More puzzles

It’s sometimes hard to define whether a puzzle can be classified as “mathematical” or not. They certainly are if they require mathematical techniques or knowledge to solve them. Today’s puzzles don’t, but I would still class them as mathematical: firstly, because they use numbers or geometrical space or logic; and secondly because they tickle the mathematical bits of your brain and so, in a very small way, hone your skills.

1. The two jugs

Solve this in under 30 seconds – excellent. In under a minute – not too bad.

Given a water supply and two jugs, one of which holds exactly 3 litres and the other exactly 5 litres, how can you measure out exactly 1 litre of water.

2.  The nine dots

The diagram shows a 3 x 3 grid of nine dots. Without taking your pencil off the paper, can you draw 4 straight lines such that all the dots are connected.

3.  The farmer’s daughter

Based on an original puzzle in Edward de Bono’s ground breaking book on lateral thinking.

Long ago, in a remote village, a farmer owed a large amount of money to the village moneylender, and was unable to pay back the debt. The moneylender suggested that, instead of the money, he would be happy to marry the farmer’s daughter instead. The horrified farmer refused the offer. So the money lender suggested a game of chance. “I’ll put a black pebble and white pebble in my bag, and ask your daughter to pick one out. If she refuses, you will go to jail. If it’s black, I will forgive you your debt, but take your daughter in marriage. If it’s white I will forget the debt, and your daughter will be free.” The path they were standing on was made of black and white pebbles, and the moneylender picked up two and put them in his bag – but the daughter managed to spot that he had picked up two black pebbles. What is her best course of action?

Answers

1.  It’s tempting to fill the 5 litre jug and pour it into the 3 litre, leaving 2 litres – and trying to find a way to split that in half. But the solution is to fill the 3 litre, pour it into the 5 litre; then fill the 3 litre again, and pour as much as you can into the 5 litre – leaving exactly 1 litre left behind.
2. Our minds tend to try and keep us within the limits of the diagram. But to solve this one, the lines need to extend outside the grid:
   
3. Most of us in this situation would presumably point out the moneylender’s “mistake”, and get him to pick up a black and a white pebble, thus giving us a half chance of success. But the daughter used lateral thinking to turn the situation to her advantage. She took a pebble out of the bag, then “accidentally” dropped it onto the path. Pointing out that it was now impossible to see what it was, mixed up with all the other pebbles, she took the second pebble out of the bag. It was black of course, so “the first one I picked out must have been the white one.” The moneylender could hardly admit his dishonesty, so the farmer and his daughter were free.