Over the course of the game, which is the most visited square on any Monopoly board? Is it the GO square where you start, the Free Parking diagonally opposite, or perhaps Virginia Avenue or the Boardwalk? The answer is in fact the Jail square. Why? Well, you could just throw the dice and find yourself just visiting, or you might find that the dice takes you to the square diagonally opposite, where a policeman tells you to go to jail. You might even be unlucky and pick up one of the Chance or Community Chest cards that send you straight to jail. And if that wasn’t enough ways to send you down, if you throw a double, you get to go again, but if you throw three doubles in a row, then rather than being rewarded for your impressive feat of dice rolling, that too is punished with a three-turn sentence.
As a result, on average, players find themselves visiting the Jail square about three times more often than most other squares on the board. That isn’t much help to us at the moment, because you can’t buy the jail. But here is where the math comes to the fore: where are players most likely to land after being in jail? The answer depends on the most likely throw of the dice when they leave that square.
Each die can land equally on one of the six faces. With two dice, that gives 6 x 6 = 36 different possible throws, each equally likely. But when you analyze those possibilities, you find that a score of 2 or 12 is very unlikely, because there’s only one way to make either of these combinations, whereas there are six ways to make a total score of 7.
Figure 3.8
So, there’s a 6 in 36 or 1 in 6 chance of getting a 7, and scores of 6 and 8 are the next most likely. A throw of 7 from jail gets you to a Community Chest square, which you can’t buy, but the two orange properties on either side (Tennessee Avenue and St. James Place) are the next most likely stops.
If you are lucky enough to land in the orange region of properties, these are the ones to buy and stack with hotels while you sit back and collect the rent that all your opponents will have to pay you as the dice take them out of jail and straight to your lair.
THE NUMBER MYSTERIES GAME SHOW
This is a game for two players. Take 20 envelopes and number them from 1 to 20. Player 1 writes down 20 different sums of money on pieces of paper and puts one in each envelope. Player 2 then chooses an envelope and is offered the sum of money inside. He can accept the money or choose a different envelope. If he chooses a different envelope, he can’t go back and claim a previous prize.
Player 2 continues opening envelopes until he is happy with the prize he has. Player 1 then reveals all the prizes. Player 2 scores 20 points if he claimed the top prize on offer. He scores 19 points if he got the second best prize, and so on.
All the envelopes are now emptied, and player 2 writes down 20 different sums of money on pieces of paper and puts one in each envelope. Player 1 must now try to get the best prize she can. Once she settles on an envelope, she scores points in the same way player 2 did. The winner is whoever has the highest score. This doesn’t mean the highest amount of money, but the highest number of points.
The intriguing aspect of this game is that you don’t know what the range of prizes is: the top prize might be $1, or it might be $1,000,000. The question is whether there’s a mathematical strategy that will help you to increase your chances of winning. Well, there is. It’s in a secret formula that depends on e—not the psychedelic kind, but the mathematical kind. The number e = 2.71828 . . . is probably one of the most famous numbers in the whole of mathematics, second only to the enigmatic π, and crops up wherever the concept of growth is important. For example, it is intimately related to the way the interest accumulates in your bank account.
Imagine that you have $1 to invest and are looking to see what different interest-rate packages the banks are offering. One will pay 100 percent interest after one year, which would increase your investment to $2. Not bad, but the next bank offers to pay 50 percent interest every half-year. After six months, that would give you $1.50, and after a year, $1.50 + $0.75 = $2.25—a better deal than the first bank. A third bank is offering 33.3 percent added every third of a year, which comes out at $(1.333)3 = $2.37 after 12 months. As you slice the year into smaller and smaller chunks, this compounding of the interest works to your advantage.
By now, the mathematician in you will hopefully have realized that the bank you really want is the Bank of Infinity, which divides the year into infinitely small units of time, because this will give you the maximum balance you can achieve. Although the balance increases the more you divide the year, it doesn’t become infinite but tends instead toward this magic number, e = 2.71828 . . . . Like π, e has an infinite decimal expansion (indicated by the “ . . .”), which never repeats itself. It turns out to be the key to helping you win the Number Mysteries game show.
