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Pigeonhole Principle

Pigeonhole Principle. Cihat İmamoğlu 24.11.2009 / CMPE220 / Haluk Bingöl. What is?. A basic, almost obvious mathematical principle that states for n items and m boxes, if n > m, then at least one of the boxes must contain more than one item.

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Pigeonhole Principle

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  1. Pigeonhole Principle Cihat İmamoğlu 24.11.2009 / CMPE220 / Haluk Bingöl

  2. What is? • A basic, almost obvious mathematical principle that states for n items and m boxes, if n > m, then at least one of the boxes must contain more than one item. • A fancier explanation: there does not exist an injective function on finite sets whose codomain is smaller than its domain.

  3. What is? • Generalized pigeonhole principle: if n objects are to be put into m containers, then at least one container must hold no fewer than ceil(n / m) objects.

  4. What for? • One of the basic combinatorial principles • Used extensively in mathematical proofs • Arises frequently in computer science • Hash table collisions • Parallel computing • ...

  5. Example 1 • The claim: In this class, there are at least two people who were born on the same month. • The proof: Number of people currently in the class is (at least “has always been”) more than 12, number of months. Therefore, by the pigeonhole principle, the claim is valid.

  6. Example 2 • The claim: For every positive integer n, there is a multiple of n that has only zeros and ones in its decimal expansion. • The proof: Consider the n+1 integers: 1, 11, 111, ..., 11..1 (where the last integer in this list is the integer with n+1 ones). Because there exists n possible remainders in division by n, and there are n+1 elements in given number list, then, by pigeonhole principle, there must be at least two that gives the same remainder. The difference of these will be divisible by n, and will consist only of zeros and ones.

  7. Example 3 • The claim: Consider an equilateral triangle whose sides are all 2. Among five points chosen inside the triangle, there exists at least two with distance smaller than or equal to 1. • The proof: Let’s divide the bigger triangle into 4 smaller equal triangles whose sides are all 1. Then at least two of the chosen points must lie in the same subtriangle, where no two points can be farther than 1.

  8. Exercises • Show that for any N positive integers, the sum of some of these integers (perhaps one of the numbers itself) is divisible by N. • Show that among any n + 1 positive integers not exceeding 2n, there must be an integer that divides one of the other integers. • Show that there exists a power of 7 whose decimal representation ends with 0001.

  9. “Bir koltuğa iki karpuz sığmaz.”

  10. References • Matematik Dünyası, 2003 Kış, “Çekmece ya da Güvercin Yuvası İlkesi”, Haluk Oral • "Pigeonhole principle." Wikipedia, The Free Encyclopedia. 17 Nov 2009, 23:56 UTC. 21 Nov 2009 <http://en.wikipedia.org/w/index.php?title=Pigeonhole_principle&oldid=326439761>. • Discrete Mathematics and Its Applications, 6th Ed., Kenneth H. Rosen • Sonlu Matematik, Refail Alizade, Ünal Ufuktepe

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