The constraints I have set on this book would preclude my discussing these questions even if it were not the case that they far exceed my range of vision. The reader is encouraged to consult the works that have been written on each of these. Volumes have been written, yet the questions are still unanswered. That is why they are great questions. The first four questions are purely and arguably the four greatest unanswered mathematical questions. The others are scientific (empirical) questions that have strong mathematical components (though question 5 is difficult to categorize).
1.What is the most simple algorithm that identifies the primes? This question is inextricably entwined with Riemann’s Prime Number Hypothesis, a speculation about the distribution of primes that is a wonderful next step for the reader with a lively curiosity about mathematics. (This is the question I would ask God if I could ask just one question.)
2.P = NP (the NP completeness problem)?
3.Is there an infinity larger than the rational numbers but smaller than the real numbers (the continuum hypothesis)?
4.Is there an even number that is not the sum of two primes (Goldbach’s conjecture)?
5.Why does mathematics work in the real world?
6.What is the topology of the universe?
7a.Will the universe close?
7b.If the universe closes, will time go backward, and if it does, what does this mean?
8.Why does time have direction?
9.Is there a way around the antirealism implied by Bell’s Theorem and the relevant experiments? If not, which assumption should we give up?
10.What mediates between the small (quantum world) and the large (classical world)? In other words, what is the state of Schrödinger’s cat?
11.What goes on inside a black hole? Does it have an “other side”?
12.Does it rain in Indianapolis in the summertime?
You already know that there is a way to play tic-tac-toe that guarantees you never lose. When you were very young, you probably found tic-tac-toe to be a challenging game. But you soon discovered a strategy that permitted you to avoid losing; the strategy enabled you to tie the game at worst, and to win if your opponent made a bad move.
You accomplished this intuitively, but you could have done so by making a matrix in which you wrote down every move your opponent could make, every response you could make to your opponent’s move, every response your opponent could make to your response, etc. By simply following a path that ended with a tie you would guarantee a tie even if your opponent always made his best possible move.
All of this is true of chess (and checkers). It has been proven that there is a guaranteed nonlosing strategy for every game that has complete information (i.e., each player knows the state of the game at each moment, and there are no hidden cards, as there are in poker) and in which chance plays no role (e.g., games using dice). Such games guarantee a draw (tic-tac-toe), a victory for the player who goes first (some versions of nim), or a victory for the player who goes second (other versions of nim). So, you might ask, why do they have chess championships?
Because, while we know that there is a correct way to play chess (i.e., a strategy that avoids losing), we have virtually no idea what that strategy is. (It is often the case in mathematics that it can be proven that there is an answer to a question without anyone having any idea what the answer is.) In chess, we do not know whether the strategy results in a draw, a victory for the player who goes first, or a victory for the player who goes second.
The reason we do not know this is that, while the total number of possible states of the “board” in tic-tac-toe is relatively small, the number of possible states in chess is, while finite, approximately 10118, a number unimaginably much greater than that required to count all of the electrons in the universe. Indeed, Seth Lloyd, professor of mechanical engineering at Massachusetts Institute of Technology, has estimated that the number of states the universe has ever been in—that is, the number of physical events that have taken place, including every change in the state of every particle—is only 10120.
However unknowable the master strategy for chess, a master strategy for tic-tac-toe is easily stated.
As a child, nearly everyone learns how to play tic-tac-toe without losing. However, it is not so easy to articulate the strategy one has intuited. This is the best Mike Mayers—a dab hand at numbers (and my brother-in-law)—and I can do. Elegant it is not, but it is the best we can do. See if you can remove one of these rules (or part of one) and still guarantee a draw. Following these rules guarantees that you will not lose (i.e., you will draw), even if your opponent plays a perfect game (and sometimes win if your opponent does not follow the strategy). It does not matter whether you go first or second.
Note that these rules do not guarantee you will win every game in which a victory is possible, only that you will never lose. An opponent who randomly selects moves or who plays badly will often give you an opportunity to win that will require that you make moves not given by these rules, while following these rules will result in a draw. However, a set of rules that guarantees victory in such cases would be far too unwieldy to be of use, and the correct moves in such cases will be intuitively obvious.
