RECREATIONAL MATH - The Handy Math Answer Book

The Handy Math Answer Book, Second Edition (2012)

RECREATIONAL MATH

MATH PUZZLES

What is a puzzle?

A puzzle is a mathematical problem that produces a solution often in the form of rearranging pieces (often geometric) or filling in the blanks (such as a crossword puzzle). Puzzles do not typically require superior mathematical knowledge, but many originate from more advanced mathematical or logistical problems. They also include board games (such as chess) and brain teasers.

What are some types of puzzles?

There are numerous categories of puzzles. The following lists the most common:

· Puzzles derived from board games include chess-type problems (chess and eight queens puzzle)

· Logic puzzles (paint by numbers)

· Mechanical puzzles (Rubik’s Cube)

· Computer puzzle games

· Cube games

· Solitaire-type puzzles

· Brain teasers

· Riddles

· Tiling puzzles (including jigsaw, polysquare, and Tangram puzzles)

· Whodunits

· Word puzzles (anagrams, cracking a code, crossword puzzles, find-the-word puzzles, and verbal arithmetic puzzles)

What is a cryptarithmetic puzzle?

Some of the most challenging verbal arithmetic puzzles are called cryptarithmetic puzzles. These number puzzles (often called cryptarithms) are made up of mathematical equations whose digits are represented by letters or symbols; the goal is to identify the numerical value of each letter. In such a puzzle, each letter represents a unique digit, but there are rules. As in ordinary arithmetic notation, the leading digit of a multi-digit number must not be zero; also, the puzzle usually has only one solution.

Image

Playing games with puzzles that often demand mathematical skills has been a popular pastime for generations.

Cryptarithmetic puzzles are most often divided into two types. An alphametic cryptarithm is one in which the letters are used to represent distinct digits. These are derived from related words or meaningful phrases in the puzzles. A digimetic cryptarithm is one in which the digits are used to represent other digits. One of the most famous cryptarithmetic puzzles was developed by Englishman Henry Ernest Dudeney (1857–1930) and published in a 1924 issue of the Strand Magazine:

Image

In this puzzle (developed by Englishman Henry Ernest Dudeney [1857-1930] and published in 1924 in the Strand Magazine), the addition the addition sum of each letter represents a digit, with all the letters being different digits. These letters represent the numbers as follows: O = 0, M = 1, Y = 2, E = 5, N = 6, D = 7, R = 8, and S = 9. Or

Image

What is a Tangram?

Some puzzle forms go back thousands of years, such as the Tangram, which is also known as a dissection puzzle. The Tangram is of Chinese origin—literally called the Seven-Board of Cunning—but the word itself is of English origin, and is built from the words “tang” (thought to be a synonym for “Chinese” in the Cantonese dialect) and “gram.” It is thought that the Pythagorean theorem was discovered in Asia before Pythagoras’s time with the help of Tangram pieces (for more about the Pythagorean theorem, see “Geometry and Trigonometry”).

The Tangram consists of a square divided into seven pieces called Tans, all of which must be arranged to match particular designs, usually a square. The seven pieces include five triangles of various sizes, one square, and one parallelogram. All the shapes must be used in the final form and none can overlap.

What are tiling and dissection puzzles?

Tiling puzzles are two-dimensional shapes that are reassembled into a larger given shape without overlaps. The best examples of these are dissection puzzles; the most common ones are those in which an object is converted to another by making a finite number of cuts, then reassembling the pieces. (Most of the cuts are represented by straight lines, but not always; in addition, sometimes the cut object can be reassembled into two or more shapes.)

What is a stomachion?

A stomachion is a dissection puzzle similar to a Tangram. It uses 14 pieces in the puzzle, each of varying polygonal shapes and arranged into a 12 by 12 square grid. Each of the pieces has an area that is an integral fraction: for example, 24, 12, 9, 6, or 3, of the total area of the square, which is 144 units. The object of a stomachion is to arrange the pieces into interesting, and often recognizable shapes, such as people, animals, and objects.

This puzzle was also an ancient game known to the Greek mathematician Archimedes (c. 287-212 B.C.E.), and thus is also called the Loculus of Archimedes (“Archimedes Box”). It is doubtful Archimedes invented the puzzle, but he did explore its geometric aspects. It also now appears that in contemplating the different solutions to this puzzle, Archimedes actually was anticipating the branch of mathematics we now call combinatorics.

Image

This is an example of a stomachion, a type of puzzle in which a geometric shape is dissected into several smaller shapes that can then be rearranged.

What are some other examples of dissection puzzles?

There are several other example of dissection puzzles, including the following:

· The haberdasher problem: Puzzle inventor Henry Ernest Dudeney (1857–1930) of England was instrumental in developing many more puzzles than the cryptarithmetic puzzle described above. One of the most famous of his geometrical puzzles is the “haberdasher’s problem,” which asks how an equilateral triangle can be cut into four pieces and reassembled to form a square. (His model used hinges that would move the pieces into place.)

· Pythagorean square puzzle: In this dissection puzzle, the two squares on the left are combined to form a single large square on the right.

· T-puzzle: The T-puzzle is a dissection puzzle that forms the letter T. Four pieces are used to create the capital letter.

What is the 15 Puzzle?

The 15 Puzzle was introduced in 1878 by American amateur mathematician Samuel Loyd (1841–1911). He called it the “Boss Puzzle” and later the “15-16 Puzzle.” It is one of the most famous puzzles in his book Sam Loyd’s Cyclopaedia of 5,000 Puzzles, Tricks and Conundrums, published in 1914 after his death by his son, Sam Loyd. This puzzle has 16 squares; 15 of them are numbered from 1 to 15 and placed in a 4 by 4 configuration, with one position, the 16th, left open. The idea was to reposition the squares from a given arbitrary arrangement by sliding them from place to place until they were in numerical order (1, 2, 3, and so on). For some initial starting points, the rearrangement was possible; for others, it was not.

But Loyd offered a twist to the puzzle—he switched the positions of the squares numbered 14 and 15—and offered $1,000 to anyone who could solve the puzzle. Working out the puzzle became a craze in America, with reports of companies prohibiting employees from playing during office hours—it was as popular as playing computer solitaire is today. Even in Europe, the craze grew. Deputies in Germany’s Reichstag played the puzzle, and in France it was claimed to be a greater curse than alcohol or tobacco. But Loyd knew no one could solve the puzzle, much less remember all the steps taken to try and get to a solution, because there was no solution!

Image

An example of a Latin square.

What is a Latin square?

A Latin square is considered to be an n by n (n × n) array filled with n different letters (called Latin letters). These letters occur exactly once in each row and column. The name “Latin square”—considered in the field of combinatorics—was first mentioned by Leonhard Euler who used the Latin characters as symbols. (For more about combinatorics, see “Foundations of Mathematics”; for more about Euler, see “History of Mathematics” and “Mathematical Analysis.”)

Image

In the haberdasher problem, an equilateral triangle is divided and rearranged into a square.

Image

The Pythagorean square puzzle tests your geometry skills by asking you to take two square shapes and reconfigure them into one larger square.

Image

In the T-puzzle, several shapes must be assembled into the letter T.

