Bogglers Solutions

By Scott Kim|Friday, June 01, 2001

Electoral Confusion

1) 85 votes. Six states vote 6 to 5 in favor of the winner, while the other five states are 11 to 0 in favor of the loser.

2) All but one state. Ten states are 6 to 5 in favor of the winner, while the remaining state is 11 to 0 in favor of the loser. In this case the loser receives 61 votes, while the winner receives 60. By the way, the same could theoretically happen in U.S. elections: One candidate could win 49 states, each by one vote, and lose the remaining state by 50 or more votes.

3) 119 of the 121 votes. There are just three states: two states with one person each who votes for the winner and one state with 119 people, all of whom vote for the loser.

4) 90 votes. There are just two states: a state with 61 people, 31 of whom vote for the winner, and a state with 60 people, all of whom vote for the loser. In general the largest number of votes the loser can receive approaches 75 percent of the total as the population increases.

5) 30 votes. It takes just five votes to win a state if the vote is 5 to 4 to 2 (or 5 to 3 to 3), and it takes six states to win the presidency. In general it takes just over one sixth of the total votes to win: one third of the votes in half of the states.



Flavor Face-Off

This puzzle is based on the work of election theorist Donald Saari of the University of California at Irvine (see "May the Best Man Lose," Discover, November 2000). Saari has studied alternative voting methods, such as approval voting, in which you vote for all the candidates of whom you approve. His conclusion: Every voting method, including simple majority, introduces its own bias.

1) The pair comparisons rank the three flavors as B > C > A— the opposite of the result obtained by just tallying the top choices.

2) 60, 31, and 30, for a margin of 29 between the winner and the second-place finisher.

3) Nine is the smallest possible population, with four, three, and two citizens preferring one each of the three rankings.

4) The simplest explanation is that a third of the population votes for each of the following three preference rankings. Note that every preference ranking travels around a circle in the order Pensicola to Semicola to Texicola and back to Pensicola.

Pensicola > Semicola > Texicola
Semicola > Texicola > Pensicola
Texicola > Pensicola > Semicola




Tennis Mismatch

1) The gold medal winner is always the best player. The worst rank the silver medal winner could have is #8.

2) The gold medal winner is always the best player. The worst rank the silver medal winner could have is #5.

3) The smallest possible number of days is seven: #1 plays #2, #2 plays #3, and so on until #7 plays #8. The largest possible number of days is nine: seven days to determine the #1 player using a ladder, as in problem 2, then two more days to determine the best player from among the three players that the #1 player beat.






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Got new solutions for the puzzle? Want to see other people's solutions? Talk to the puzzle master in his discussion forum at www.scottkim.com.

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