Impossible Puzzles

A farmer has to row a wolf, a goat and a cabbage across a river, in a little boat that holds him and just one of them. Left alone without him, the wolf will eat the goat, and the goat will gobble up the cabbage. How can he get everything across safely, and in how few trips?

You have nine balls that look exactly the same, but one of them is a tiny bit heavier. You have a balance scale but no weights. How few times can you use the scale and still be sure to find the heavy one?

Downstairs there are three light switches, and only one of them works the lamp upstairs, which you can't see from down there. You're allowed to walk upstairs just once. Can you work out which switch it is?

Four friends reach a rickety rope bridge in the middle of the night, with just one torch between them. The bridge holds two people at a time, nobody may cross without the torch, and a pair walks at the slower one's speed. Alone they take 1, 2, 5 and 10 minutes to cross, so how fast can all four get over?

A broken calculator has only one working number key, the four, and it dies after you press it four times. You still have + − × ÷, brackets, √ and !, and you may glue two 4s into 44. Can you make it show 7, and in how many different ways?

A giant will only let you past if you bring him exactly four litres of water. You have a tap, a three-litre jug and a five-litre jug, and neither jug has any markings on it. How can you measure exactly four litres, and in how few steps?

A chess queen is the most powerful piece on the board: she attacks every square along her row, her column and both diagonals, all the way to the edge. Can you squeeze four queens onto a tiny board, four squares by four, so that none of them is under attack? And how many different ways are there?

On a strange island, knights always tell the truth, knaves always lie, and you can't tell them apart by looking. Two islanders, A and B, stand on the path to the treasure, and A grins: "We are both knaves." B says nothing at all, so can you still work out who is who?

Nine dots sit in a neat square, three rows of three. Join all nine with four straight lines drawn in one go, without lifting your pencil from the paper. People have stared at this for hours, so is it even possible?

A hundred coins lie on the table, and exactly ten of them show heads. You're blindfolded and can't feel which side is up, but you may move and flip any coins you like. Can you split them into two piles with the same number of heads?

Two trains head toward each other on the same long track, and a fly sits on the nose of one of them. The fly zooms to the other train, turns around, zooms back, and keeps bouncing between them until the trains meet. With all those trips getting shorter and shorter, how far does the fly travel in total?

I'm thinking of a whole number between one and a hundred. Each time you guess, I'll only tell you "higher" or "lower". How many guesses do you need to be sure of catching my number, even on your unluckiest day?