Image

A 15 Puzzle might not be a fair challenge of a person’s skills, if the numbers don’t begin in the correct order.

Other characters can be used in a Latin square, as can numbers in numerical order. Latin squares are used most often in such fields as science, engineering, statistics, and even computer science; in other words, fields in which the researcher desires to control the variation in an experiment related to rows and columns in the field. For example, in plant genetics or forestry, data can be put into a Latin square to determine the growth characteristics of a plant, or to see which plants are growing best where in a forestry situation.

What are logic puzzles?

In general, a logic puzzle involves the description of an event or contest. It is derived from the mathematical field of deduction: Clues are provided and the puzzle player has to piece together what actually happened by using clear and logical thinking (thus the name).

One of the most famous logic puzzle pioneers was writer, photographer, mathematician, and illustrator Charles Lutwidge Dodgson (1832–1898), otherwise known as Alice’s Adventures in Wonderland author Lewis Carroll. In his book The Game of Symbolic Logic he introduced several games, asking the reader to solve a puzzle, such as: “some games are fun,” “every puzzle is a game,” thus, “are all puzzles fun?” Puzzles such as this are known as syllogisms, in which the reader is given a list of premises and asked what can be deduced from the list. (For more information about syllogisms, see “Foundations of Mathematics.”)

Are there contests for puzzle players?

Yes, there are many contests for puzzle players—from crosswords to regular puzzles, and from national to local contests. While most of us are content to work out a crossword puzzle at home on a lazy Sunday afternoon, there are those people who love to compete with others to solve a crossword or other puzzle. Thus, puzzle tournaments and contests were born.

There is one contest that is famous all over the globe: sponsored by the World Puzzle Federation, it is called the World Puzzle Championships. The United States team has competed in this forum for many years, solving some of the toughest puzzles found in any contest. (For more about the World Puzzle Federation, see http:// www.worldpuzzle.org; for more about the Team USA, see http://wpc.puzzles.com.)

One of the most well-known U.S. contests is the American Crossword Puzzle Tournament, which started in 1978 and is the first such contest since the 1930s. The contest had just five puzzles; points were awarded for correct letters, not words, in the grid, making it difficult to judge. Today, the grading and puzzles are more streamlined, making it popular; in fact, close to a thousand people now compete in the ACPT each year. For 30 years, the competition was held in Stamford, Connecticut; by 2008, because of the increase in participants and growing popularity of puzzles, it was moved to Brooklyn, New York. (For more about the ACPT, see http://www.crosswordtournament.com; for more about puzzles, see “Mathematical Resources.”)

What is an example of a logic puzzle?

Most of us have encountered logic puzzles before, usually on mathematics tests given in grade and high school. They often contain numbers (the mathematics connection) and a (often seemingly convoluted) sequence of events to which the reader has to determine the outcome.

For example, the following will jar the memory banks of everyone who has ever seen such logic puzzles on an exam or elsewhere: One weekend, three people check into a bed and breakfast. They pay $30 ($10 each) to the B&B owner and go to their room. The owner remembers that there is a special deal that weekend, and the actual room rates come to $25. He gives $5 to his brother (who works with him) and tells him to return the money to the guests. On the way to the room, the brother realizes that $5 would be difficult to share between three people—5/3 = 1.666 …, an uneven number—so he pockets $2 and gives $1 to each person. Therefore, each person paid $10 and got back $1; this means they paid $9 each, totaling $27. The brother has $2, so the entire total is $29. Where is the remaining dollar?

The solution? Deduction comes in handy here. It’s the owner who’s important to pay attention to, not the brother. Overall, each guest paid $9 (a total of $27), the owner now has $25, and the brother has $2; thus, the brother’s amount should be either added to the owner’s money (25 [owner] + 2 [brother] + 3 [guests] = 30), or subtracted from the guests’ amount of $27 (27 [guests] -2 [brother] + 5 [refund from owner] = 30)—not added to the guests’ amount. This proves that not only is the owner’s brother a bit dishonest, but also how easy it is to con some people out of a few dollars if they aren’t used to solving logic puzzles!

There are also more local tournaments, such as the Boston Crossword Puzzle Tournament. In 2011, it was held at Harvard University and drew over 150 participants from Harvard and the general public. They turned out to solve four puzzles developed by Boston-area constructors of puzzles; the on-stage final was held with the three top finishers all competing to solve a fifth puzzle.

Another local puzzle contest took place in 2011. Called the Bay Area Crossword Puzzle Tournament, it was held as a benefit for the nonprofit California Dictionary Project . The four puzzles used for the competition were developed by the crossword puzzle editor of the New York Times, Will Shortz (1952-), who donated unpublished puzzles.

What are the number of possible positions for a Rubik’s Cube?

Rubik’s Cube was invented in the 1970s by the Hungarian architect, inventor, and mathematician Ernö Rubik (1944-), who also invented a number of other puzzles, including Rubik’s Clock. The cube measures 3 by 3 by 3, with a total of 26 subcubes on the outside. All the subcubes are hinged, making them easy to turn (by a quarter turn in either direction) in any of the planes on the cube. Initially, each of the six sides are painted a certain color; the object is to move the cube planes in a random way, then return the cube so that each side has a single color again.

What are the possible number of positions of a Rubik’s cube? Mathematicians need to use factorials (symbolized with the ! sign; for more information about factorials, see “Algebra”) in order to find out the many iterations, as seen in the following equation:

Image

The number of positions turns out to be 43,252,003,274,489,856,000, or more than 43 quintillion turns.

What is the St. Ives problem?

The St. Ives problem is one of deduction and reasoning. The centuries-old original poem states:

While on my way to St. Ives,

I met a man with seven wives.

Each wife had seven sacks;

Each sack had seven cats;

Each cat had seven kits.

Kits, cats, sacks, wives;

How many were going to St. Ives?

Image

The popular Rubik’s Cube puzzle requires a player to spin sections of the cube in order to make the colors on all six sides match.

By this time, most people start adding and multiplying, trying to come up with the answer. But in reality, it’s a trick question: The narrator is on the way to St. Ives; the group he or she met along the way were leaving, not going to, St. Ives. Therefore, the number “going to St. Ives” equals (at least) one: the narrator.

Of course, there are some mathematicians who can’t leave well enough alone, and have calculated the total number of cats, kits, sacks, and wives, based on a geometric series. According to this equation, 2,801 were going to St. Ives, if the man, his wives, their cats, etc. had turned around.

What is the popular game of Sudoku?

Sudoku (pronounced soo-doo-koo, not the incorrect sa-doo-ko) is a very popular mathematically based puzzle game that has somewhat mysterious origins. According to puzzle expert Will Shortz, the modern version of Sudoku was probably developed by Howard Garns (1905–1989), a retired architect and freelance puzzle constructor. In 1979, Garns published something called “Number Place” in a Dell magazine, a popular publisher of number, letter, and word games. But it took just over two decades for the game to be noticed in any detail. At that time, it was brought out by the Japanese puzzle company Nikoli under the name Sudoku, which means “single number” in Japanese. But in reality, it actually took until around 2005 to truly make Sudoku a household name (Garns died in 1989, so he never saw his invention become popular).

