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### 1 Introduction

The following is a variant of a well-known mathematical puzzle. A man is on a train from London to Cambridge and has a bottle of water with him. Prove that there is at least one moment on the journey when the volume of air in the bottle, as a fraction of the volume of the bottle itself, is exactly equal to the fraction of his journey that he has completed. (For instance, the bottle might be two fifths full, and therefore three fifths empty, at the precise moment when he is three fifths of the way from London to Cambridge. Note that we do not assume that the bottle is full at the start of the journey or empty at the end.)

The solution, if you have not seen this sort of question before, is surprisingly ...

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