MIT6_042JS10_lec27_prob

MIT6_042JS10_lec27_prob - Massachusetts Institute of...

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Unformatted text preview: Massachusetts Institute of Technology 6.042J/18.062J, Spring 10 : Mathematics for Computer Science April 12 Prof. Albert R. Meyer revised April 6, 2010, 982 minutes In-Class Problems Week 10, Mon. Problem 1. Solve the following problems using the pigeonhole principle. For each problem, try to identify the pigeons , the pigeonholes , and a rule assigning each pigeon to a pigeonhole. (a) Every MIT ID number starts with a 9 (we think). Suppose that each of the 75 students in 6.042 sums the nine digits of his or her ID number. Explain why two people must arrive at the same sum. (b) In every set of 100 integers, there exist two whose difference is a multiple of 37. (c) For any five points inside a unit square (not on the boundary), there are two points at distance less than 1 / 2 . (d) Show that if n + 1 numbers are selected from { 1 , 2 , 3 ,..., 2 n } , two must be consecutive, that is, equal to k and k + 1 for some k ....
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MIT6_042JS10_lec27_prob - Massachusetts Institute of...

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