Sudoku doesn’t require any calculations or arithmetic skills, but is actually a game of placing numbers in squares using simple rules of logic and deduction. (Because of this, it is a game loved by young and old alike.) The overall game is played on a large grid that contains nine 3-by-3 grids that are partially “solved” by the person developing the puzzle. The objective is to fill in the remaining numbers in the entire 9-by-9 grid so that each column, row, and each of the nine 3-by-3 (called boxes, blocks, regions, or sub-squares) contain all the digits from 1 to 9. There is usually only one solution; when completed, the result is essentially a type of Latin square (see above) with the squares being individual sections that make up the entire 9-by-9 grid.

There are definite rules, too. For example, the same single number cannot appear twice in the same 9-by-9 playing-grid row or column; and the same number cannot appear twice in any of the nine 3-by-3 blocks of the entire playing grid. As long as the puzzle comes with at least 17 digits already placed on the grid, it is considered a solvable, one solution Sudoku. If there are less than 17 digits, then the puzzle has more than one possible solution, and therefore the game cannot be solved properly.

Image

Sudoku is a popular modern puzzle in which players must figure out which numbers fall within 9 grids, with each 3-by-3 grid containing the numbers 1 through 9 without repetition.

It may seem mathematically simple, but the 9-by-9 grid means there are 81 numbers that have to fall into the correct sequence, which means there are millions of possible combinations. Mathematically, the total number of possible digit combinations on a standard Sudoku grid is 6,670,903,752,021,072,936,960. But most mathematicians argue that many of these combinations are identical, only seen on the grid backward or rotated. Figuring out the number again shows there are 3,359,232 possible combinations. Thus, in a way, that is the total number of possible Sudoku puzzles. If you’ve only solved a half dozen, you have a long way to go!

What are some off-shoots of Sudoku?

Like a television program that creates a spin-off program highlighting a character from the original program, most successful games, especially mathematical puzzles, produce off-shoots of their own, and Sudoku is no exception.

One of the most popular variants is Hypersudoku, which is also known as NRC Sudoku, Windoku, Hyper-Sudoku, or 4-Square Sudoku. As it sounds, this game has the same layout as the normal Sudoku, but with additional interior blocks that have to contain the numbers 1 to 9. The problems to solve include the overlapping squares, which means the player has to scan and know what numbers fit into the squares and overlap squares.

Sudoku game iterations also come as overlapping and multiple grids. One that is popular is the Japanese Gattai 5 (“merged five”) Sudoku (also called Samurai SuDoku or High Five, depending on the publication in which the puzzle is found), in which five 9-by-9 grids overlap at the corner regions, forming a shape called a quincunx. Not to be left out—and as found in most Latin square puzzles (see above)—some Sudoku games come with letters rather than numbers. One of the more popular games is called Wordoku, in which the puzzle is solved using a certain amount of letters within the grid.

There are dozens of other Sudoku spin-offs. For example, there is the Sudoku version of Rubik’s Cube, called Sudoku Cube (for more about the Rubik’s Cube, see this chapter); and even a three-dimensional version of Sudoku invented by Dion Church in 2005. And, of course, there are books, magazines, and sundry websites all dedicated to not only regular Sudoku, but all the off-shoots you can imagine.

Are there any Sudoku puzzle championships?

Yes, there are plenty of championships to compete against other Sudoku players, too many to list here. One such competition is the International Sudoku Tournament, such as the one in 2011 held in Beijing, China (the former and current national Sudoku champions Sarah Jane Cua and Timothy James Tan topped the youth category). There is also the World Sudoku Championship, first held in March 2006 in Lucca, Italy; in 2011, it was held in Hungary. There is also the Sudoku National Championship in the United States. In addition, there are more local championships, such as the Indian Sudoku Championship, which is held to qualify contestants to represent their country in the World Sudoku Championship. Most of the competitions offer cash prizes, a chance to go on to other tournaments, or just the satisfaction of winning.

How has Sudoku improved the world of genome sequencing?

Researchers discovered that the logic of a Sudoku mathematical puzzle can be used to understand genome-sequencing, and thus, the growing field of genetics. The researchers called their discovery “DNA Sudoku” in which a 2,000-year-old Chinese mathematical theorem was combined with several concepts from cryptology. In the past, only a single DNA sequence could be seen at one time; then a method to determine several hundred at a time was developed.

But the DNA Sudoku outdoes them all, allowing tens of thousands of DNA samples to be combined, along with their sequences determined (with the four bases in DNA, represented by the letters A for adenine, G for guanine, C for cytosine, and T for thymine) all at once. Not only does it speed up sequencing, it reduces the amount of money spent on the task, from upwards of $10 million to only tens of thousands of dollars with the DNA Sudoku method. Not only does it save money, it also speeds up sequencing when exploring how certain DNA sequences are associated with genetic diseases such as TaySachs. This way, individuals who carry the disease-causing mutations can easily be identified.

The DNA Sudoku researchers’ goal was to speed up the sequencing of many samples (called multiplexing). To do this, each sample is tagged with a “barcode,” a short string of DNA letters identifying the sample. From there, the samples were sequenced, with the barcode tags telling the researchers which sequence belonged to what sample. But making a single barcode for each sample was time consuming and costly, especially since there are thousands of samples to examine. The researchers then decided to mix the samples in certain patterns, making “pools” of samples with one barcode, all based on the Chinese remainder theorem. They named the entire method DNA Sudoku because of its resemblance to the logic and combinatorial number-placement rules of the Sudoku game we all know and love to play.

MATHEMATICAL GAMES

What is a game?

A game is a recreational activity that involves a “conflict” resulting in gains and losses between two or more opponents (although some games can involve one player acting alone). In general, all games must have a goal that the players are trying to reach, and the opponents must follow strict, formal rules that determine what the players can or can’t do within the game. If any of the rules are broken during the game, it is often referred to as a foul—or, at its worst, cheating.

The study of games is also a branch of mathematics and logic that is called game theory. In game theory, games can be simple and solved with mathematics that result in a complete “solution” (result of the game). It also includes the analysis of more complex games, such as cards, chess, and checkers, and can even be applied to real-world situations in economics, politics, and warfare.

What is gambling?

Gambling is the act of playing a game for stakes—it is thought of as the art of taking chances. It is also often called betting. A bet is the amount of money, or other object of value, that is risked in a wager. Most people gamble with the hope of winning a certain stake, usually a cash payment. But in order to get such a payoff, the gambler must risk money or valuables, betting these items on the outcome of a game, contest, or other event. All of this depends on the outcome of activities that are partially or wholly dependent upon chance. What is chance? Mathematically speaking, chance is a measure of how likely it is that an event will occur—a probability. (For more about chance and probability, see “Applied Mathematics.”)

What are betting odds and how are they determined?

Betting odds are usually written in the form r:s, in which r is the “chances for” and s is the “chances against.” This can be stated as: “r to s” or “chances for to chances against”; or “s to r” or “chances against to chances for.” But note that odds of 1:1 are said as “one to one” not “one out of one.” Odds are usually calculated as follows:

total chances = chances for + chances against;
or chances against = total chances - chances for

Using a 52-card deck as an example, the odds of drawing a king from the deck are 1 in 12—or 4:(52 - 4) = 4:48, which equals 1:12.

