# Tag Archives: seating

## Planes, manes and pigeonholes

All aboard?

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 $N+1$ objects to put in $N$ boxes then one of those boxes must contain at least two of the objects.

Or if you put $M$ pigeons in $N$ pigeonholes (or envelopes in pigeon-holes) then, if $M>N$, 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.