Number 6: Differences between Numbers in Pigeon Holes

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Homework Help Overview

The discussion revolves around a combinatorial problem related to the pigeonhole principle, specifically focusing on the differences between numbers in defined pairs. The original poster is attempting to analyze a specific case involving ordered pairs and their implications in the context of the problem statement.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • The original poster presents a reasoning process involving the distribution of numbers into pigeonholes and questions the validity of their logic regarding the differences between numbers. Other participants express uncertainty and seek clarification on the problem's structure and the original poster's reasoning.

Discussion Status

Contextual Notes

Participants are navigating the constraints of the problem as presented, with some suggesting the possibility of an error in the problem itself. The original poster's approach involves specific assumptions about the distribution of numbers that are under scrutiny.

ehrenfest
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Homework Statement


http://math.stanford.edu/~vakil/putnam07/07putnam1.pdf
I am working on number 6.

Consider the the ordered pairs (1,12),(2,13),...,(89,100). There are 89 of them. These are the pigeon holes.

There are 55 numbers between 1 to 100 and each of them flies to one or two pigeon holes. So split all of the pigeons not equal to 1 and 100 in half to get at least 108 pigeons. Then we have 108 pigeons flying to 89 holes. So there must be two numbers that differ by 11.

What is wrong with my logic?


Homework Equations





The Attempt at a Solution

 
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Maybe there is something wrong with the problem instead...
 
anyone?

This problem is really bugging me!
 
please--I am probably just making a stupid mistake and I want to fix it!
 

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