I was recently travelling on a budget flight with a friend, with no assigned seating. Walking up to the queue, we wondered whether we would be able to sit together. As people joined behind us, my friend, who whole-heartedly detests maths in any disguise, pointed out that we certainly could. To be so full-up to mean that we couldn’t, each row of three would need at least two people sitting in it. Since we could see from the queue that we had more than a third of the passengers still left to board after us, then as long as we weren’t trampled in a boarding stampede, we could certainly sit together.
This was actually a subconscious application of the pigeonhole principle, one of the most intuitive theorems that mathematicians use. It states that if you have objects to put in boxes then one of those boxes must contain at least two of the objects.
Or if you put pigeons in pigeonholes (or envelopes in pigeon-holes) then, if , at least one of the holes has at least two pigeons. So far, the type of maths you wouldn’t be shocked to see explained on television by a primary-coloured puppet.