MIT6_042JF10_assn02

MIT6_042JF10_assn02 - 6.042/18.062J Mathematics for...

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6.042/18.062J Mathematics for Computer Science September 14, 2010 Tom Leighton and Marten van Dijk Problem Set 2 Problem 1. [12 points] DeFne a 3-chain to be a (not necessarily contiguous) subsequence of three integers, which is either monotonically increasing or monotonically decreasing. We will show here that any sequence of Fve distinct integers will contain a 3-chain . Write the sequence as a 1 ,a 2 ,a 3 ,a 4 ,a 5 . Note that a monotonically increasing sequences is one in which each term is greater than or equal to the previous term. Similarly, a monotonically decreasing sequence is one in which each term is less than or equal to the previous term. Lastly, a subsequence is a sequence derived from the original sequence by deleting some elements without changing the location of the remaining elements. (a) [4 pts] Assume that a 1 < a 2 . Show that if there is no 3-chain in our sequence, then a 3 must be less than a 1 . (Hint: consider a 4 !) (b) [2 pts] Using the previous part, show that if a 1 < a 2 and there is no 3-chain in our sequence, then a 3 < a 4 < a 2 . (c) [2 pts] Assuming that a 1 < a 2 and a 3 < a 4 < a 2 , show that any value of a 5 must result in a 3-chain . (d) [4 pts] Using the previous parts, prove by contradiction that any sequence of Fve distinct integers must contain a 3-chain . Problem 2. [8 points] Prove by either the Well Ordering Principle or induction that for all nonnegative integers, n : n ( ) 2 i 3 = n ( n + 1) . (1) 2 i =0 Problem 3. [25 points] The following problem is fairly tough until you hear a certain one-word clue. The solution is elegant but is slightly tricky, so don’t hesitate to ask for hints! During 6.042, the students are sitting in an n × n grid. A sudden outbreak of beaver ﬂu (a rare variant of bird ﬂu that lasts forever; symptoms include yearning for problem sets and craving for ice cream study sessions) causes some students to get infected. Here is an example where n = 6 and infected students are marked × .

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2 Problem Set 2 × × × × × × × × Now the infection begins to spread every minute (in discrete time-steps). Two students are considered adjacent if they share an edge (i.e.,
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MIT6_042JF10_assn02 - 6.042/18.062J Mathematics for...

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