Pigeonhole principle strong form – theorem: let q 1 , q 2 , , q n be positive integers if q 1 + q 2 + + q n − n + 1 objects are put into n boxes, then either the 1st box contains at least q 1 objects, or the 2nd box contains at least q 2 objects, , the nth box contains at least q n objects. The pigeonhole principle 1 pigeonhole principle: simple form theorem 11 if n+1 objects are put into n boxes, then at least one box contains two or more objects proof trivial example 11 among 13 people there are two who have their birthdays in the same month example 12 there are n married couples. By the pigeonhole principle, two or more must belong to the same suit 8 if you have 10 black socks and 10 white socks, and you are picking socks randomly, you will only need to pick three to find a matching pair the three socks can be one of two colors by the pigeonhole principle, at least two must be of the same color.
A new proof of the weak pigeonhole principle alexis maciel department of mathematics and computer science clarkson university potsdam, ny 13699-5815 usa complexity of the weaker forms of the pigeonhole principle, which we define as those for which n the weak pigeonhole principle is connected to how much.
Pigeonhole principle (general form): if more than \(k \cdot n\) objects are placed into \(n\) boxes then at least one box must contain more than \( k \) objects the case of \( k = 1 \) corresponds to the naive pigeonhole principle stated earlier.
Peng shi, duke university the pigeonhole principle, simple but immensely powerful 4/13 how to apply the pigeonhole principle in general, it may not be so clear how to apply the principle.