CHAPTER 1 THE LoRE of NumbERs

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1 Chapter 1 The Lore of Numbers CHAPTER 1 The Lore of Numbers 1.1 Ancient Systems of Numeration 1.2 From Hindu-Arabic to the Numbers of Technology 1.3 Magical Number Patterns 1.4 Number Puzzles and Alphametics 1.5 Computation Shortcuts and Estimation Chapter Summary Chapter Review Chapter Test The numerals 0 through 9 are an intrinsic part of our society and are accepted and used throughout the world. Has this always been the case? Have all cultures always used these numerals? Were these the first numerals created? In this chapter, we answer those questions and give you insights into the lore of numbers. You will see that many cultures developed entirely different systems using symbols that might seem entirely foreign to us. You will be introduced to some of the magic of numbers, be intrigued with computation shortcuts and Vedic Multiplication, and be challenged by number puzzles and adventures. page 1

2 Research Projects Chapter 1 The Lore of Numbers 1. Arithmetic Symbols: One of the important advantages of the Hindu-Arabic system is that it allowed the common person to perform arithmetic calculations. From 1400 to 1700, the symbols, +,,,,,and techniques for performing the operations they symbolize were developed. Who were some of the people involved, what were their contributions, and when were they made? 2. Numerology: What is numerology? How does the system of numerology work? Using your name and birth date, perform some numerological calculations. What do you think about the results? 3. Magic Squares: Magic squares are introduced in Section 1.3. Discuss at least two methods for creating magic squares of various sizes. Who developed these methods? Show how to generate magic squares using these methods. Math Projects 1. Four Play: Using four 4 s create problems that would give all the integer answers from 1 to 20. You can use addition, subtraction, multiplication, division, exponents, square roots, and the correct order of operations to create the problems. For example, you could create an answer of zero in the following ways. 0= = 44 ( 4 4) 4 0= = Notice that each problem uses four 4 s to obtain the answer of zero. Now you must obtain answers for 1 to 20 using four 4 s. 2. My Number System: In this chapter, you will be exposed to various systems of numeration. With that background, design your own system of numeration using your own symbols and scheme for representing numbers. Give examples and explanations of the components of your system. Represent your age, phone number, birth date, height in inches, and weight in the system. 3. Original Alphametics: In Section 1.4, you will be introduced to Alphametics. Create four of your own Alphametic puzzles. Be clever in the words used in your alphametic puzzles and find solutions to your puzzles. page 2

3 1.1 Ancient Systems of Numeration Section 1.1 Ancient Systems of Numeration It seems appropriate that early in our journey through mathematics, we should study math s basic building blocks, numbers. Ever since prehistoric times, human beings have concerned themselves with numbers. Prehistoric men and women were almost totally consumed with survival. However, it is believed that they possessed a basic sense of numbers and had the ability to distinguish between more, less, and equal. Their language might have been lacking in the words to represent numbers, but anthropologists agree that, at the very least, they possessed a visual number sense and an awareness of form. As humans progressed, they became civilized. They grew crops, domesticated animals, wove cloth, made pottery, and lived in villages. Their number sense also grew. Though we have no written records, archaeological findings and opinions of anthropologists suggest that by the beginning of recorded history (3000 B.C.), human beings had developed the ability to tally and count. A tally is a mark that represents the object being counted. This process of tallying took the form of scratches on the ground or on cave walls, knots on ropes or vines, piles of pebbles or sticks, and notches on pieces of bone or wood. For example, to count the number of days between full moons, you could make a mark each evening on the wall of your bedroom until the next full moon appeared. Such a tally might look like this: ///// ///// ///// ///// ///// /// However, if the number of objects to be counted is very large, the tally method becomes very cumbersome. For example, if you wanted to record the population of the United States (296 million in 2005) using the tally system shown above and the tallies were typed on both sides of standard letter-sized paper, you would end up with a pile of more than 22,000 pieces of paper. As society became more involved in measurement, commerce, and taxes, more efficient means of representing the number of tallies were needed. A number is a quantity that answers the question How much? or How many? The symbols that are used to represent numbers are called numerals. In common usage, the terms number and numeral are used interchangeably, but the number is really the abstract concept, the amount or value, and the numeral is a symbolic representation of that amount or value. For example, the amount of hands shown below can be represented by the numeral 10 in the Hindu-Arabic system, X in Roman numerals, in the Attic Greek system, 1010 in the binary system, and so on. IIIII IIIII The set of symbols and the method used to represent numbers is called a system of numeration. In this section we will examine the major systems that have been created: Egyptian (c B.C.), Babylonian (c B.C.), Ionic Greek (c. 450 B.C.), Mayan (c. 300 B.C.), Attic Greek (c. 300 B.C.) Roman (c. 200 B.C.), Chinese (c. 200 B.C.), and Hindu-Arabic (c. 825). We do not expect you to be an expert in ancient systems of numeration. We merely want to expose you to this very important area of mathematics. Furthermore, we do not expect you to memorize all the symbols in this section. As you read this section and solve the problems, you will need to refer frequently to the lists of symbols for a given system of numeration. page 3

