The First Odd Number
By John P. Robertson
The integers from one to ten billion are written out in formal English and listed in alphabetical order. For example, “forty-six,” “one thousand twenty-four,” and “two hundred twenty-nine.” Punctuation and spaces are ignored in the alphabetization. What is the first odd number in the list?
Sailing Club Election
The puzzle was about a club election with three candidates—Alice, Bob, and Carol. Voters ranked the three candidates in order of their preference. The first preferences resulted in an exact three-way tie. The second preferences also gave a three-way tie. Alice observed that because the club has an odd number of members, a vote on two candidates cannot end in a tie. She offered that the club vote first on a two-way contest between Bob and Carol, after which she would face the winner of that contest. Carol complained that this would give Alice a better chance of winning than she or Bob. The question was, “Is Carol right?”
Roger Bovard's solution is as follows. Under her proposal, Alice is sure to win. The proposed contest between Bob and Carol would be decided by the distribution of second choice votes on ballots listing Alice as the first choice.Without loss of generality, assume Bob is the winner. Then the second proposed contest between Alice and Bob would be decided by the distribution of second choice votes on ballots listing Carol as the first choice.
The details are: each candidate gets K first place votes and K second place votes. There are 3K members, which means K is odd. Ballots with Alice as the first choice have J second choice votes for Bob and K – J second choice votes for Carol. Because Bob is the winner of the first proposed contest, J > K – J. The ballots with Carol as the first choice would have the remaining K - J second place votes for Bob and J second place votes for Alice. Since J > K - J, Alice would win the second proposed contest.
Rob Thomas suggests that Alice is a cheater who should be reported to the ABCD!
Solutions were also sent in by Rose Barrett, John Jansen, Stuart Klugman, David Oakden, David L. Ruhm, Eric Savage, Steffen Siegel, Jason Stubbs, and David Uhland.