Question

From a Course on the Introduction to Abstract Algebra

4. Use pigeonhole principle to show that ·Out of 15 people there are at least two that have their birthdays in the same month. ·Out of five points inside a square with the side length two there are at least two that are at most of distance of V2.

Answer 1

pigeonhole principle says that if n items (pigeons) are put into m boxes, where n > m, at least one box must contain more than one item (i) number of people (birthdays) n = 15 number of months (boxes) m = 12 means at least one month have birthdays of two people (ii) divide the square of side length 2 into four equal squares of side length 1 now if we start putting these 5 points (pigeons) into 4 smaller squares (boxes), we will have at least two points in same square (say A) maximum distance between these two points, = diagonal of smaller square A = squareroot 2 that means there are at least two points that are at most of distance of squareroot 2