4 Systems Using Addition And Subtraction 1.1 Ancient Systems of Numeration Egyptian Hieroglyphic System The most ancient type of numeration system using addition is the Egyptian hieroglyphic system. The numerals used in this system are shown in the table. The value of a number is simply obtained by finding the sum of the values of its numerals. 1 Staff 10,000 Pointing Finger 10 Heel Bone 100,000 Tadpole 100 Spiral 1,000,000 Astonished Man 1000 Lotus Blossom 10,000,000 Sun Example 1 Find the amount represented by the following. Solution: Using the table, add the values of each symbol. 1, 000, , , , , ,000, , , = 1,302,345 Example 2 Write 1753 as a numeral in the Egyptian hieroglyphic system. Solution: 1753 = , which in Egyptian hieroglyphics gives the following. In such an additive system, the symbols can be placed in any order. 123 = = = page 4

5 1.1 Ancient Systems of Numeration Attic Greek System Another example of a system of numeration that uses an additive grouping scheme was found in records in Athens, Greece, around 300 B.C. The Attic system uses the following numerals ,000 50,000 Example 3 Find the amount represented by the following Attic numeral. Solution: Using the chart above, add the value of each numeral. 10, , = 20,819 Example 4 Write 6376 in the Attic system of numeration. Solution: Using the table above, we get the following = = Roman System Around 200 B.C., the Romans also developed a system of numeration that used grouping symbols with addition and subtraction when the numerals were written with 4 s and 9 s that is, 4 (IV), 9 (IX), 40 (XL), 90 (XC), 400 (CD), 900 (CM), and so on. With these numbers, the 1 I 5 V 10 X 50 L 100 C 500 D 1000 M 5000 V 10,000 X 50,000 L 100,000 C 500,000 D 1,000,000 M numeral representing a smaller number is placed before the numeral that represents a larger number. This indicates that the smaller numeral is to be subtracted from the larger one. Example 5 What amount is represented by DMVCCLIV in Roman numerals? Solution: Using the table above, we get the following. page 5

6 1.1 Ancient Systems of Numeration Example 6 D = 500,000 MV = = 4000 CC = = 200 L = 50 IV = 5 1 = 4 Thus, DMVCCLIV = 500, = 504,254. Write 1989 in the Roman numeration system. Solution: 1989 = = M 900 = CM 80 = LXXX 9 = IX This gives 1989 = MCMLXXXIX. Systems Using Addition and Multiplication Traditional Chinese System The traditional Chinese system of numeration appearing in the Han dynasty around 200 B.C. also used grouping symbols and addition of numerals to represent numbers. However, instead of repeating a symbol when many of the same symbols are needed, multiplication factors are placed above the numeral. Numerals are written vertically with the following symbols. 一二三四五六七八九十 百千万 ,000 page 6

7 Example 7 Find the amount represented by the following. 三千六百一十八 Solution: 三千六百一十八 = = = Ancient Systems of Numeration This gives a total of Notice that each digit of the numeral 3618, except for the units digit, is represented by two symbols in the Chinese system. Example 8 Write 453 in the traditional Chinese system. Solution: 453 = 四百五十三 Ionic Greek System The Greeks (c. 450 B.C.) used the 24 letters of their alphabet along with three ancient Phoenician letters for 6, 90, and 900, to represent numbers. The symbols used to represent numbers in the Ionic Greek system are shown in the chart below. page 7

