MATH Section 6.2

Consider seven integers selected from the first 10 positive integers. Click and drag the steps to their corresponding step numbers to show that if seven integers are selected from the first 10 positive integers, there must be at least two pairs of these integers with the sum 11. (You must provide an answer before moving to the next part.) Explanation Group the first ten positive integers into five subsets of two integers each, each subset adding up to 11. Select seven integers from these sets. By the pigeonhole principle at least two of them come from the same subset. Leave the two integers in the same subset (group). Then there are five more integers and four groups. Again, by the pigeonhole principle, there are at least two integers in the same group. Hence, two pairs of integers from two groups are obtained. In each case, these pairs of integers have a sum of 11, as desired.

1. Group the first ten positive integers into five subsets of two integers each, each subset adding up to 11. 2. Select seven integers from these sets. By the pigeonhole principle at least two of them come from the same subset. 3. Leave the two integers in the same subset (group). Then there are five more integers and four groups. Again, by the pigeonhole principle, there are at least two integers in the same group. 4. Hence, two pairs of integers from two groups are obtained. In each case, these pairs of integers have a sum of 11, as desired.

A bowl contains 10 red balls and 10 blue balls. A woman selects balls at random without looking at them. What is the minimum number of balls she must select to be sure of having at least three blue balls? Explanation The woman needs to ensure that there are atleast three blue balls. If she selects atleast 13 balls, then at most 10 of them are red, so at least three are blue. On the other hand, if she selects 12 or fewer balls, then 10 of them could be red.

13

A drawer contains 12 brown socks and 12 black socks, all unmatched. A man takes socks out at random in the dark. Select the least number of socks that he must take out to be sure that he has at least two black socks. Explanation He needs to select 14 socks in order to ensure that there are at least two black socks. If he does so, then at most 12 of them are brown, so at least two are black. On the other hand, if he takes 13 or fewer socks, then 12 of them could be brown, and he might not get two black socks.

14

A drawer contains 12 brown socks and 12 black socks, all unmatched. A man takes socks out at random in the dark. Select the least number of socks that he must take out to be sure that he has at least two socks of the same color. Explanation The two colors are the pigeonholes. We want to know the least number of pigeons needed to ensure that at least one of the pigeonholes contains two pigeons. By the generalized pigeonhole principle, the answer is three. If three socks are picked, at least ⌈3/2⌉ = 2 must have the same color. On the other hand, two socks are not enough, because one might be brown and one might be black.

3

How many numbers must be selected from the set {1, 2, 3, 4, 5, 6} to guarantee that at least one pair of these numbers add up to 7? Explanation We apply the pigeonhole principle by grouping the numbers cleverly into pairs (subsets) that add up to 7, namely {1, 6}, {2, 5}, and {3, 4}. If we select four numbers from the set {1, 2, 3, 4, 5, 6}, then at least two of them must fall within the same subset, since there are only three subsets. Two numbers in the same subset are the desired pair that add up to 7. We also need to point out that choosing three numbers is not enough, since we could choose {1, 2, 3} and no pair of them add up to more than 5.

4 numbers from the given set, since there are exactly three subsets of two integers from the given set that add up to 7

A bowl contains 10 red balls and 10 blue balls. A woman selects balls at random without looking at them. How many balls must she select to be sure of having at least three balls of the same color? Explanation There are two colors: these are the pigeonholes. We want to know the least number of pigeons needed to ensure that at least one of the pigeonholes contains three pigeons. By the generalized pigeonhole principle, the answer is five. If five balls are picked, at least ⌈5/2⌉ = 3 must have the same color. On the other hand, four balls are not enough, because two might be red and two might be blue.

5

Let (xi, yi), i = 1, 2, 3, 4, 5 be a set of five distinct points with integer coordinates in the xy plane. Show that the midpoint of the line joining at least one pair of these points has integer coordinates. Explanation There are four possible pairs of parities: (odd, odd), (odd, even), (even, odd), and (even, even). Since we are given five points, the pigeonhole principle guarantees that at least two of them will have the same pair of parities. The midpoint of the segment joining these two points will therefore have integer coordinates.

By the pigeonhole principle, if we have 5 points, then at least two of these points will have the same parity (both odd or both even). The midpoint of the segment joining these two points will therefore have integer coordinates.

Consider seven integers selected from the first 10 positive integers. Is the conclusion in the part (a) of the question true if six integers are selected rather than seven? Explanation The conclusion of part (a) is not true for all groups (or sets) of six numbers that are chosen from the first 10 positive integers. For example, the set {1 ,2, 3, 4, 5, 6} has only one pair, 5 and 6, from the set whose sum is 11.

No

There are 30 students in a class. Choose the statement that best explains why at least two students have last names that begin with the same letter. Explanation This follows from the pigeonhole principle, with k = 26. There are 30 students in a class: these are the pigeons. There are 26 letters: these are the pigeonholes. If 26 students have last names that begin with the 26 different letters then the remaining four students must have last names that begin with any of these letters. By the pigeonhole principle, at least two students will have last names that begin with the same letter.

Using the pigeonhole principle, in any class of 30 students there must be at least two students who have last names that begin with the same letter since there are only 26 letters in the alphabet.

Consider a set of six classes, each meeting regularly once a week on a particular day of the week. Choose the statement that best explains why there must be at least two classes that meet on the same day, assuming that no classes are held on weekends. Explanation There are six classes: these are the pigeons. There are five days on which classes may meet (Monday through Friday): these are the pigeonholes. Each class must meet on a day (each pigeon must occupy a pigeonhole). By the pigeonhole principle, at least two classes must meet on a day of the week.

Using the pigeonhole principle, in any set of six classes there must be at least two classes that meet on the same day because there are only five weekdays for each class to meet on.