Solution to: Cork in the Canal
If you have written down a full paper of mathematical formulas, you have been thinking too complicatedly...
It is obvious that the cork does not move relatively to the water (i.e. it has the same speed as the water). Therefore, if the swimmer is swimming away from the cork for half an hour (upstream), it will take him another half hour to swim back to the cork again. Because the swimmer is swimming with constant effort, his speed is constant relatively to the speed of the water. You can look at it as if the water in the river does not move, the cork does not move, and the swimmer swims a certain time away from the cork and then back. Consequently, in that one hour time, the cork has floated from 1 kilometer up stream to the bridge.
Conclusion: The water in the canal flows at a speed of 1 km/h.