What is the difference between probability and odds?

Probability is usually expressed as a fraction (sometimes as a percentage). For example, if there are ten pieces of fruit in a jar—three apples and seven oranges—then the probability of taking out an orange is 7/10 (or seven chances of an orange out of a total of ten chances).

On the other hand, odds are expressed as the number of chances for (or against) versus the number of chances against (or for). Thus, if there are three chances of picking an apple and seven chances of picking an orange, the odds are 7 to 3 against you picking an apple. Just reverse this to find the odds in favor; or, in this example, the odds would be 3 to 7 in favor of picking an apple.

In order to convert the odds to probability, just add the chances. Thus, if the odds against a horse winning the Kentucky Derby are 4 to 1, that means that, out of 5 (or 4 + 1) chances, the horse has one chance of winning. That makes the probability of the horse winning 1/5, or 20 percent.

How do raffles work?

Most raffles one encounters usually have to do with the local library, animal shelter, or other nonprofits trying to raise money. In the majority of cases, a simple raffle entails buying a ticket for a chance to win a prize. For example, a fundraiser raffle ticket costing one dollar will provide a chance to win a new car, big screen television, a dinner at a local restaurant, or seats to a concert—in other words, there is potential for a big pay off at a minimal investment.

The basics of raffle drawings are simple. A group buys the tickets and purchases the prizes (or they are donated). People buy the tickets at a set price (and, as everyone knows, the better the prizes, the more tickets sold). The more tickets you buy, the better the chance to win. Thus, raffles are considered to be a game of chance. (But note: Because raffles are games of chance, many states have tight regulations and laws that need to be followed—even for nonprofit organizations—so check with your local government laws if you run a raffle.)

There is another type of raffle called a 50-50 raffle, in which 50 percent of the money taken in by the raffle sales is offered as the prize, while the other 50 percent is collected by the group giving the raffle. In most cases, a 50-50 raffle has a time limit, and only a single raffle ticket is randomly chosen to win. Some groups also make other stipulations, such as the winner has to be present to win, or a certain amount of tickets need to be sold in order to pick a winner.

Image

The odds of winning a state lottery are always the same because lotteries choose from the same group of numbers with each drawing.

Why is it difficult to win a lottery?

According to one state lottery site, a lottery “is a plan that provides for the distribution of money, property, or other reward or benefit to persons selected by chance from among participants some or all of whom have given a consideration for the chance of being selected.” In other words, a person buys a chance at winning a certain sum of money. But in reality—as with many games of chance—the odds are not in the participants’ favor. With most lotteries, such as a “lotto-type” lottery, a person has a better chance of being in a car or plane accident, or being hit by lightning, than winning. But that doesn’t stop many people. Every year, billions of dollars are spent on state lotteries in the United States alone.

The reason for this “dream of winning” is simple: It’s how this game of chance is perceived. Many people believe that if they just keep playing the same number(s), eventually that number or numbers will be chosen. What they often don’t realize when playing a lottery is the idea of replacement. For example, choose a 52-card deck to represent the lottery, with the queen of hearts as the winning card. Say the first picked card from the deck is the king of diamonds and it’s not reshuffled back into the deck. From there, we continue to pick cards. After each choice, if the cards are not put back into the deck, eventually, our chances of picking the queen of hearts gets better and better. After all, the choices of cards in the deck become less. If it is not picked yet, the last card will at some point be the queen of hearts.

But a regular lottery does not reshuffle the numbers. Instead, lotteries choose from the same group of numbers each week, which makes it even more difficult to win. There may be repetitions in winning numbers, but the odds of winning are the same each time the lottery is played.

What are the odds of winning the powerball lottery?

It was eventually going to happen: A number of lottery-offering states got together to have lotteries with huge amounts of prize money. The resulting powerball lotteries have been very lucrative—not for the players, but for the states. Such gigantic sums of money tempt quite a few people to take the risk, with many buying hundreds of tickets in an attempt to better their odds.

But does it work? Not really. There is a way to determine the odds of such “lotto-type” lotteries in which numbered balls (or numbers) are randomly chosen to represent a winning number. This is usually expressed as: n! / (n - r)! r!, in which n is the highest numbered ball and r is the number of balls chosen. (The n! is “n-factorial”; for more information about factorials, see “Algebra.”)

In math, this type of equation is called a combination. For example, if there are 50 balls and 5 are chosen, there are 50 possible numbers that can come up first, leaving 49 that can come up second, and so on. The equation becomes:

Image

or the chances of winning are about 2 million to 1.

Do all states have the same lottery odds?

No, not all states have the same lottery odds. Each state not only has its own lottery names and numbers, but also a specific budget to spend and offer as winnings. There are more than 35,000 lottery games in over 40 states, with over 50 billion dollars in sales each year, which does not even include spending in casinos, racing, or charity events. Adding those in would raise the average annual amount spent on gaming to around an amazing $90 billion. All this money changing hands—and the millions of people who like playing the odds—means that state lotteries and gaming will be here for a long time.

As of this writing, Delaware has the second most per-capita lottery spending in the country, at about $633.66, with the winnings per dollar spent at 15.5 cents. In other words, the state’s residents spend a great deal of money with little return; in fact, no one in the state has won a million dollars or more since the summer of 2009.

Yet another state excels when it comes to the lottery: in Massachusetts, the per-capita lottery spending is $806.57; the winnings per dollar spent is 77.3 cents. To date, it is reported that the state has taken in more than $4 billion in lottery revenues, which represents an average expenditure of more than $800 per adult. The winnings per dollar of 77 cents is the best return in the country; and the state’s lottery website boasts that more than a dozen people have won $1 million or more in the month of March 2011 alone.

But no matter how you slice it, playing the lottery (or any gaming activity) in any state is not the best odds. The states are in it to make money, not to give out money. For example, according to one statistic, if the odds of winning a California Super Lotto game were 1 in 18 million (which is feasible), and if one bought 50 lottery tickets a week, the chances of winning comes down to once every 6,923 years, which is why most people should not quit their day jobs and play the lottery anywhere. The odds are not, and never will be, in your favor, no matter how many states advertise people who win it big.

CARD AND DICE GAMES

What are card games?

A collection of cards (or a deck) is a set of n rectangular pieces, usually made of heavy coated paper or cardboard, that hold special varying markings on one side and a uniform, identical pattern on the other. The special markings make each card unique, with each marking representing something playable in a certain card game.

Why isn’t it easy to win a powerball lottery
if there are so many numbers to choose from?

Don’t think powerball lotteries give a person an edge. For example, a power-ball lottery can be one in which 5 out of 50 balls (or numbers) are drawn, with an extra powerball pulled out of a different number of balls. This is not like figuring the odds for a “drawing 6 out of 50 balls” contest, but is actually two separate lotteries: one with 5 out of 50 balls and one with the powerball group.

The probability of matching the first five balls is determined as above. But the powerball is separate. For instance, say the powerball is taken from a group of 36, making the powerball group’s odds 36:1. The probability of winning the entire jackpot can be determined by adding this to the results of the five-ball draw. Now the odds become even higher: The 50/5 drawing + a powerball of 36 = (2,118,760 × 36):1 = 76,275,360:1 or about 76 million to one. And there are even worse odds if more balls are in the powerball group.

