Section 4.2

• 2. Pigeonhole principle with k=26.
• 4. There are d possible remainders when an integer is divided by d (0, 1, ..., d-1). By the pigeonhole principle, if we have d+1 remainders, at least 2 must be the same.
• 6. Pigeonhole principle with k=|T|.
• 8. midpoint of segment whose endpoints are (a,b) and (c,d) is ((a+c)/2, (b+d)/2). Coords of fractions will be integers only if a & c , b & d have the same parity. There are 4 possible parities. Given 5 points the pigeonhole principle guarantees at least 2 will have the same pairs of parities. Midpoint will therefore have integer coords.
• 10. 26 (similar to 8 above).
• 12. see 11 in text.
• 14. Look at pigeonholes {1001,1001}, {1002,1003},...,{1098,1099}. There are 50 sets in the list. If we have 51 numbers in the range 1000 - 1099 then 2 must come from the same set.
• 16. Maximal length increasing sequence: 5, 7, 10, 15, 21. Decreasing sequence: 22, 7, 3.
• 18. Theorem 3, with n=10.
• 20. Let the 5 people be A,B,C,D,E. Suppose the following pairs are friends: AB,BC,CD,DE,EA. The other 5 pairs are enemies, therefore there are no 3 mutual friends and no 3 mutual enemies.
• 22. Let A be one of the people. He has either 10 friends or 10 enemies since having 9 or fewer would account for less than 20 people. Assume A has 10 friends. By exercise 21, there are either 4 mutual enemies among these 10 people or 3 mutual friends. In the former case, together with A, we have our 4 mutual enemies. In the latter case, together with A, we have our 4 mutual friends.
• 24. There are 99,999,999 possible +ve salaries less than $1 M i.e. from $0.01 to $999,999.99. By the pigeonhole principle, given 100,000,000 workers, at least 2 will have the same salary.
• 26. Let K(x) be the number of computers that computer x is connected to. This could be 1,2,3,4,5. By the pigeonhole principle, this guarantees that at least 2 of the values of K(x) are the same, given 6 computers.
• 28. Let K(x) be the number of people that person x knows. The possible values for this are 0,1,2,...,n-1, where n>=2. The pigeonhole principle cannot be applied directly here because in this case we have n pigeons and n pigeonholes. However, note that the values 0 and n-1 cannot exist together since if someone knows no one, no one can know everyone else, therefore there are actually only n-1 possible values. Applying the pigeonhole principle to this fact shows that there are at least 2 people who know the same number of other people.
• 30. a) Solution of Exercise 29, with 24 replaced by 2 and 149 replaced by 127. b) Solution of Exercise 29, with 24 replaced by 23 and 149 replaced by 148. that there are 9 students in class.