http://zetahype.wordpress.com/2008/06/23/making-it-count-the-pigeonhole-principle/ ...a godsend to the millions who look around on slow news days and wonder “Whats goin’ on..in mathematics these days?” Wed, 23 Jul 2008 01:21:16 +0000 http://wordpress.com/ hourly 1 http://zetahype.wordpress.com/2008/06/23/making-it-count-the-pigeonhole-principle/#comment-31 Making It Count: Counting the complement « The Mumbling Mathematician Wed, 23 Jul 2008 01:21:16 +0000 http://zetahype.wordpress.com/?p=5#comment-31 [...] under: Uncategorized — zetahype @ 1:21 am Here’s the much delayed second part of the Making It Count series which was launched with much fanfare a month ago. The counting technique I will present is [...] [...] under: Uncategorized — zetahype @ 1:21 am Here’s the much delayed second part of the Making It Count series which was launched with much fanfare a month ago. The counting technique I will present is [...]

]]> http://zetahype.wordpress.com/2008/06/23/making-it-count-the-pigeonhole-principle/#comment-20 zetahype Tue, 24 Jun 2008 13:41:51 +0000 http://zetahype.wordpress.com/?p=5#comment-20 umm..i did..the party of 6 problem is the one I solved in the second example. umm..i did..the party of 6 problem is the one I solved in the second example.

]]> http://zetahype.wordpress.com/2008/06/23/making-it-count-the-pigeonhole-principle/#comment-19 Meech Tue, 24 Jun 2008 13:39:18 +0000 http://zetahype.wordpress.com/?p=5#comment-19 You were talking about Ramsey numbers but didnt introduce the Party Problem? That I would have thought would be the party piece. You were talking about Ramsey numbers but didnt introduce the Party Problem? That I would have thought would be the party piece.

]]> http://zetahype.wordpress.com/2008/06/23/making-it-count-the-pigeonhole-principle/#comment-14 zetahype Tue, 24 Jun 2008 08:03:02 +0000 http://zetahype.wordpress.com/?p=5#comment-14 You are right. I have unintentionally used a generalised version of the pigeonhole principle in the second problem: If n objects are distributed among m boxes then at least one box will have [n/m] objects, where [*] is the smallest integer greater than or equal to *. In the problem we want to distribute 5 edges between two colours, so at least one colour will have [5/2] = 3 edges. Note that this version of the pigeonhole principle allows us to introduce probabilities. If an object has an even chance of being alloted one of the m boxes (uniform probability 1/m) we ask what the probability of one of the boxes having more than one object is. Ofcourse, if n>m, the probability is 1 but it is not hard to compute the probability even otherwise. The "probabilistic pigeonhole principle" is a useful tool in solving density problems in number theory. I am sure we will encounter it later. You are right. I have unintentionally used a generalised version of the pigeonhole principle in the second problem: If n objects are distributed among m boxes then at least one box will have [n/m] objects, where [*] is the smallest integer greater than or equal to *.
In the problem we want to distribute 5 edges between two colours, so at least one colour will have [5/2] = 3 edges.
Note that this version of the pigeonhole principle allows us to introduce probabilities. If an object has an even chance of being alloted one of the m boxes (uniform probability 1/m) we ask what the probability of one of the boxes having more than one object is. Ofcourse, if n>m, the probability is 1 but it is not hard to compute the probability even otherwise. The “probabilistic pigeonhole principle” is a useful tool in solving density problems in number theory. I am sure we will encounter it later.

]]> http://zetahype.wordpress.com/2008/06/23/making-it-count-the-pigeonhole-principle/#comment-12 Anonymous Mon, 23 Jun 2008 21:49:06 +0000 http://zetahype.wordpress.com/?p=5#comment-12 I can under stand how you use the principle to the first problem but it is not obvious in the second one. It does not appear to a situation where one can use directly the pigeonhole principle that you stated? I can under stand how you use the principle to the first problem but it is not obvious in the second one. It does not appear to a situation where one can use directly the pigeonhole principle that you stated?

]]> http://zetahype.wordpress.com/2008/06/23/making-it-count-the-pigeonhole-principle/#comment-9 zetahype Mon, 23 Jun 2008 20:03:36 +0000 http://zetahype.wordpress.com/?p=5#comment-9 Well..if you must know. I chose wordpress over blogspot because I heard it allows inline Latex commands which would help me type the maths bits. Turns out it supports latex at a very basic level..so I make pdf files using latex and capture them as images and attach them at the right places in my post. I'm not terribly happy with the outcome but I'm trying different things and will make improvements. Well..if you must know. I chose wordpress over blogspot because I heard it allows inline Latex commands which would help me type the maths bits. Turns out it supports latex at a very basic level..so I make pdf files using latex and capture them as images and attach them at the right places in my post. I’m not terribly happy with the outcome but I’m trying different things and will make improvements.

]]> http://zetahype.wordpress.com/2008/06/23/making-it-count-the-pigeonhole-principle/#comment-8 Tobias Mon, 23 Jun 2008 19:14:18 +0000 http://zetahype.wordpress.com/?p=5#comment-8 why the different fonts? why the different fonts?

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