The first rule that applies determines your move:
1.On your first turn, go in the center if you can, otherwise go in a corner.
2.If you have only the center square and your opponent has (only) two diagonally opposing squares, go in a noncorner.
3.If any line has one empty square and two squares with your opponent’s mark, go in the empty square.
4.Unless there is no choice, do not complete a line that already has one X and one O.
5.At any point that you cannot follow any of these rules, go anywhere.
Fermat believed that there was no integer solution to the equation a4 + b4 + c4 = d4. This is not surprising as the smallest solution is believed to be 95.0004 + 217,5194 + 414,5604 = 422,5604. This is one of many lessons demonstrating how induction can never be proof in mathematics because the next numbers might be the proof.
John Allen Paulos, author and math professor at Temple University, gives an interesting example of a counterintuitive reality.
Say that Babe Ruth has a higher batting average for the first half of the season than L1ou Gehrig. Say also that Ruth has a higher batting average for the second half of the season.
Ruth must have a higher batting average for the season, right? Wrong. Say that for the first half of the season Ruth had 55 hits in 160 at bats for a 0.344 average and that Gehrig went 82 for 240 for 0.341. And say that for the second half of the season, Ruth went 60 in 240 for 0.250 and Gehrig went 38 in 160 for 0.238. Ruth has a higher average for each half of the season. But for the season, Ruth went 115 in 400 for 0.288, while Gehrig went 120 in 400 for 0.300.
Note that the players had the same number of at bats (i.e., the same number of chances) for the season. Had they had the same number of at bats in each half of the season, it would have been impossible for a player to lead for each half and yet come in second for the season.
To see all of this a bit more easily, here is another example. Say that for the first half of the season Don Drysdale has 9 victories and 7 losses for a winning percentage of 0.563 and that Sandy Koufax is 3 and 3 for 0.500. Say that for the second half of the season Drysdale is 3 and 0 for 1.000, and Koufax is 10–3 for 0.764. Each pitcher has 19 decisions for the season. Drysdale has a better winning percentage in each half of the season, but Koufax has a better percentage for the whole season. (Drysdale is 17–7 for 0.632; Koufax is 13–6 for 0.684.)
Consider an infinite series of numbers, say 1 + 2 + 3 + 4 … . Clearly this series has no limit; as you add numbers, the series gets bigger, without limit. Whatever number you choose—even one with a quadrillion digits—the series will eventually pass that number, and the sum of the series will forever get larger.
Now consider this series: 1/2 + 1/4 + 1/8 + 1/16 … . This series does have a limit: 1. You can forever get closer and closer (and, taking the series as a whole, consider the total to be 1), but you can never get a total past 1.
It is not always easy to tell from inspection whether a series has a limit. As Martin Gardner points out, consider the ubiquitous “harmonic series,” 1/2 + 1/3 + 1/4 + 1/8 … . The terms become increasingly smaller, leading one to suspect that the series (like 1/2 + 1/4 + 1/8 + 1/16) approaches a limit. It certainly seems to.
It takes this pokey series 12,367 terms to pass 10. Worse yet, the number of terms it takes to pass 100 is more than 1,000,000,000,000,000,000,000,000,000,000,000. But the series has no limit. Pokey or not, it just keeps on going its pokey way.
Nim is a game that has been played in various forms on at least four continents for at least four centuries. Like tic-tac-toe, it is a challenging game until one realizes that there is a correct way to play.
In the case of tic-tac-toe, there is a correct way for both players, and if both players make the correct moves, the game will always end in a tie.
In the case of nim, when one player makes the correct moves, he will always win. Whether this is the player who goes first or the player who goes second depends on the variation of nim being played.
Where the perfect strategy for tic-tac-toe is discoverable by a bright child, discovery of the correct nim strategy takes a mathematical intuition of the highest order for one without mathematical experience.
Nim is often called “the Marienbad game” because it was played, in its most familiar version and the one given here, in the movie Last Year at Marienbad. In the movie, and traditionally, matches are used, but pencil marks are far more convenient.
The correct nim strategy is given on page 31. Try playing a few games before looking at the correct strategy. Once you do, you can never again enjoy nim as a challenging game. On the other hand, in learning the strategy you will experience a deep and, for the mathematically inclined, exciting glimpse of the deep connections underlying the game.
