Help with hard math problem!

There are multiple solutions, but one is the missionary swims 1/6x towards the corner opposite the cannibal, when the cannibal will have swam 1/2x. Then the missionary changes directions and goes straight up. The missionary will only have .5 - 1/6sqrt2 to go (.382x) while the cannibal has at least 1.5x left to go. He can't possible make it there in time. 

Uh...the cannibal can't swim...

Assuming you meant the cannibal ran, then yes, that should work...

Regarding this problem: 

A) Escape the cannibal means get out of the pool and run away -- staying in the pool isn't "escaping" (I'm assuming that for the purposes of the puzzle the missionary is able to go from swimming to running at the perimeter of the pool instantaneously)

B) This IS a hard math problem... everyone is trying to solve it with straight lines and algebra. Sorry, this is a calculuus problem and it's over my head, the missionary is NOT swimming a straight line to the pool's edge, he is swimming a curved line that maximizes the distance between the running cannibal and himself at all times(*edit, okay, that gives the general idea but actually-- he maximizes the ratio between his distance from the canibal, and his distance from the wall). The solution is an equation that plots the swimming missionary's curved path. If the path intersects the perimeter the missionary escapes.

I can do the conceptual part... don't ask for the solution though.

The missionary escapes provided that he can see the location of the cannibal. Say the square is xyzt and the cannibal sits at y. Then the missionary starts swimming towards t and stops at the middle of the distance between his initial position and t. Here he looks to see which way did the cannibal chose: through x or through z. Say the cannibal chose to go through z. Then the missionary starts swimming towards the edge xt. 

The missionary swims a distance of (Sqrt(2)+1)/4, while the cannibal needs to run a distance of 2+1/4 in order to catch him (assuming the pool has edge of unit length). 

Funny problem :)

Ahhh.. it's true we can have our poor blighted missionary swim in straight lines because the cannibal is obese and slow and there's lots of slack -- .  -- My reasoning remains correct if you wanted to plot the exact path the missionary would take (assuming he doesn't want to get any closer to the cannibal than is absolutely necessary) and the calc becomes necessary if the pool is circular.

*edit* BOLLOCKS. Actually the missionary still gets away in a circular pool without resort to calculuus -- (If the missionary swims directly away from the cannibal's point of origin without adjusting course at all, the cannibal must run Pi times faster to catch the missionary  (god is clearly on the missionary's side)

In order to require calc we would need the question to be: What is the slowest the cannibal can run and still be certain of a nice meal of missionary steak.

if the missionary can run thrice as faster as the cannibal, and if the cannibal cannot swim, this means the missionary can just stay in the middle of the water avoiding the edges. Or he could jump out to the surface and outrun the cannibal. So no matter what happens, the missionary will be able to escape.

This question is more based on logical thinking and common sense...I think there is no need for the complication of complex algebraic equations.

 I think you miss-read the question.

Oh the answer is about 3.75 (The cannibal needs to run 3.75 times faster). And wouldn't want to be the missionary for the world. I hate the water.

The problem is, it will greatly slow down the missionary to be constantly looking over at the cannibal at every point in time.

This is a fairly easy example of this problem. If the missionary starts from the centre of the pool, there will always be a centre of a side or a corner that is more than three times as far from the cannibal as from him, and he should head straight for one such. Actually more simply, if the missionary heads in the directly opposite direction from where the cannibal starts, he will get to the edge first (obvious when the cannibal is at a corner or the centre of a side, needs checking for other cases).

There are similar problems with a circular pool. The first one has a cannibal who is four times as fast, and a version designed to be (almost) as hard as possible has a cannibal who is 4.5 times as fast. Both are solvable, but you need to have a solution nearer to optimal for the last one of course. 

To find the optimal solution you will need a whole bunch of brain-stuff that I don't have.

Swim along a curved path.

Assuming the cannibal follows the movement "get as close as possible to the missionary," there's a rather neat solution. If we move a small amount towards one of the sides and wait, the cannibal will try to get as close as possible to us, which is in the centre of that side. We then reverse direction towards the opposite side, and the cannibal has to move through 2 sides of the square, whereas we now only have to move through half the square, letting us leave well before the cannibal arrives .

Yes, correct. Which means the problem is not as simple as I suggested earlier. Sorry! 

The most interesting question is what is the critical speed ratio? i.e. above this ratio and the cannibal feasts, below the ratio and the missionary survives.

if the missonary swims straight for the middle of a side opposite from the cannibal the cannibal would have to chase after him on the most effcient path.

(the missonary needs to swin a 1/2 length, the cannibal runs 1 1/2 which is exactly three times a 1/2 so he can't delay)

then if one third of the way to the side the missonary swims to the other sides midpoint opposite of the cannibal he should be able to escape.

(the cannibal will be 1/2 way done one side so he now has to run 2 lengths to get to the misonaryies escaping point. where as the missonary swims           (1/3^2 + 1/2^2)^1/2=0.60~ units of time where as the cannibal would have to run 2/3 units of time.) since .6 < 2/3 the missonary can escape assuming he can instantly submerge

Interesting. Personally I've been trying to figure this one out. It's from Mihai Suba's Dynamic Chess Strategy.

The above circle is a lake. Point G is a girl in a boat. Point B is a brute who wants to catch her. The brute can only run along the shoreline, while the girl can float with her boat all over the lake. While the girl goes through a distance of a radius, the brute may cover a semi-circumference. Once she reaches a point on the shore before the brute, she must be considered to have escaped (she runs more quickly than him on land). Find out how the girl escapes.