As for state powerball lotteries, there are plenty. According to the Multi-state Lottery Association, powerball lotteries are, as of this writing, played in 42 states, Washington, D.C., and the U.S. Virgin Islands. The list below gives the Multi-state Lottery Association’s own odds for winning the powerball, but from the calculations above, their numbers are a bit off, so don’t be fooled:

Powerball Prizes and Odds

Match

Prize

Odds

5 balls + powerball

Grand Prize

1 in 195,249,054.00

5 balls

$200,000

1 in 5,138,133.00

4 balls + powerball

$10,000

1 in 723,144.64

4 balls

$100

1 in 19,030.12

3 balls + powerball

$100

1 in 13,644.24

3 balls

$7

1 in 359.06

2 balls + powerball

$7

1 in 787.17

1 ball + powerball

$4

1 in 123.48

powerball only

$3

1 in 61.74

The most common cards for games is a 52-card deck represented by four specifically colored suits (spades and clubs in black, diamonds and hearts in red), with 13 cards of each suit numbered 1 through 10, followed by several face cards—jack (J), queen (Q), and king (K). Card “1” is usually an “ace”; card 11 is represented by a “jack,” 12 by a “queen,” and 13 by a “king.” The value of the ace often changes depending on the game. For example, it can either hold a value of 1 or 11 (in blackjack) or 14 (in bridge). Such cards are also used for many gambling games, such as poker and baccarat. Interestingly enough, the investigation of the probabilities of various outcomes in card games was one of the original motivations for the development of modern probability theory. (For more about probability, see “Applied Mathematics.”)

What are the probability and odds of drawing a certain card from a deck?

The probability of an event is usually described as the chances for the event to occur over the total chances for the event to occur (chances for / total chances). In the case of a deck of 52 cards, the probability of drawing a king from the deck is 4/52 = 1/13 = 0.077, or 7.7 percent. As seen above, the odds, or the ratio of chances for to chances against (chances for : chances against) can be found by the formula: total chances equals the chances for plus chances against (or total chances = chances for + chances against, or chances against = total chances - chances for). Thus, the odds of picking a king from a deck of 52 cards is 4:(52 - 4) = 4:48 = 1:12.

What are the probabilities and odds of being initially dealt certain hands in five-card poker and bridge?

Because there are a certain number of cards in a playing deck, mathematicians have worked out the probabilities and odds of being dealt certain hands for certain games. The following explains the odds in terms of “chances against : chances for.”

In five-card poker the probability of drawing a royal flush (a poker hand with the ace, king, queen, jack, and 10 all in the same suit) is 1.54 × 10-6, with odds of 649,739:1. For a straight flush (a poker hand with consecutive cards in the same suit, but not a royal flush), the probability is 1.39 × 10-5, with odds of 72,192.3:1. Three of a kind (three cards with the same value) has a probability of 0.0211, with odds of 46.3:1, while one pair (two cards with the same value) has a probability of 0.423, with odds of 1.366:1.

In bridge 13 top honors has a probability of 6.3 × 10-12 and odds of 158,753,389,899:1. A 12-card suit, ace high, has a probability of 2.72 × 10-9, and odds of 367,484,697.8:1. Getting four aces has a probability of 2.64 × 10-3, with odds of 377.6:1.

What is the casino’s or house edge?

A casino is a gambling facility that normally includes all or a combination of the following: slot machines, video games, card games, and other games such as keno, craps, and bingo. In order to make the casino the major money-maker—not the gambler—there are certain “rules” of the casino, including the casino’s edge, otherwise known as the house edge.

The house edge is the ratio of the expected player loss to the initial amount bet; it is an exact measurement, usually expressed as a percent, of the casino’s advantage in a game. A casino earns money by paying winners at “house odds,” an amount that is slightly less than the true odds of winning the game. There are definitely exceptions to this rule, which is why there are professional gamblers, but the majority of the time, the casino has the advantage.

What are the possible numbers of distinct 5-card hands in 5-card poker and 13-card hands in bridge?

There is, of course, a mathematical way of determining the possible number of distinct 5-card hands in poker—or the various combinations of 5 cards from a 52-card deck. The formula, in which N is the number of combinations, is as follows (note: the 52 written over 5 is called a binomial coefficient; for more about coefficients, see “Algebra”):

Image

Not to be outdone, such a number can be determined for bridge games, too. In the following, N again represents the combinations:

Image

For example, if the house edge for blackjack or slots is 5 percent, then for every $100 bet initially made, the player can expect to lose $5. This not only helps the casino, but it is sometimes a way for patrons to compare one game to another, and even estimate how much they may lose.

Of course, this is not always so straightforward. Some people win and others lose in the short run, but the casino always wins in the long run. There are also exceptions because of the way a game is played, such as in craps (see below). The best scenario would be to have no house edge, or an edge against the house.

What are the odds when playing blackjack?

Blackjack, or twenty-one (sometimes called pontoon), is a game of chance based on a 52-card deck. The name comes from gambling houses and casinos that wanted to popularize the game by offering a 10-to-1 payoff for a hand consisting of the ace of spades and a black jack (club or spade). It worked, and blackjack has ever since been one of the most popular games in casinos (and as a personal card game), even though a winning hand doesn’t have to contain a jack.

In blackjack, the player plays against the dealer. Each player is dealt two cards: one face down and one face up; the dealer also deals two cards to himself or herself: one face down and one face up. The player then decides, based on how much his or her cards add up to (not to exceed 21), whether he or she wants the dealer to “hit” him or her, meaning having the dealer deal another card from the top of the deck. Numbered cards are valued according to their number; face cards (kings, queens, and jacks) are equal to 10 points; and the ace can either be 1 or 11.

The player wants to get exactly 21, or a number under 21 but more than the dealer. For example, if the player has a 9 of hearts and an ace, the amount is equal to 10 or 20 (9 + 1 or 9 + 11). If the dealer shows a total of 17 with all his or her cards, the player wins, getting twice his or her bet; if the dealer shows a total of 21 with his or her cards, the dealer wins, and the player loses his or her bet. If the dealer and player have the same amount, it is called a push, and the player keeps his or her bet. And if either the player or dealer draws two cards that equal 21, it is called a blackjack and the person with blackjack wins the hand.

Overall, no matter how good you play, blackjack is almost always in the favor of the dealer, not the player, especially when playing in a casino (for more about casinos and winning/losing, see above). There are some strategies that may help a person win, including card counting to determine the cards already used, but most of those tactics are forbidden in casinos.

What are dice?

Dice are small cubes usually used in games of chance. Each die (dice is the plural) has six sides numbered with dots from one to six. The dots are placed on the cubes so that the sum of dots on opposite sides equals seven; the total number of dots on each die equals 21. They are mostly associated with certain types of games, with the simplest involving a player, or many players, who throw (or toss or roll) the dice for the highest sum.