Notice how all this nicely sums up both the gains and losses that come with modernization. The discovery of the correct strategy for nim was a solution to a problem that had gone unsolved for centuries. At the same time, the solution meant the loss of a game that had provided pleasure and social contact for millions and had enabled thousands to make a living.
Draw the following layout with a pencil.
l l l l l
l l l l
l l l
l l
l
Two players alternate turns. On each turn, a player crosses out marks on one line only. He may cross out as few or many as he wishes, but he must cross out at least one. The player who is forced to cross out the last mark loses.
Notice that position on a line does not matter. For example, if the player going first wishes to take away two marks from line three, it makes no difference whether he takes away the two on the right of line three or the two on the left of line three. Thus, for ease of play, it is traditional to take away marks from the right. So this player would cross out the marks indicated by x’s.
l l l l l
l l l l
l x x
1 1
1
Here are a few layouts to get you started.
l l l l l l l l l l l l l l l
l l l l l l l l l l l l
l l l l l l l l l
l l l l l l
l l l
l l l l l l l l l l l l l l l
l l l l l l l l l l l l
l l l l l l l l l
l l l l l l
l l l
l l l l l l l l l l l l l l l
l l l l l l l l l l l l
l l l l l l l l l
l l l l l l
l l l
l l l l l l l l l l l l l l l
l l l l l l l l l l l l
l l l l l l l l l
l l l l l l
l l l
Read “How to Play Nim,” on page 29, before reading this section.
This method works for all forms of nim. The 5-4-3-2-1 layout used in this example (the most commonly used layout) always results in a win for the player who goes first, assuming that he follows the strategy given.
In the case of some alternative layouts (for example 7-5-3-1) or rules (for example, the person who takes the last match wins), the player who goes second always wins. The strategy remains the same, simply be certain to let your opponent have the first move. (There is nothing he can do to avoid losing.)
Here we also assume that the goal is to force your opponent to cross out the last mark (i.e., the player who crosses out the last mark loses). This is the more commonly used variation. If you are playing a variation in which the player who crosses out the last mark wins, simply follow the strategy given, but play to “lose.”
Count the number of marks in each row and memorize the number associated with the number of matches in a line.
Layout = Number
l l l l l = 101
l l l l = 100
l l l = 11
l l = 10
l = 1
Whenever there are three marks in a row, that row equals eleven. Thus, if the first player crosses out two marks from the top row, the value of the top row goes from 101 to 11. (For the moment, do not worry about why five matches equals 101, four matches equal 100, etc.)
If you forget the code, you can reconstruct it easily. Begin with one match or mark, which equals 1. Two matches or marks will equal the next (ten-based) number that can be made of just 0s and 1s, which is 10. Three matches or marks equals the next (ten-based) number that can be made of just 0s and 1s, which is 11. Continue this to 100 and 101.
Always leave your opponent with a total of all even digits (not simply an even number; i.e., 210 is not a good total to leave your opponent). However, when at the end of a game you find that you must leave your opponent with all remaining rows containing only one digit each, leave him with an odd total). This will always be possible.
Assume you go first. Consider your first move. The total of a full layout is 223 (101 + 100 + 11 + 10 + 1). You want to leave your opponent with 222. (Any other possible total will have an odd digit.) You can cross out one mark in the top row (making 101 become 100), one mark in the middle row (making 11 become 10), or the mark in the bottom row (removing 1). You cannot remove, say, one from the second row (making 100 become 1) because the total will then be 124, which has an odd digit.
To play the game and always win, you need only remember or reconstruct the code (lllll = 101; llll = 100; lll = 11; ll = 10; l = 1) by the 1-10-11-100-101 method described. To see deeply into the structure and beauty of the mathematics, study closely the derivation of the code.
The five greatest women mathematicians are arguably, in chronological order, Hypatia, Maria Agnesi, Sophie Germaine, Sonya Kovalevsky, and Emmy Noether, clearly the greatest female mathematician. Only Kovalesky ever married, and hers was a platonic marriage of convenience. Make of this what you will, but it is unlikely that it represents a coincidence.