The Pidgeonhole Principle Adeel Khan (chess64) April 9, 2006 Sources: Mathematical Circles by Fomin, Genkin and Itenberg, Mathematical Excalibur, Volume 1, Issue 1 by Hong Kong University of Science and Technology, Problem Solving Strategies by Engel, and Mathematical Database at http://www.eng.mathdb.org/e main.html Problems. 1. There are 100 pidgeonholes. How many pidgeons must you have to guarantee that if all the pidgeons are put into a pidgeonhole, there is one pidgeonhole with at least 2 pidgeons? 2. There are 100 pidgeonholes. How many pidgeons must you have to guarantee that if all the pidgeons are put into a pidgeonhole, there is one pidgeonhole with at least 10 pidgeons? 3. (Mathematical Circles) One million pine trees grow in a forest. It is known that no pine tree has more than 600000 pine needles on it. Show that two pine trees in the forest must have the same number of pine needles. 4. (Johns Hopkins Math Tournament 2006) Find the smallest n so that no matter how n points are placed in a unit square, there exists a pair of points separated by a distance no greater than 2 √ 3 . 5. (Mathematical Excalibur) Suppose 51 numbers are chosen from 1, 2, 3,..., 99, 100. Show that there are two which do not have any common prime divisor. 6. (Problem Solving Strategies) A chessmaster has 77 days to prepare for a tournament. He wants to play at least one game per day, but not more than 132 games. Prove that there is a sequence of successive days in which he plays exactly 21 games. 7. (Mathematical Excalibur) Show that among any nine distinct real numbers there are two, say a and b, such that 0 < a - b 1+ ab < √ 2 - 1. Solutions. 1. If there were 100 pidgeons, there could be one pidgeon in each pidgeonhole, so the smallest number of pidgeons that works is 101 . In general, if there are n pidgeons and n + 1 pidgeonholes, there is one pidgeonhole with at least 2 pidgeons. 2. If there were 900 pidgeons, there could be nine pidgeons in each pidgeonhole. Therefore, if we have 901 pidgeons, there must be at least one pidgeonhole with at least 10 pidgeons. In general, If we have kn + 1 pidgeons and n pidgeonholes, one pidgeonhole must contain at least k + 1 pidgeons. This is the pidgeonhole principle. It is also called Dirichlet’s box principle, but pidgeonhole is easier to remember. 3. Consider the pidgeonholes to be the number of pine needles on the trees: 0, 1, 2,..., 600000, and the pidgeons are the trees. (If a tree is in the 1337th pidgeonhole, it has 1337 pine needles.) By the pidgeonhole principle, since there are more than 600001 pidgeons, there must be a pidgeonhole with at least 2 pidgeons in it, or at least 2 trees must have the same number of pine needles.
Sources: Mathematical Circles by Fomin, Genkin and Itenberg,
Mathematical Excalibur, Volume 1, Issue 1 byHong Kong University of
Science and Technology, Problem Solving Strategies by Engel, and
Mathematical Databaseat http://www.eng.mathdb.org/e main.html
Problems.
1. There are 100 pidgeonholes. How many pidgeons must you have
to guarantee that if all the pidgeons are putinto a pidgeonhole,
there is one pidgeonhole with at least 2 pidgeons?
2. There are 100 pidgeonholes. How many pidgeons must you have
to guarantee that if all the pidgeons are putinto a pidgeonhole,
there is one pidgeonhole with at least 10 pidgeons?
3. (Mathematical Circles) One million pine trees grow in a
forest. It is known that no pine tree has more than600000 pine
needles on it. Show that two pine trees in the forest must have the
same number of pine needles.
4. (Johns Hopkins Math Tournament 2006) Find the smallest n so
that no matter how n points are placed in aunit square, there
exists a pair of points separated by a distance no greater than
2
3.
5. (Mathematical Excalibur) Suppose 51 numbers are chosen from
1, 2, 3, . . . , 99, 100. Show that there are twowhich do not have
any common prime divisor.
6. (Problem Solving Strategies) A chessmaster has 77 days to
prepare for a tournament. He wants to play at leastone game per
day, but not more than 132 games. Prove that there is a sequence of
successive days in which heplays exactly 21 games.
7. (Mathematical Excalibur) Show that among any nine distinct
real numbers there are two, say a and b, suchthat
