly opposite from the monster’s starting point. To reach this point, the monster would have to travel π times as far as you (i.e., a distance of π*r* versus *r*). Since π > 3, you would thus be safe.

However, since the monster runs four times as fast as you, the simple strategy above will result in you getting eaten. Can you think of a more complicated strategy that does work in this case?

If that’s too easy, then can you escape if the monster runs five times faster than you? What is the maximum speed (e.g., five times, six times) that the monster can run such that you can still escape?

## About Irving Dai

Irving is a third-year graduate student studying topology and geometry at Princeton University. His mathematical interests include gauge theory and related Floer homologies. In his spare time he plays the violin (occasionally, and usually badly). He is fond of cats.

Neat puzzle. My solution would be:

Keeping swiming in an ever increasing spiral till one reaches say r/2 away from the opposite bank to where the monster is standing. Then head radially off to the bank. The monster would be 4x faster, which is 4xr/2 = 2r, which is less than pi*r, the distance it would have to cover to reach the point exactly opposite.

Well the similar system for higher speeds, only one would have to spiral longer till a corresponding shorter distance to the bank is reached.

Sorry for the mistakes! Once again

Neat puzzle. My solution would be:

Keep swimming in ever increasing circular spiral about the center till one reaches say r/2 away from the opposite bank to where the monster is standing. Then head radially off to the bank. The monster would be 4x faster, which is 4xr/2 = 2r, which is less than pi*r, the distance it would have to cover to reach the point exactly opposite.

Well the similar system for higher speeds, only one would have to spiral longer till a corresponding shorter distance to the bank is reached.

Nice idea! However, it isn’t clear that while swimming in a circular spiral, you can always manage to stay on the opposite side of the monster. For example, suppose that you are at a radius of r/2. Then you can check that the monster’s angular velocity is greater than your angular velocity.

If the circle’s radius is r and your distance from the centre is 4 you can do it.

The fastest the monster can run while this strategy works is π+1 times as fast as you. Proving that the monster can always catch you if it can run >π+1 times as fast, I’m not sure about.

SPOIILERS!!!

If the circle’s radius is r and your distance from the centre is less than r/4 then your angular velocity is greater than the monster’s, so you can keep it on on the opposite side of the large circle to your smaller one. So you can work your way out to r/4 distance from the centre while keeping the monster on the opposite side, then make a dash, so you only have 3r/4 distance to run while the monster has to run πr, so as 4π/3 > 4 you can do it.

The fastest the monster can run while this strategy works is π+1 times as fast as you. Proving that the monster can always catch you if it can run >π+1 times as fast, I’m not sure about.

Nice solution!

I really loved your article and I’m still not sure about the maximum speed of the monster. I’ve been looking for good sites that provide all sorts of riddles to entertain me while i’m at work. I think riddles help keep our minds fresh and can also provide a good laugh. Thanks for the blog.

http://www.riddlesandanswers.com/tag/long-riddles/