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- 1. J.M.S.S. Silva
- 2. Pigeon-hole Principle If n (> m) pigeons are put into m pigeonholes, theres a hole with more than one pigeon.
- 3. Alternative Forms• If n objects are to be allocated to m containers, then at least one container must hold at least ceil(n/m) objects.• For any finite set A , there does not exist a bijection between A and a proper subset of A .• Let |A| denote the number of elements in a finite set A. For two finite sets A and B, there exists a 1-1 correspondence f: A- >B iff |A| = |B|.
- 4. History The first statement of the principle is believed to have been made by Dirichlet in 1834 under the name Schubfachprinzip ("drawer principle" or "shelf principle") Also known as Dirichlets box (or drawer) principle
- 5. General Problems There 750 students in the a batch at UOM. Prove that at least 3 of them have their birthdays on the same date ? ○ 366 * 2= 732 < 750 ○ Thus at least 3 students have the birthday on the same date.
- 6. Problems on Relations There are 50 people in a room. some of them are friends. If A is a friend of B then B is also a friend of A. Prove that there are two persons in the room who have a same number of friends. In league T20 tournament of 16 cricket teams, every two teams have to meet in a game. Prove that at any time there are two teams which played equal number of matches.
- 7. Solution Case 1 Case 2 There exists a person There does not exists with 49 friends (he is a a person with 49 friend of all other people) friends Then there cannot be a The no of friends vary person with 0 friends between 0 – 48 . The no of friends vary There are 50 people between 1 – 49 . and only 49 values. There are 50 people and only 49 values.
- 8. Problems On Divisibility Prove that there exists a multiple of 2009 whose decimal expansion contains only digits 1 and 0.
- 9. Answer Consider 2010 numbers - 1,11,111,1111, … ,1111…111. Each of these numbers produce one of 2009 remainders - 0,1,2,3 ,…,2008 We have 2010 numbers and 2009 remainders By pigeon-hole principle some two numbers have the same remainder .Let those 2 numbers be A and B (A>B) Consider A-B. which is a multiple of 2009. - In the form of 11…1100…000
- 10. Problems On Divisibility Prove that of any 52 natural numbers one can find two numbers n and m such that either their sum m+n or difference m-n is divisible by 100. Consider sets {0},{1,99},{2,98}….{49,51},{50} There are 51 sets By pigeon hole principle at least 1 set should have 2 members If we consider any set above if they have 2 members in the set, m+n or m-n is divisible by 100
- 11. Problems on Geometry 51 points are placed, in a random way, into a square of side 1 unit. Can we prove that 3 of these points can be covered by a circle of radius 1/7 units ?
- 12. Answer To prove the result, we may divide the square into 25 equal smaller squares of side 1/5 units each. Then by the Pigeonhole Principle, at least one of these small squares should contain at least 3 points. Otherwise, each of the small squares will contain 2 or less points which will then mean that the total number of points will be less than 50 , which is a contradiction to the fact that we have 51 points in the first case !
- 13. Answer - continue Now the circle circumvented around the particular square with the three points inside should have Radius=Sqrt(1/100+1/100 ) =Sqrt(1/50) <Sqrt(1/49)=1/7 1/10
- 14. Applications Lossless data compression cannot guarantee compression for all data input files. The pigeonhole principle often arises in computer science. For example, collisions are inevitable in a hash table because the number of possible keys exceeds the number of indices in the array In probability theory, the birthday problem, or birthday paradox pertains to the probability that in a set of randomly chosen people some pair of them will have the same birthday
- 15. Applications The proof of Chinese Remainder Theorem is based on pigeon-hole principle Let m and n be relatively prime positive Integers. Then the system: x = a (mod m) x = b (mod n) has a solution.
- 16. References http://en.wikipedia.org/wiki/Pigeonhole_principle http://en.wikipedia.org/wiki/Johann_Peter_Gustav_ Lejeune_Dirichlet http://en.wikipedia.org/wiki/Lossless_data_compre ssion#Limitations Article on "What is Pigeonhole Principle?" by Alexandre V. Borovik, Elena V. Bessonova. Article on "Applications of the Pigeonhole Principle" by Edwin Kwek Swee Hee ,Huang Meiizhuo ,Koh Chan Swee ,Heng Wee Kuan , River Valley High School
- 17. Thank You

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