Dice have been around for more than 3,000 years, with evidence found in ancient Egyptian tombs, Chinese burial chambers, and the ruins of Babylon. The Greeks and Romans were avid users of dice and associated games, and they have been popular from the Middle Ages on. Dice have been made from many materials, including ivory, bone, wood, metal, and eventually plastic.

Not all dice are square. The cube (or hexahedron) belongs to the group of five Platonic solids, or solids formed by regular polygons. Thus, other Platonic solids have also been made into dice, including shapes (polyhedra) such as tetrahedrons, octahedrons, and dodecahedrons, which are used for certain types of games. (For more information about Platonic solids and polyhedra, see “Geometry and Trigonometry.”)

How do you play yahtzee?

Yahtzee is a derivation of the name “yacht game.” It was invented by a wealthy Canadian couple to play on their yacht. They soon contacted a person who made his fortune selling Bingo games in the 1920s, American game entrepreneur Edwin S. Lowe (1911–1986), who then spread the word about the yahtzee game (even to the point of having yahtzee parties), and the rest is favorite-game history.

What are the odds when playing craps?

Craps is probably the most popular game of chance in the world; it is also illegal to play in many places. But it has a long history: It was played in ancient Greece and Rome and was even a mainstay of some old 1930s and ‘40s movies. Craps can be played using a wall and a pair of dice. It is a popular casino game in places such as Las Vegas—and even on the Internet—with betting on craps involving a complex equation.

Its popularity no doubt comes from its simplicity. In craps, a player throws two dice; their number (roll) is the total of the dots on the top faces of the dice. If the initial roll is a 7 or 11 (called a natural), the player wins. If the number 2, 3, or 12 comes up—called craps—the player loses, but keeps the dice. If the sum of the dice adds up to the number 4, 5, 6, 8, 9, or 10, that number becomes the thrower’s “point.” The player then continues to shoot until he or she throws the point number again, in which case the gambler wins and retains the dice. But if the player shoots a sum of 7 before he or she can roll the point value, he or she loses and gives the dice to the next player.

Craps is truly a game of chance, with the probability mathematics of craps fairly straightforward. For example, take the probability of winning on a roll-by-roll basis, in which P(p = n) is the probability of rolling a point n. The resulting numbers show that the probability of winning is 244/495, or the shooter wins about 49.2929 percent of the time.

The game itself is one of chance: Each player rolls five 6-sided dice three times, with each die having a set amount (see table below for the amounts relegated to each dice throw). Besides the table, the dice thrower can have all dice showing a single number, four and three of a kind, four or five numbers in a row (“small straight” and “large straight,” respectively), two of one number and three of another (“full house”), or yahtzee (five of the same number). The word “chance” is used to describe any roll. The roller of the dice also gets 35 bonus points if the total from the first six categories (on the list below) is greater than 62.

The following lists all the possible iterations of tossing the dice and how many points are rewarded for each:

Aces (sum of 1s)

Twos (sum of 2s)

Threes (sum of 3s)

Fours (sum of 4s)

Fives (sum of 5s)

Sixes (sum of 6s)

3 of a kind (sum of all the dice)

4 of a kind (sum of all the dice)

Full house (25)

Small straight (30)

Large straight (40)

Yahtzee (50)

Chance (sum of all the dice)

Image

In the game of yahtzee, players role five dice, trying to get various combinations to score points.

According to mathematician P. Woodward, there are 1,279,054,096,320 possible outcomes of throwing the dice in yahtzee.

There is also another form of yahtzee called triple yahtzee in which players try to get each type of roll three times over the course of the game, not just once. Each number of the toss is put into a single, double, or triple column; the value is then multiplied by the stated weight when scores are totaled.

As of this date, the company that manufactures the game, Hasbro, claims that more than 100 million people around the globe play yahtzee. If this is true, there are many games out there for everyone to play!

SPORTS NUMBERS

What is sabermetrics?

Sabermetrics is the study of baseball using objective evidence, such as baseball statistics. It uses scientifically based data and various interpretation methods to explain why teams win and lose. Sabermetrics was taken from the acronym SABR, or the Society for American Baseball Research, and was coined by baseball historian, statistician, and writer Bill James (1949–).

What are some of the statistics used in baseball?

There are numerous statistics used in baseball, including batting and pitching statistics. Batting statistics can be divided into several numbers. The batting average (AVG) is the number of hits a player makes divided by the number of times at bat. It does not include walks or sacrifice hits. The runs batted in (RBI) is the number of runners who scored on a player’s hit, base on balls, or sacrifice. The on-base plus sluggage (OPS) is a good measure of a hitter’s ability. This statistic combines getting on-base (on-base percentage, or OBP) and advancing runners (slugging percentage, or SLG). It is also more accurate thanks to the adjustment 1.2 × OBP + SLG, which compensates for the fact that SLG has a wider range than OBP.

Pitching statistics include the ERA and WHIP. The earned run average (ERA) is the earned runs times the innings in a game (most commonly nine) divided by the innings pitched. The walks plus hits per inning pitched (WHIP) records the bases on balls (walks) plus hits divided by innings pitched. It’s a good way to measure the approximate number of walks and hits a pitcher allows in each inning that he pitches. It then compares that amount to other pitchers to formulate a pitcher’s index.

Image

The mathematical study of the American pastime of baseball is called “sabermetrics.”

What are some statistics used in football?

There are several mathematical statistics used in football. The touchdown percentage is the touchdown passes divided by the pass attempts. The passer (quarterback) rating is determined by four elements and their statistical calculations: percent of completions, average yards gained per attempt, percentage of touch-down passes, and the percent of interceptions. The average is 1.0, the bottom is 0, and the maximum anyone can receive in any category is 2.375 (this is difficult; for a passer to gain 2.375 in completion percents, he would have to complete 77.5 percent of his passes). Those passers who earn a 2.0 rating or better are exceptional. For example, the National Football League ranks this at 70 percent in completions, 10 percent in touchdowns, 1.5 percent in interceptions, and 11 yards average gain per pass attempt.

What is the significance of all those numerals on car windshields?

Yes, numbers are even associated with cars. For example, car windshields often come with necessary numbers, such as the car’s registration, vehicle identification numbers (VIN), or even the inspection sticker numbers. But what about other numbers associated with cars, such as NASCAR drivers’ numbers? There are a lot of people who remember their favorite drivers by their racing numbers. Some that come to mind are #2 Rusty Wallace; #3 for the late Dale Earnhardt; #8 Dale Earnhardt, Jr.; #24 Jeff Gordon; and #88 Dale Jarrett.

Rushing is a statistic commonly heard after any football game on the after-game special. It is the average yards per carry (AVG), a number measured by the total yards divided by the attempts. In punting the net punting average (NET) is the gross punting yards, minus the return yards, minus 20 yards for every touchback, divided by the total punts.

What are some basketball statistics?

One of the main ways a National Basketball Association coach evaluates a player’s game performance is by efficiency, which is determined by using the formula: (points + rebounds + assists + steals + blocks) - (field goals attempted - field goals made) + (free throws attempted - free throws made) + turnovers.

Field goal percentage (FG percent) is determined by the field goals made (FGM) divided by the field goals attempted (FGA). Finally, the free throw percentage (FT percent) is determined by the number of free throws made (FTM) divided by the free throws attempted (FTA).

What is a point spread?

A point spread is the predicted scoring difference between a game’s two opponents; it is used by bookmakers to equalize two teams for betting purposes. For example, if a person chooses the team that is favored, the gambler has to win by more than the point spread in order to get credit for a correct pick. If the underdog team is chosen, the bettor must lose by less than the spread number in order to get credit for a correct pick. If a point spread is listed as “off,” this means there is no official point spread for the game.

For example, if a person picks Army to win, and the team is favored to beat Navy by five points (called a spread of five points), Army must win the game by six or more points in order for the bettor to win. If Army wins by exactly five points, the game is called a “push,” and no one wins.

How does a person determine the odds in horse racing?

The track makes its money in a certain mathematical way: When the odds of the horse race are converted to probabilities, they usually add up to more than 1, giving the track the advantage. For example, say a race track has 12 races with four horses, and the odds of winning each race are as follows (note: race tracks don’t use the same horse in each race—this example is for illustrative purposes only):

· Horse 1—1:1; probability = chances for (1) / total chances (1 + 1) = ½ or 6/12

· Horse 2—2:1; 1 / (1 + 2) = 1/3 or 4/12

· Horse 3—3:1; 1 / (1 + 3) = ¼ or 3/12

· Horse 4—5:1; 1 / (1 + 5) = 1/6 or 2/12

If a person puts $1 bets on horse number 2, he or she would have to win 4 out of 12 races to break even. But note: All these numbers add up to 15/12, or 1.25, a higher number than 1, so as long as no horse wins more than its probability, the house wins.

There’s another way of looking at this type of betting. In order for the gambler to do “better” than the track, he or she has to win 15 times in 12 races—a physical and mathematical impossibility—which is why the track always makes money. Of course, a gambler may bet on certain low-probability horses that win more races than expected, earning a few more dollars along the way, but don’t bet on it.

What are some numbers associated with the World Cup?

There are many numbers used in the analysis of most sports, and the World Cup football games are no exception. For example, in 2010, mathematicians from the University of London collected ball-passing data from all of the FIFA World Cup games, and then predicted which teams would win the matches. They put together a network of passes between players throughout the tournament, then analyzed how the networks compared between the teams. This was a straight use of graph theory, the same mathematical ideas used to analyze different types of networks, such as the Internet computer network, modeling what would happen if parts of the network were suddenly removed.

With the football game, each player in the network was given a score called the “centrality,” which measured how useful they were to the entire network (team). The higher the centrality score, the bigger the impact if the player were to be removed. From this, the researchers determined that the Spanish players made the highest number of passes in that year’s tournament—almost 40 percent more than the German team—giving them a higher centrality score. In contrast, the Dutch team was more on the offensive side, with a very low number of passes between players, and thus a low centrality score. The Germans seemed more balanced, with a higher number of passes; and the English team seemed almost even, with no single player more important than another.

Overall, the mathematical formula was right on the money: In 2010, Germany beat England and Spain beat the Netherlands in semifinal play, and the Spanish won the FIFA World Cup final against Germany.

How do you figure out a handicap in golf and bowling?

In several sports games, players will have what is called a handicap, or a numerical measure of a person’s playing ability to create an even playing field for all players. This means that if you play with an expert in the game, your handicap will usually make up for the differences in expertise, or lack thereof.

For example, in the game of golf, a person’s handicap is used to calculate a net score from the number of strokes that are actually played. Although there is no such thing as a handicap in professional golf, for the amateur, it is handy to have; and it’s intended to be an indication of a player’s potential, not his or her average score. The score is calculated based on the player’s golfing ability from his or her recent history of rounds; the golfer can essentially “take a stroke” (or deduct a stroke) from his or her score at a certain hole. (A golfer’s handicap can change as the player improves with practice.) At the end of the game, the golfer’s net score is figured out by subtracting the player’s handicap from the number of strokes actually taken (the gross score); the golfer without the handicap’s score is taken as he or she plays a regular round of golf. The person with the lowest score at the end of the play is the winner.

Bowling is also a sport in which an individual player can have a handicap. For the same reason as in golf, a handicap in bowling makes for a more even playing field; typically, a league or tournament using such a system will vote on an average that is usually higher than the best bowler on the team. This is the number that the handicap will be based on, or the “base average.” For example, if a bowler’s score is 140, and the base average is 210, the handicap would be 70 pins, or 210 – 140 = 70. If the bowler scored 70 for a game, he or she would add 70 to the score, making it 140. Most often, bowlers in a league cannot take the full handicap value, only a portion. For example, if the league only accepted 80 percent of the handicap—and, from the example above, 70 × 80 percent = 56—the bowler can only add a 56 pin handicap to the final score, or 196 (140 + 56 = 196).

JUST FOR FUN

What is number guessing?

Number guessing is a game (some say trick). Some people can do number guessing in their heads, as long as the numbers are kept small. Have a person think of any positive integer n (not zero or any negative numbers) and apply the following steps:

1. For example, a person chooses the number 25 and keeps that number to him- or herself. Ask the person to compute 3 × n. (3 × 25 = 75)

2. Ask if the number is odd or even. (odd)

3. If the number is even, tell him or her to divide the number by 2; if the number is odd, tell him or her to add 1 to the number, then divide that number by 2. (75 + 1 = 76/2 = 38)

4. Tell him or her to take that number and multiply by 3, then divide by 9. (38 × 3 = 114/9 = 12.666 …)

5. Take that number and multiply by 2 (12.666 … × 2 = approximately 25). The answer should be the original number—or close to it.

What is a magic square?

Image

In these 3-by-3 Chinese magic squares, the numbers are arranged so that when added vertically, horizontally, or diagonally, they are always equal to 3 times the center number.

A magic square is an array of numbers in an n by n square that contains positive integers—from 1 to n2—with each number occurring only once. The numbers in the squares are arranged so that the sum in any horizontal, vertical, or main diagonal direction is always the same. This is shown in the following formula: n(n2 + 1)/2.

Magic squares are often divided into orders; for example, a three-order magic square means three boxes per row and three boxes per column. In reality, “magic” squares are actually matrices (for more information about a matrix, see “Algebra”); they can be odd-order (such as a 3-by-3, or 5-by-5 matrix) or even-order (such as a 4-by-4 or 6-by-6 matrix) magic squares. Perhaps the simplest magic square is the 1-by-1 square, with the only entry being the number 1.

Such magic squares have been known for centuries. For example, the Chinese knew about the three unique normal squares of order three. In Chinese literature dating from as early as 2800 B.C.E., a magic square known as the Loh-Shu, or “scroll of the river Loh,” was invented by Fuh-Hi, who is thought of as the mythical founder of the Chinese civilization.

What is Pascal’s triangle?

Pascal’s triangle, as the name implies, is a collection of numbers in the shape of a triangle. Each number in the triangle is the sum of the two directly above. For example, in the accompanying illustration, the 6 on line 5 is the sum of the pair of 3’s above; the next line is 1, 10 (1 + 9), 45 (9 + 36), 120 (36 + 84), and so on. Although the triangle was known to both the Chinese and the Arab cultures for several hundred years before, it was named after the person who brought it to the forefront of mathematics: French mathematician Blaise Pascal (1623–1662). (For more information about Pascal, see “History of Mathematics.”)

