Five or Six Reasons Why Parity Puzzles Are Fun

May 2, 2010 — -- In belated recognition of April being Math Awareness Month, my column this May will deal with parity.

The notion refers to the evenness or oddness of a number, say April the fourth month versus May the fifth. Despite its simplicity parity plays an important role in many areas of mathematics.

It also lends itself to some nice little puzzles, including Rubik's cube and the 15 puzzle. Here are five or six easy examples. The sixth one is fuzzy and involves politics and the Supreme Court, so it doesn't really count.

The answers to the puzzles appear at the end of the column, but don't peek first ... Unless, of course, you feel like peeking.

1. A loose leaf notebook consists of 100 sheets of paper. Number them, front and back, from 1 to 200. Tear out any 25 of the sheets, and add up the 50 page numbers on them. Can you choose the sheets so that the sum of the 50 numbers is 2010?

2. Consider the sum of the first 10 numbers: 1+2+3+4+5+6+7+8+9+10. Can you change some of the plus signs to minus signs so that the resulting sum is 0? For example, 1+2-3-4-5-6+7- 8+9+10=3. This is close, but not 0.

3. Before you is a regular 8-by-8 checkerboard with two diagonally opposite squares missing. Also before you are 31 dominoes, each 2 squares long, 1 square wide.

Since each domino covers two squares when placed on the checkerboard, 31 dominoes should be enough to cover this mutilated checkerboard. (A regular board has 8x8=64 squares, so this one has 64-2=62 squares, and 2x31=62.) So, can you cover this mutilated checkerboard with the 31 dominoes?

4. Two equally matched teams, A and B, play in a best-of-seven World Series. The probability that team A (or team B) wins any given game is 50%, and the first team to win four games wins the Series. The series ends at that point. Is it more likely that the series will end in six games or seven?

5. A dozen prisoners are told that they'll be lined up in the morning, each facing the backs of those ahead of them in line. They're also told that either a red hat or a blue hat will be placed on each of them. They won't be able to see the color of their own hat, but will be able to see the color of the hats of those in front of them.

After lining up the prisoners in this way, the guards will go to the last person in line and ask him what color his hat is. If he gives the correct color, he will be released, but if he answers incorrectly, he will be killed. Then the next to last person will be asked the color of his hat and released if he answers correctly and killed if not, and so on up the line.

The prisoners are told of this impending procedure the night before and try to decide on a strategy that will save as many of their lives as possible. What should they decide?

6. The Supreme Court has nine justices. The majority rules, and on many issues, the court splits 5-4, with the five more conservative justices prevailing. President Obama will decide soon on whom he will nominate to replace Justice Stevens, one of the four more liberal justices. What is his best shot at changing the court and its rulings in the short term.

Answers

1. The sum of the page numbers on opposite sides of a sheet of paper must be odd because the pages are numbered consecutively making one of them odd and the other even and thus their sum odd. If we add up 25 such odd numbers, we'll always get an odd number, but 2010 is even.

2. First notice that the sum of the first ten numbers is 55. That is, 1+2+3+4+5+6+7+8+9+10=55. Consider now what happens if we change +3 to -3. The sum is decreased by 6. If we change +8 to -8, the sum is decreased by 16.

In fact, whenever we change a sign from + to -, we decrease the sum by an even number. Starting at the odd number of 55 and subtracting even numbers from it will always lead to an odd number, but 0 is even.

3. Since this is a checkerboard, alternate squares are colored red and black, and so the two missing diagonally opposite squares are both the same color, say red. A normal checkerboard has 32 red and 32 black squares, but this one has only 30 red and 32 black squares. This makes covering it with 31 dominoes impossible since each domino will cover one red and one black square.

4. In order for the series to end in either six or seven games, it is necessary for it go more than five games. Thus after five games one of the teams, say it's A, is ahead three games to two. (If one of the teams were ahead four games to one, the series would be over after five games.)

So it's three games to two in favor of A. If A wins, the series lasts exactly six games, and if B wins, the series goes into the seventh game. Since each team is equally likely to win, it's equally likely that the series will go six or seven games.

5. The best strategy is for the last person in line (prisoner number 12) to answer "red" if he sees an even number of red hats in front of him and blue if he sees an odd number. Unfortunately for him, he'll be right only 50% of the time, but his answer enables all of the other prisoners to survive.

If the last prisoner (12) answers "red," the prisoner in front of him (number 11) knows that 12 saw an even number of red hats. If 11 also sees an even number of red hats in front of him, he knows that his hat is blue and answers accordingly. If 11 sees an odd number of red hats in front of him, he knows his hat is blue and answers accordingly. And so on for prisoners number 10, 9, 8, and so on.

6. Since Obama wishes to move the court in a more liberal direction, the only way this might come about in the short term is to pick not necessarily the most eloquent or learned judge, but rather that judge who would be most likely to persuade Justice Kennedy, the swing justice among the five more conservative justices. This would switch the court's vote in certain cases from 5-4 to 4-5.

Simplistic and nowhere near as clear-cut as the straight math problems, this analysis is at least tenuously related to the issue of parity. Incidentally from what I've read about her, Judge Diane Wood would be the most likely replacement for the recently retired Justice Stevens to bring this change to the court.

John Allen Paulos, a professor of mathematics at Temple University in Philadelphia, is the author of the best-sellers "Innumeracy" and "A Mathematician Reads the Newspaper," as well as, most recently, "Irreligion: A Mathematician Explains Why the Arguments for God Just Don't Add Up." He's on Twitter and his "Who's Counting?" column on ABCNews.com appears the first weekend of every month.