The mathematical analysis of this game implies that you should first calculate , which is about 0.37. You should start by opening 37 percent of the envelopes, or about seven of them. Continue to open envelopes, but stop at the one whose contents beats all the envelopes you’ve opened so far. The math implies that one out of three times this will ensure that you end up with the top prize on offer. This strategy isn’t just useful in playing the Number Mysteries game show. In fact, many decisions we make in our lives can be reached by adopting this tactic.
Remember the first boyfriend or girlfriend you had? You probably thought he or she was amazing. Perhaps you dreamed romantically of spending your life together, but then had that nagging feeling that maybe you could do better. The problem is that if you dump your current partner, there’s generally no way back, so at what point should you just cut your losses and settle for what you’ve got? House hunting is another classic example. How many times do you see a fantastic house on your first visit, but then feel you need to see more before you commit yourself, only to risk losing the first great house?
Amazingly, the same math that helps you win the Number Mysteries game show can give you the best chances of landing the best partner or the best house. Let’s say you start dating at the age of 16 and decide that you’ll aim to have found the love of your life by the time you reach 50. And let’s assume that you get through partners at a constant regular rate. The math says that you should survey the scene for the first 37 percent of the time you’ve set yourself, which takes you to about the age of 28. Then you must choose the next partner who’s better than all the people you’ve dated up to that point. For one in three people, this will ensure that they end up with the best partner possible. Just be sure not to reveal your method to the love of your life!
HOW TO WIN AT CHOCOLATE ROULETTE
Even if you know your math, games like Monopoly or the Number Mysteries game show still rely on chance. Here’s a simple game for two players, which illustrates how math can guarantee you a win every time. Take 13 bars of chocolate and a red-hot chili and put them in a pile on the table. Each player, in turn, takes one, two, or three items from the pile. The aim is to force your opponent to take the chili.
Figure 3.9 Chocolate roulette.
Provided you go first, there is a strategy that will always leave your opponent with the chili. However many bars of chocolate your opponent takes, you always remove the number of bars that makes the total taken during the round add up to four. For example, if your opponent takes three bars of chocolate, you take one, making four bars in total. If your opponent takes two, you also take two.
The trick is to arrange the bars of chocolate in rows of four (do this in your head; otherwise, you give the game away). There are 13 bars to start with, so that’s three piles of four with one bar remaining (plus the chili, of course). Your opening move, then, is to take the one bar of chocolate leftover. After that, you carry on as just described: in response to your opponent’s move, you take a number that adds to four. In this way, the combination of your opponent’s move and your move removes one of the piles of four chocolates each time. After three rounds, your opponent is left with just the chili on the table.
Figure 3.10 How to arrange the c
hocolates to guarantee a win.
The strategy does depend on you going first. If your opponent goes first, it only takes one slip for you to be back in the winning position. For example, if your opponent takes more than one bar of chocolate as his opening move, he’s already started eating into the first pile of four chocolates, so you take the rest of the pile as before.
You can extend the game by starting with a different number of bars of chocolate or by varying the maximum number of bars you are allowed to take in one turn. The same math of dividing the bars into groups will enable you to concoct a winning strategy.
There is another variant of this game, called nim, which uses a slightly more sophisticated mathematical analysis to guarantee you your win. There are four piles this time. One pile has five bars of chocolate, the second pile has four, the third pile has three, and the final pile consists of just the chili. This time, you are allowed to take as many bars of chocolate as you want, but they can only be taken from one pile. For example, you could take all five bars of chocolate from the first pile, or just one bar of chocolate from the third pile. Again, you lose if your only choice is to take the chili.
