This is an old one but is a wonderful example of a puzzle with an unexpected, counter-intuitive solution.
‘Two 50ml glasses are filled to the brim, one with wine and the other with water. A teaspoon (5ml) of water is transferred to the wine glass and thoroughly mixed in. Then a teaspoon of the mixture is transferred back to the water glass. Question: is there now more water in the wine, or wine in the water?’
The more you think about it, the more you can be convinced either way! At first you think there is more water in the wine: neat water was transferred to the wine, but a mixture transferred back. On the other hand, nearly all the spoonful being transferred back is wine and some water is being left behind.
The astonishing solution is that the amount of water in the wine is in fact the same as the amount of wine in the water; not ‘just about’ the same, but exactly the same. The simple way to understand this is to consider the situation at the end of the two transfers. Each glass now has exactly 50ml of liquid, the same as at the start. So, however much wine is in the water glass must be matched by the amount of water in the wine glass. Let’s try it with some numbers.
a) 5ml of water is transferred to the wine glass. There is now 45ml of water in the wine glass, 50ml wine + 5ml water in the wine glass. Let’s suppose that when we take the spoonful of the mixture it contains 4ml wine and 1ml water. When that is transferred back to the water glass, there will then be 46ml of wine in the wine glass + 4ml water; and 46ml of water in the water glass + 4ml wine.
Does it matter what size the glasses are? No, and in fact they could be different size glasses to start with. Does it matter how big the spoon is? No, you can transfer any amount, as long as you then transfer the same amount back. Does it matter how much the mixture is stirred? Not at all – it doesn’t even have to be stirred at all.
While I’m thinking about glasses, how about this one:
‘There are six glasses in a row, the first three are full, the next three are empty. How can you move just one glass and end up with them alternately full and empty?’
Answer: lift the second glass up, empty it into the fifth, and put it back again!
Previous puzzle blogs: