1. A certain convention numbered 100 politicians. Each politician was either honest or dishonest. We are given the following facts:
i) At least one of the politicians is honest.
ii) Given any two politicians, at least one of them is dishonest.

From this, how many of the politicians are honest, and how many are dishonest?
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2. A man travelled upstream from his dock in a motor-boat. As he passed under a bridge one mile from the dock his emergency paddle fell overboard, a loss which he didn't discover until ten minutes later. He immediately turned and went back to retrieve the paddle, and did so directly opposite his dock. If he travelled at a constant speed through the water, and lost no appreciable time turning round, what was the rate of the river's current?
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1. Only one of the politicians is honest. If even one more was honest, there would a chance to pick a set of honest politicians. I also may point out the premises here are fictitious, because there's no such thing as an honest politician.

2. I despise math problems disguised as physics, but I'm gonna guess the rate of the current was zero, since the man's speed was constant.

Touché! Although it is so true!

Q1. I'm inclined to agree with you about politicians' honesty, Gil Galad, but these little puzzles don't necessarily relate to the real world.
Q2. If the current's rate was zero he would have recovered the paddle in the same spot as he dropped it; that is, under the bridge.

Q1. We are told that at least one is honest. So let's pick any one honest politician (call him Frank). Now pick any one of the remaining 99, call him Fred. By condition ii), at least one of the two men - Frank, Fred - is dishonest. It isn't Frank, so it must be Fred. And since Fred arbitrarily represents any of the other 99 politicians, then each of the 99 must be dishonest. Only one therefore - Frank - is honest.


Q2. The paddle does not move relative to the water; and the boat has a constant speed relative to the water. No matter what the boat's speed it will take as long (10 minutes) to get back to the paddle as it did to get away from it. Since the paddle (and the water) moved 1 mile in that total period of 20 minutes, the current must have been 3 mph.