Help with hard math problem!

 Yeah, but I'm bored and have nothing better to do. :/

h = sqrt[ ( 1/2x ) ^ 2 + (1/2x) ^ 2 ] = root 2 x

this is wrong... it's (1/√2)x or (1/2)(√2)x

The missionary goes to the water's edge on any side of the pool.  When the cannibal gets near, the missionary pulls him in and baptizes him. 

Problem solved.

Oinquarki is right. The missionary's procedure could be to go halfway to the opposite corner, then turn straight toward the side opposite the one the cannibal chose to run down. Thus the missionary swims one-quarter of the square root of two (1.414/4 or just over .35) plus one-quarter of a side length (or .25), equals about .60 of a side length. The cannibal must run 2.25 side lengths to catch him, but 2.25/3=.75 side length. He misses the cannibal by a lot, which means that other paths are possible for the missionary.

Think about what you're saying.  The cannibal is not going to be daft enough to ever choose a path longer than 2 sides of the square. 

LOL!  Sorry, m.  At first, I missed yours--surely a veritable marvel of concision--when I was replying.  ;)

ha ha ha..

yes he can escape but he might lose an arm in the process

or he could just get better at swimming

Example a 10x10 pool.  The swimmer is 5feet away from each side wall and therefore about 7 feet from the corner ( 5^2+5^2=c^2...c=7).  If the swimmer goes straight to the farthest corner he will be caught(his speed is 1ft/s and canibal is 3f/s. so 7 seconds is how long it would take to get to the corner and it would take 20/3 seconds for the canibal( or just under 7 seconds).  but if the swimmer goes only halfway to the corner (3.5feet) he will have went about 2.5feet towards both sides. he now starts to go straight towards the side farthest from the canibal.  so he could travel a distance of 6 feet in 6 seconds(3.5+2.5).  the canibal would have to go about 22.5 feet and that would take 7.5 seconds.  the missionary wants the shortest route which would be 5 feet, but if he goes that way he will be caught because the canibal can take the best route.  so what the missionary does is makes progress towards the shortest route, while making the canibal try and stop the route farthest from the canibal( to the opposite corner).  the canibal runs ten feet without getting any closer to the other side of the pool while the swimmer gets closer to both sides of the pool.

Three further problems. Let the pool be circular instead of square.

Problem 1: the cannibal is 3 times as fast (trivial)

Problem 2: the cannibal is 4 times as fast (not so easy)

Problem 3: the cannibal is 4.5 times as fast (possible, but a bit harder)

Any takers for these "circular pool" problems?

[P.S. the best solution works with a speed ratio of up to a fraction more than 4.6]

Uh...this could be wrong, but if the missionary swims a spiral path, keeping as far away from the cannibal at all times, while slowly getting closer to the edge??

Yes, understood it, but I think you need to be more precise. In my opinion, the circular pool problem is easier because of the symmetry  (I have a precisely optimal solution of that problem). The corners in the square pool make the arguments about optimal strategy more difficult.

But for inspiration (and to answer Wasabi_Kid), in the circular pool problem there are two different phases. First when the missionary is still close to the centre of the pool he moves so as to stay exactly opposite the cannibal. This becomes impossible when his distance from the centre is past a fraction that is the same as the speed ratio (for a particular angular speed, actual speed is proportional to the radius). From this point the missionary pokes his nose out and the cannibal chases him one way or the other. The missionary picks a particular point on the shore and heads straight for it, with the cannibal getting nearer in angle all the way. Which point he heads for is the main issue in the third of my questions.

By analogy, I would expect a similar thing to occur here, complicated by the corners. The missionary first heads along a diagonal. The cannibal has to choose a direction to follow. There may be a stage here where there is some jockeying for position while the missionary tries to get a bit away from the centre without the cannibal getting any "closer". What this means is not as easy to define as with a circular pool (there it is the angle between them). Finally, the missionary makes a bee-line for an optimal location on the boundary (which depends on the ratio of speeds, I infer from the solution of the circular pool problem).

That should help us find the absolutely optimal solution and prove it.

[So Wasabi_Kid, you got a good idea for the circular pool problem, but there is some more to be done to make it an answer!]

Sacrificing one arm will surely make it much easier to swim in spirals.

the very first thing came to my mind is that this depends on dimensions of the pool - regardless of the shape of the pool what matters is who can get to who faster so the distance as well as the speed matters - i might be totally off the track here but i am no math genius :)

what about this variation?

    An escaped prisoner finds himself in the middle of a square swimming pool. The guard that is chasing him is at one of the corners of the pool. The guard can run faster than the prisoner can swim. The prisoner can run faster than the guard can run. The guard does not swim. Which direction should the prisoner swim in in order to maximize the likelihood that he will get away?