The way to win this game is to know how to write numbers in binary rather than decimal form. We count in 10s because we’ve got ten fingers. Once you’ve counted up to 9, you start a new column and write 10 to indicate one lot of ten and no units. But computers like to count in 2s, or what we call binary. Each digit represents a power of 2 rather than a power of 10. For example, 101 represents one lot of 22 = 4, no 2s, and one unit. So 101 is the number 4 + 1 =5 in binary. The table here shows the first few numbers written in binary.
Decimal
Binary
0
1
1
1
2
10
3
11
4
100
5
101
6
110
7
111
8
1000
9
1001
Table 3.2
To win at nim, you need to convert the number of bars of chocolate in each pile into binary. The first pile has 101 bars, the second 100 bars, the third 11 bars. Writing this last number as 011 and putting the three numbers on top of each other give us this:
101
100
011
Notice that the first column has an even number of 1s; the second, an odd number of 1s; and the third column, an even number of 1s. The winning move each time is to remove bars of chocolate from one pile in such a way that each column ends up with an even number of 1s. So in this case, remove two bars from the third pile of three to reduce the number of bars to 001.
Why will this help you win? Well, at every turn, your opponent will be forced to leave at least one of the columns with an odd number of 1s. Your next move is to take bars of chocolate to make them all even again. Because the number of bars of chocolate is constantly going down, at some point, one of you will remove bars of chocolate so that the piles have 000, 000, and 000 in them. Who does that? Your opponent always leaves an odd number of 1s in at least one of the piles, so it must be you that makes this move. Your opponent gets left with the chili.
This strategy will work however many bars of chocolate you put in each pile. You can even increase the number of piles.
WHY ARE MAGIC SQUARES THE KEY TO EASING CHILDBIRTH, PREVENTING FLOODS, AND WINNING GAMES?
Lateral thinking is a handy talent when it comes to doing math. By looking at things from a different angle, the answer to a difficult conundrum can suddenly become transparently obvious. The skill lies in finding the right way to look at the problem. To illustrate this, here’s a game that, at first glance, is tricky to keep track of but becomes much easier when we come at it from a different direction. To play the game, you can visit the Number Mysteries website to download and cut out the props you’ll need.
Each contestant has an empty cake stand on which can fit 15 slices of cake. The object of the game is to be the first to fill the cake stand with exactly three chunks of cake chosen from nine chunks of different sizes, the smallest consisting of just one slice and the largest consisting of nine slices. Contestants take turns choosing one of the chunks.
Figure 3.11 Choose three chunks of cake to fill your cake stand before your opponent fills his.
The aim is to get three numbers from 1 to 9 that add up to 15 while at the same time keeping track of what your opponent is doing so you can defeat his attempts. So if your opponent has chosen chunks with three and eight slices, then you need to stop him from making 15 by taking the chunk with four slices. If the chunk you wanted has been taken, you’ll have to find a different way to get to 15 by using the chunks you’ve already chosen and the remaining ones. But you must always use precisely three chunks to fill the stand—filling the stand with two chunks of nine and six slices is not a winning move, nor is filling it with four chunks of one, two, four, and eight slices.
Once you start playing, it soon gets quite difficult to keep track of all the different ways that you and your opponent might fill the cake stand. But the game becomes much easier once you realize that what you’re playing is actually another classic game in disguise: tic-tac-toe. Instead of the classic 3 x 3 grid in which you place Os and Xs, trying to get three in a row before your opponent does, this game is played out on a magic square:
Table 3.3
The most basic magic square is a way of arranging the numbers from 1 to 9 in a 3 x 3 grid so that the numbers in the columns, rows, and diagonals all add up to 15. This arrangement provides all the possible ways to get 15 by adding three different numbers chosen from 1 to 9. By playing the cake game as tic-tac-toe on this magic square, anyone who gets three in a row will have gotten three numbers that add up to 15 before his or her opponent has.
