Lecture 4 21 127 Concepts of Math 09 05 2012 Administrivia EXCEL Tutoring Work in groups get help with course material from a group tutor Very helpful Sign up in Cyert Hall B5 Walk in Tutoring Tutoring for 21 127 will be available on Wednesday nights from 8 30 11 00 p m in both the Donner Reading Room and the Mudge Reading Room As usual the service is free of charge and students can drop in for help without an appointment Inductive Arguments continued Remember that we are working on inductive arguments where some instance of a fact depends on previous instances We want to emphasize that this is not always an algebraic fact or a formula Sometimes it s a geometric fact or something more abstract One example is the Takeaway game you played in recitation This is also explained in Chapter 2 of the textbook Regions Consider n lines on an infinite plane two dimensional surface such that no two lines are parallel and no more than two lines intersect at one point How many distinct regions do the lines create Start with n 1 Obviously R 1 2 Next n 2 We find R 2 4 Next n 3 We find R 3 7 Now let s try to find a pattern Suppose we already know R n Can we find R n 1 in terms of that Take n lines We know they make R n regions What happens when we add a new line It is not parallel to any of the existing lines and it does not intersect any existing intersection points Thus it must intersect every one of the n existing lines This creates n new intersection points Look at the segments these points create There are n 1 such segments Note two of them are infinite rays while the rest are finite segments Every one of those segments splits an existing region into two new regions Thus R n 1 R n n 1 This gives us a recursive formula Solving the recursion we find that R n 1 R n R n n 1 R n 2 n 1 n R 2 3 R 1 2 n n n 1 n 1 n 1 n 1 n 1 Since we know R 1 2 we can say R n 1 2 2 3 n n 1 2 n 1 X k 1 1 k 1 1 n 1 X k 1 k and this is a sum we have investigated before Recall that R n 1 1 Pn k 1 k n n 1 2 Therefore n 1 n 2 2 One final simplification we would like to make is to replace n 1 with n throughout the equation because it makes more sense to have an expression for R n Note For what values of n is this valid R n 1 n n 1 2 Tilings Given a 2 n array of squares how many different ways can we tile the array with dominoes Start with n 1 Obviously T 1 1 Next n 2 We have T 2 2 Next n 3 We can draw all the cases and find T 3 3 Next n 4 We can draw all the cases and find T 4 5 At this point we don t want to draw any more Let s find a pattern Suppose we know T n Can we identify T n 1 from that Consider a board of size 2 n 1 How many tilings are there We know that some of the tilings we want to count are made up of a tiling of a 2 n board with one extra domino tacked on vertically Is that all of them Nope Some of the tilings of a 2 n 1 board have two horizontal dominoes in the last two columns How many such tilings are there That s right T n 1 Do we know that value Well we need to suppose that we do If so we can write T n 1 T n T n 1 Can we get this recursive formula started and off the ground We need to know two initial cases Knowing T 1 and T 2 tells us T 3 Then T 2 and T 3 tell us T 4 And so on This is the familiar Fibonacci Sequence It does have a closed form solution 1 T n 5 n 1 1 5 1 2 5 n 1 1 5 2 Games Let s play a game We ll call it Twenty Sum Rules There are two players On each player s turn they must add 1 or 2 or 3 to the previous total The starting point is 0 The winner is the one who says 20 Let s play together Who wins Does it matter who goes first What is a winning strategy This is a mathematical game in the sense that we can analyze it logically It s not necessarily a fun game in real life though Notice that Player 2 is at an advantage The goal number 20 is a multiple of 4 No matter what Player 1 does Player 2 can return the total to a multiple of 4 This will continue until they reach 20 What we are actually arguing for is the following claim 2 If n is a multiple of 4 then Player 2 can get to n during the playing of the game What does it mean to be a multiple of 4 We are really making an inductive argument on a variable k For every natural number k Player 2 has a winning strategy when playing the Sum Game to 4k This fits more nicely into our usual consideration of induction where k increases by one each time For the initial case k 1 we can just enumerate all three cases no matter what Player 1 does 1 or 2 or 3 say he plays x Player 2 responds with 4 x Inductively we suppose that Player 2 wins when playing to 4k When they play to 4 k 1 4k 4 we know that Player 2 will get to 4k Then Player 1 says x either 1 or 2 or 3 so the total is 4k x Then Player 2 says 4 x so the total is 4k 4 Player 2 wins 3
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