Question

Pignohole principal explain

Answer 1

In mathematics, the pigeonhole principle states that if items are put into containers, with, then at least one container must contain more than one item.



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Answer 2

Answer: The pigeonhole principle is one of the simplest but most useful ideas in mathematics, and can rescue us here. A basic version says that if (N+1) pigeons occupy N holes, then some hole must have at least 2 pigeons. Thus if 5 pigeons occupy 4 holes, then there must be some hole with at least 2 pigeons. It is easy to see why: otherwise, each hole as at most 1 pigeon and the total number of pigeons couldn't be more than 4. (This proof shows that it does not even matter if the holes overlap so that a single pigeon occupies 2 holes.)

So, if I divide up the square into 4 smaller squares by cutting through center, then by the pigeonhole principle, for any configuration of 5 points, one of these smaller squares must contain two points. But the diameter of the smaller square is Sqrt[2]/2, so these two points must be closer than 3/4, as claimed. The pigeonhole principle made what seemed like a slippery argument airtight.