Lecture 36 21 127 Concepts of Math 12 03 2012 Administrivia Homework 10 is posted Due on Thursday Extra Credit projects are due on Friday Pigeonhole Principle Any problem that asks you to prove there must be some objects with a certain property or to guarantee the existence of something that probably indicates an application of the Pigeonhole Principle To address such a problem try to identify the pigeons we are placing and the boxes into which we are placing them The pigeons are usually hinted at by the question in the problem statement What is the type of object whose existence you are trying to guarantee That will probably be your pigeon type See the first example below we want to prove there is a certain sum so we try making our pigeons the values of the numbers After that you should think about how many boxes you will need to guarantee something happening Usually once you identify this number the way to actually choose what those boxes are should become apparent Example The numbers on our clock have fallen off and been replaced in random order Oh no Prove that some consecutive string of 3 numbers has a sum of at least 20 Take the 12 positions are partition them into the usual 4 quadrants regardless of what numbers are there now How large small can the sums of these triplets be Well the total sum of all 12 numbers is 1 2 3 12 12 X k 1 k 13 13 12 78 2 2 Thus we can think of having 78 pigeons to distribute to these 4 boxes Some box must get at least 78 4 19 5 pigeons Since we must distribute whole pigeons this means some box gets at least 20 of them Example Amongst any set of m distinct natural numbers none of whom is itself a multiple of 10 there are at least two such numbers whose sum or difference is a multiple of 10 Find the smallest value of m such that this claim is valid Try out some small cases Pick 1 2 3 4 5 That works so far so m has to be at least 6 Is m 6 the right number Let s try to prove it Suppose we have an arbitrary set of 6 natural numbers Assign them to the Pigeonhole Boxes that are categorized by their last digit i e place n in a box based on the smallest positive value of x satisfying x n mod 10 as follows 1 9 2 8 3 7 4 6 5 If we have 6 numbers then some box has two numbers That means those numbers either have last digits that sum to 0 modulo 10 or else those numbers have the same last digit so their difference is 0 modulo 10 Either way we have a sum or difference that is 0 modulo 10 i e a multiple of 10 Hooray Graph Theory and Ramsey Theory We used the Pigeonhole Principle in Chapter 1 of the textbook to show that there are guaranteed to be 3 mutual friends or enemies amongst 5 people Reread that section of the textbook to see where and how we 1 used the Pigeonhole Principle This problem falls under the area of Ramsey Theory This branch of math seeks to answer questions of the form How big do I need this object to be to guarantee it contains within it some sub object that has property X This is a very rich area mathematically speaking There are many questions that are easy to state but seemingly impossible to answer Do some Googling and see what you can find Try thinking about these types of problems on your own and see what you can come up with Probability We are now moving on to talk about Probability It is quite natural to work on Combinatorics first because we can learn how to count things Now we start to wonder about how likely certain outcomes are given a certain process In some nice cases this amounts to just counting the outcomes we want amongst the total number of possibilities Of course this intuitive notion of probability entirely depends upon some underlying assumptions which don t always necessarily hold This next example is meant to i blow your minds and ii show you why we need to be careful about defining what probability means because our intuitions can often lead to very strange and undesirable consequences Motivating Example Bertrand s Problem Consider a circle Draw a random chord A chord is just a straight line that goes through the circle and intersects its circumference at two points What is the probability that this chord is longer than the side length of an inscribed equilateral triangle For illustration s sake consider a circle with radius r 1 Then an inscribed equilateral triangle has side lenghts 3 Can you prove this Try it We offer the following three procedures for generating a random chord Notice that each of them leads to a completely different although individually quite reasonable result about what the odds are that a random chord has the desired property Yowza Random Endpoints Method From the circumference of the circle choose two points randomly Connect them with a straight line to make the chord Analysis After we ve chosen the first random point on the circumference we can rotate the inscribed triangle around so that one of the vertices coincides with the chosen point This symmetry applies because we only care about the length of the chord not where it lies There are now three regions of the circumference separated by the vertices of the triangle Two of those regions are unfavorable they yield a short chord but the region opposite the already chosen point is favorable it yields a long chord Thus the probability is 1 3 2 Random Radial Point Method Choose a random angle from 0 to 360 and consider the radius corrseponding to that angle Choose a random point along this radius Draw the chord that is perpendicular to the chosen radius at the chosen point Analysis Choose the random angle Again by symmetric considerations we can rotate the inscribed equilateral triangle and circle around so that the radius is pointing straight down and is perpendicular to one of the sides of the triangle Notice that the triangle s side bisects the radius Can you prove this Try it This means that half of the points on this radius the upper half yield a short chord while half of the points the bottom half yield a long chord Thus the probability is 1 2 Random Interior Midpoint Method Choose a point randomly from the interior of the circle Identify the chord that has this point as its midpoint Analysis Consider the inscribed equilateral triangle Consider the circle inscribed inside that triangle Notice that this circle has radius 21 This follows from the fact given in the previous method s analysis the triangle s sides bisect radii of the larger circle Any point chosen from this smaller circle will correspond to a long chord Any point chosen