IB Maths Challenge: Can you solve this puzzle?
All you IB Maths wizards will enjoy this puzzle set by our resident IB Maths expert and study guide author, Ian Lucas. Part of our series of subject challenges, use your IB maths skills and try not to tie your brain in knots, but most importantly have a bit of study fun. Enjoy!
Mathematical puzzling may be seen as a recreational indulgence with no connection to "real" mathematics – just as recreation rarely feels connected to maths! In fact, pretty well every puzzle can illuminate an area of mathematics, whether number, algebra, probability, geometry – and everything inbetween. Mathematical puzzles have been around for at least 1000 years, and many have been posed where the solution has led to new discoveries. For example, the modern study of probability came about from a discussion between French mathematicians Fermat and Pascal, trying to solve a puzzle set by gambler Chevalier de Méré. And the solution by Euler of the famous "seven bridges of Königsberg problem" led to Graph Theory and Topology. 1
Here is a wonderful example of a puzzle with an unexpected, counter-intuitive solution.
Start with two identical glasses, one containing 50ml wine and the other 50ml 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: pure 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 surprising 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.
Suppose 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. Now 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.
1 You will find a description of the Königsberg Bridges problem in my revision guide for Mathematics: Applications and Interpretation HL.
If you enjoyed that challenge, be sure to check out our subject pages for IB Maths Analysis and Approaches as well as IB Maths Applications and Interpretation for more helpful articles, study tools, and brain teasers provided by Ian and our other contributors.
