Could a fish in a sealed ball, move the ball? If you had a glass ball filled with water, completely sealed and containing a fish, could the fish move the ball?
 A: Yes, with gravity and a generous definition of "moving".. it would be the same principle as the toys where you can control a sphere using a radio control (or using your iphone). The fish swims along the edge and gravity pulls it back down, which starts a rotation of the water and by friction to the sphere starts the rolling motion of the sphere on the ground or other surface. Obviously the water/sphere friction will probably be miniscule, but at least it is possible in theory :)
A follow-up question would of course be if it's possible to move a hermetically sealed sphere freefloating in vacuum and without gravity or any other appreciable fields intersecting it. If you solve this, I'm pretty sure NASA will want to talk to you (or the fish)!
A: Yes, the fish could (theoretically at least) move the ball.
A quick way to visualize why this is so is to imagine yourself sealed in a 7-foot diameter ball of air.  You could easily move the ball by walking in the direction you wanted it to go.  Of course you would have a much easier time of it, since your density is much higher relative to the surrounding fluid (air).  The fish's density is much closer to the surrounding fluid (water), so it would have a harder time of it.
A: I think the accepted answer is flawed as it assumes the fish is a different density to the water. But if fishes were heavier than water then they would sink, while if fishes were lighter than water they would float, and clearly neither of these things happen!
In fact, fishes have an organ called the swim bladder which is used to keep the fish at the exact same density as the surrounding water, so that there is no buoyancy to fight against. If a fish wants to rise it lowers its density slightly by filling the bladder with gas. This increases the fish's volume but doesn't change the mass by much, so has the effect of decreasing its density. Conversely, if the fish wants to descend it increases its density by releasing some gas and decreasing the size of the bladder.
So the fish won't budge the centre of mass by swimming as normal. But it can still move the ball if it wants... Like this.
It first releases all the gas from it's swim bladder so that it becomes more dense than the water and sinks to base of the ball, where it is now stuck like a terrestrial being. The problem is now equivalent to that of a hamster in a ball. The fish can drag itself up the side of the ball with its fins, just as a hamster would, causing to ball to roll forward.
An alternative strategy is for the fish to expand its swim bladder instead to float to the top of the ball, and then swim forward. But note that this would in fact move the ball backwards as the fish is now less dense than the surrounding water, so it would be harder for the fish to see where it was going.
EDIT: On further reflection, I note that if the ball is indeed completely filled with water at the start of the experiment, then this second strategy is likely to be difficult if not impossible for the fish. This is because it will struggle to increase its volume due to the incompressibility of water. Thus the first strategy of making itself more dense by releasing gas from the swim bladder is likely its only viable option.
A: Yes, the fish can move the sealed ball, but the centre of mass of the fish and the ball will not move.
