These are just some hints.

1. A cask of kvas. Think fractions. Alternatively, what about the lowest common multiple of 10 and 14?

2. The twelve people. Can't help. Just give it a try.

3. A fake coin. This is all about a magic number three...

4. Dangerous crossing. When a slow creature, like C or D, walks together with a fast one (A or B), isn't this a waste of time?

5. The seven bridges. Put a point on each bank and a point on each island. Connect them by possible paths. Now, look if the number of paths coming together at those points are even or odd...

6. 64-2. Think of the colours!

7. Merchant's problem. Begin with a simpler one. You have only two weights which allow you to weigh 1,2,... pounds. What are they? What's the maximal number of amounts you can weigh with them?

8. A tricky catch. A famous physicist Paul Dirac is said to have solved the problem immediately. He said the answer was -2. Indeed, throw one fish away, -2-1=-3, find one third of what's left, (-3):3=-1, take this third away, -3-(-1)=-2, and there is again -2 fish in the bucket, for the other fishermen. Dirac's solution may not look too surprising, given that in 1930 he put forward a theory which predicted the existence of antimatter! (Paul Dirac won the 1933 Nobel prize in Physics. ) However, the problem has a true positive solution which you can find by doing things backwards.

9. Crossing ladders. This problem can be reduced to the equation (4-x)^-1/2+(9-x)^-1/2=1, where x is related to the alleyway width w as w=10x², which leads to a quartic equation for x.