According to one legend, the first magic square appeared in 2000 BC inscribed on the back of a turtle that crawled out of the River Lo in China. The river had badly flooded, and the emperor Yu ordered a number of sacrifices to appease the river god. In response, the river god sent forth the turtle, whose pattern of numbers was meant to assist the emperor in controlling the river. Once this arrangement of numbers had been discovered, Chinese mathematicians started trying to construct bigger squares that worked the same way. These squares were believed to have great magical properties and became widely used for divination. The Chinese mathematicians’ most impressive achievement was a 9 x 9 magic square.
There is evidence that the squares were taken to India by Chinese traders who dealt not only in spices but also in mathematical ideas. The way the numbers weaved in and out of the squares resonated strongly with Hindu beliefs of rebirth, and in India, these squares were used for anything from specifying perfume recipes to an aid for childbirth. Magic squares were also popular in medieval Islamic culture. Their much more systematic approach to mathematics led to clever ways of generating magic squares, culminating in the thirteenth-century discovery of an impressive 15 x 15 magic square.
One of the earliest outings of magic squares in Europe is the 4 x 4 magic square that appears in Albrecht Dürer’s engraving Melancholia. Here, the numbers from 1 to 16 are arranged so that the rows, columns, and diagonals all add up to 34. As well as that, each of the four quadrants—the four 2 x 2 squares into which the big square can be split—and the 2 x 2 square at the center also sum to 34. Dürer even arranged the two numbers at the middle of the bottom row to give the year he made the engraving: 1514.
Figure 3.12 Albrecht Disrer’s magic square.
Magic squares of different sizes were traditionally associated with planets in the solar system. The classic 3 x 3 square was associated with Saturn, the 4 x 4 square in Melancholia is Jupiter’s, while the largest—a 9 x 9 square—was assigned to the moon. One suggestion for Dürer’s use of the square is that it reflected the mystical belief that Jupiter’s joyfulness could counteract the sense o
f melancholy that pervades the engraving.
Another famous magic square can be found at the entrance to the flamboyant Sagrada Familía, the still unfinished cathedral in Barcelona designed by Antoni Gaudi. The magic number for this 4 x 4 square is 33, the age Christ was when he was crucified. This square isn’t quite as satisfying as Dürer’s square because the numbers 14 and 10 appear twice, at the expense of 12 and 16.
Magic squares are something of a mathematical curiosity, but there is a problem about them that mathematicians have been unable to unravel. There is essentially only one 3 x 3 magic square. (The qualification “essentially” means that what you get by rotating or reflecting a magic square doesn’t count as a different one.) In 1693, the Frenchman Bernard Frénicle de Bessy listed all 880 possible 4 x 4 magic squares, and in 1973, Richard Schroeppel used a computer program to calculate that there are 275,305,224 5 x 5 magic squares. Beyond that, we only have estimates for the number of possible magic squares of 6 x 6 and bigger. Mathematicians are still looking for a formula that will give the exact numbers.
WHO INVENTED SUDOKU?
The spirit of sudoku can be found in a puzzle that grew out of mathematicians’ fascination with magic squares. Take the court cards (kings, queens, jacks) and aces from a standard pack of cards and try to arrange them in a 4 x 4 grid so that no row or column has a card of the same suit or rank. This problem was first posed in 1694 by the French mathematician Jacques Ozanam, who might be regarded as the man who invented sudoku.
One mathematician who certainly caught the bug was Leonhard Euler. In 1779, a few years before he died, Euler came up with a different version of the problem. Take six regiments, with six soldiers in each regiment. Each regiment has a different colored uniform: they might be red, blue, yellow, green, orange, and purple. The soldiers in each regiment have different ranks—say, a colonel, a major, a captain, a lieutenant, a corporal, and a private. The problem is to arrange the soldiers on a 6 x 6 grid so that in any column (or row) you don’t see a soldier of the same rank or in the same regiment in that column (or row). Euler asked the question for a 6 x 6 grid because he believed that it was impossible to arrange the 36 soldiers satisfactorily. It was not until 1901 that the French amateur mathematician Gaston Tarry proved Euler correct.
The Number Mysteries: A Mathematical Odyssey through Everyday Life Page 13