How can one visualize Pascal’s triangle using algebra?

One way of looking at Pascal’s triangle is through its connection to algebra. For example, expand (or remove the brackets around) the expression (1 + x)2 = (1 + x)(1 + x) = 1 + 2x + x2. The same can be done with a cube; for example, (1 + x)3 = (1 + x)(1 + x)(1 + x) = (1 + x)(1 + 2x + x2) = 1 + 3x + 3x2 + x3; and even the expression (1 + x)4, which equals 1 + 4x + 6x2 + 4x3 + x4.

The coefficients (the numbers in front of the x’s) in the results are the connection. For the first example, the coefficients are 1, 2, 1; for the second one, 1, 3, 3, 1; and for the last expression, the coefficients are 1, 4, 6, 4, 1. These, of course, are the third, fourth, and fifth lines from Pascal’s triangle.

What are some “life questions” you can figure out using math?

There are many questions you can explore about your own body and age with mathematics. For example, approximately how many Sunday nights can you expect to sleep until you are 100 years old? Just take 100 years, minus your current age, and multiply that result by 52 (weeks in a year with a Sunday). For example, if you are 25, the answer would be (100 - 25) × 52 = 3,900. How many of those will be good night’s sleep is up to you.

Image

In Pascal’s triangle, each number is equal to the two numbers directly above it.

How can I prove that 1 = 0?

There is a way to show that one equals zero, and it includes an interesting “proof”:

Consider two non-zero numbers x and y such that x = y. If that is so, then x2 = xy. Subtracting y2 from both sides gives: x2 - y2 = xy - y2. Then dividing by (x - y) gives x + y = y; and since × = y, then 2y = y. Thus 2 = 1; the proof started with y as a non-zero, so subtracting 1 from both sides gives 1 = 0.

The problem with this proof? If x = y, then x - y = 0. Notice that halfway through the “proof,” the equation was divided by (x - y), which makes the proof erroneous.

To calculate the number of times your heart has beaten since you were born, you need the help of a watch or clock. First, time your heartbeats per minute (to find out how to count your pulse, see “Everyday Math”); then multiply beats per minute × 60 minutes (in an hour) × 24 hours (in a day) × 365.25 days (in a year) × your age. For example, 72 heart beats × 60 × 24 × 365.25 × a person who is 30 = 1,136,073,600 beats since the person was born. Of course, this is an approximation, since the heart usually beats slower at night, and it speeds up when you see the bill for your latest car repair.

Figuring out how much air you breathe during a lifetime is another fun mathematical calculation. If you optimistically decide you want to eventually be 100 years old, and the average person inhales about one pint (or 0.47 liters) of air per breath, you can do the math. First take the number of breaths you take while at rest per minute (say about 21 per minute) × 0.47 liters × 60 minutes × 24 hours × 365.25 days × 100 years old = 519,122,520 liters. Again, this is only an approximation.

What are some famous (or infamous) math sayings?

It’s interesting to note that Proclus (410-485 C.E.) wrote, “Wherever there is number, there is beauty.” And he was definitely not the last to mention math in a sentence. There are plenty of mathematic jokes and sayings out in the world. Here are a few:

· “Old math teachers never die, they just reduce to lowest terms.” Anonymous.

· “How many mathematical logicians does it take to change a light bulb? None. They can’t do it, but they can easily prove that it can be done.” Anonymous.

· “Time equals money.” Anonymous.

· “As far as the laws of mathematics refer to reality, they are not certain, and as far as they are certain, they do not refer to reality.” Albert Einstein (1879–1955).

· “Mathematics is a game played according to certain simple rules with meaningless marks on paper.” David Hilbert (1862–1943).

· “The Nobel Prize in mathematics was awarded yesterday to a California professor who has discovered a new number. The number is “bleen,” which he says belongs between six and seven.” George Carlin (1937–2008)

What are some sayings with numbers in them?

Probably the most commonly known phrases over time have dealt with math, or should we say, numbers. The list of sayings that include numbers can go on and on! The following lists only a few (we know you can add—for all Carl Sagan fans—“billions and billions” more):

One is the loneliest number.

It’s one per customer.

It takes two to tango.

A bird in hand is worth two in the bush.

Two’s company; three’s a crowd.

Three strikes and you’re out.

The third time’s a charm.

He’s six feet under.

It’s six of one, half a dozen of another.

She’s behind the eight ball.

She’s in seventh heaven; he’s on cloud nine.

A stitch in time saves nine.

A cat has nine lives.

It’s better to be judged by twelve than carried by six.

Nine times out of ten.

What is the Monty Hall Problem?

Let’s Make a Deal, a once-popular television game show with host Monty Hall (who was the master of ceremonies from 1963 to 1986), is the origin of the Monty Hall Problem. The same type of problem is also represented in a card game called “three-card monte.”

The Monty Hall Problem is similar to the show. A contestant is given a choice of three doors. Behind one is a car; behind the other two there is nothing. The host asks the contestant to pick a door. After the pick, the host opens one of the unpicked doors—one the host knows is empty. The host then suggests a switch. The big question, and the problem, becomes does the contestant switch his or her choice or keep the originally chosen door?

The mathematical, statistical answer is yes, switch the doors. Why? Because the original probability that the contestant picked the correct door does not change—it is still 1/3. But if the contestant does switch, the probability becomes 2/3. Most people believe that once one of the doors is eliminated, the probability between the remaining doors becomes 50-50, but it does not.

Image

This astronaut on the Moon weighs only 17 percent of what he would weigh if he were standing on the Earth.

How can people calculate their age and weight on other planets?

It’s certainly possible to find a person’s age and weight on another planet without having to travel there. To figure out weight, one just needs to know the amount of the gravitational pull on another planet, moon, or other space body. For example, based on the chart below, if a person weighed 100 pounds on Earth, he or she would weigh 38 pounds on Mercury (100 × 0.38).

A person’s age on Earth depends on how many times the planet has orbited the Sun during his or her lifetime. For example, a person 30 years old has traveled around our star 30 times. In order to work out how old a person is on another planet, the person’s age has to be divided by the period of rotation (in terms of Earth years). Thus, if the same 30-Earth-year-old person lived on Saturn, he or she would be just over 1 Saturn-year-old (30/29.5 = 1.02 Saturn-year-old). On Mercury, he or she would be 124.5 Mercury-years old (30/0.241 = 124.5).

Calculating Your Age and Weight on Other Space Bodies

Space Body

Time to Orbit
around the Sun*

Gravity (Weight)
Compared to Earth

Mercury

87.9 days

38%

Venus

224.7 days

91%

Earth

1 year

100%

Moon

1**

17%

Mars

1.88 years

38%

Jupiter

11.9 years

254%

Saturn

29.5 years

93%

Uranus

84 years

80%

Neptune

164.8 years

120%

Pluto

248.5 years

0.01%

Sun

N/A

2800%

* In Earth days or years.

**Of course, the Moon travels with the Earth around the Sun, so it orbits the Sun once per Earth year. However, the Moon also orbits the Earth 13.37 times per year.