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Pigeon hole principle


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Pigeon hole principle

  1. 1. J.M.S.S. Silva
  2. 2. Pigeon-hole Principle  If n (> m) pigeons are put into m pigeonholes, theres a hole with more than one pigeon.
  3. 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. 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. 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. 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. 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. 8. Problems On Divisibility Prove that there exists a multiple of 2009 whose decimal expansion contains only digits 1 and 0.
  9. 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. 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. 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. 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. 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. 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. 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. 16. References Lejeune_Dirichlet 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. 17. Thank You