8 1.1 Ancient Systems of Numeration 1 α 10 ι 100 ρ 2 β 20 κ 200 σ 3 γ 30 λ 300 τ 4 δ 40 µ 400 υ 5 ε 50 ν 500 φ 6 ς 60 ξ 600 χ 7 ζ 70 ο 700 ψ 8 η 80 π 800 ω 9 θ 90 Q 900 Τ To represent a number from 1 to 999, write the appropriate symbols next to each other; for example, πδ = 84 and ωκγ = 823. To obtain numerals for the multiples of 1000, place a prime to the left of the symbols for 1 to 9 to signify that it is multiplied by For example, α = 1000 β = 2000 ζ = 7000 θ = 9000 Other techniques were used to represent amounts above thousands. This system allows numbers to be written in a compact form but requires memorization of many different symbols. Example 9 What amount is represented by θφπε? Solution: Example 10 θ = 9000 φ = 500 π = 80 ε = 5 This gives = Write 2734 in the Ionic Greek system. Solution: 2734 = = βψλδ Systems Using Place Values Babylonian System The most advanced numeration systems are those that not only use symbols, addition, and multiplication, but also give a certain value to the position a numeral occupies. The Babylonian system (c B.C.) is an example of this type of system. This system is based on 60 and uses only two symbols formed by making marks on wet clay with a stick page 8

9 1.1 Ancient Systems of Numeration Groups of numerals separated from each other by a space signify that each group is associated with a different place value. Groups of symbols are given the place values from right to left. The place values are..., 60 3 = 216,000, 60 2 = 3600, 60 1 = 60, 60 0 = 1. For example, means = = 43, = 45, 083. The Babylonian system, however, did not contain a symbol for zero to indicate the absence of a particular place value. Some Babylonian tablets have a larger gap between numerals which indicates a missing place value. Example 11 Find the amount represented by. Solution: The groupings represent 2, 11, 0 (because of the larger gap), and 34, which gives , , , = 471, 634 Example 12 Represent 4507 in the Babylonian numeration system. Solution: The place values of the Babylonian system that are less than 4507 are 3600, 60, and 1. To determine how many groups of each place value are contained in 4507, we can use the division scheme shown below = ) (1 group of 3600) ) (15 groups of 60) ) 7 0 (7 ones) = page 9

10 1.1 Ancient Systems of Numeration Mayan System In about 300 B.C., the Mayan priests of Central America also developed a place value numeration system. Their system was an improvement on the Babylonian system because it was the first to have a symbol for zero. The Mayan system is based on 20 and 18. It uses the numerals shown below The numerals are written vertically, with the place value assigned from the bottom of the numeral to the top of the numeral. The positional values from the bottom to the top are: 20 0 = 1, 20 1 = 20, l = 360, = 7200, = 144,000,... Instead of the third position having a place value of 20 2, the Mayans gave it a value of = 360. This was probably done so that the approximate number of days in a year, 360 days, would be a basic part of the numeration system. Example 13 Find the amount represented by the following. Solution: = = 28,800 = = 0 = 7 20 = 140 = 12 1 = 12 28,952 Example 14 Write 17,525 in the Mayan numeration system. Solution: The place values less than 17,525 in the Mayan system are 7200, 360, 20, and 1. To determine how many groups of each place value are contained in 17,525, we use the division scheme below. page 10

11 1.1 Ancient Systems of Numeration ) ) ) (2 groups of 7200) (8 groups of 360) (12 groups of 20) (5 ones) These ancient systems are the forerunners to the system used throughout the world today, the Hindu-Arabic System. In the next section, we will examine this system and see its advantages over the ancient systems. 1.1 Explain Apply Explore Explain 1. What is meant by a system of numeration that uses addition and subtraction? Give some examples. 2. What is meant by a system of numeration that uses addition and multiplication? Give some examples. 3. Compare numeration systems that use addition and subtraction with those that use addition and multiplication. What are the advantages and disadvantages of each type of system? 4. What is meant by a system that uses place values? Give some examples. 5. Compare numeration systems that use addition and multiplication with those that use place values. What are the advantages and disadvantages of each type of system? 6. What are some advantages of a system that uses place values over a system that does not use place values? 7. After studying seven ancient systems of numeration, you can better understand the advantages of the Hindu-Arabic system we currently use. What are some of these advantages? 8. Explain the statement For any number, there are many numerals. Apply What amount is represented by each Egyptian numeral? 9. ) page 11

12 1.1 Ancient Systems of Numeration 12. What amount is represented by each Attic numeral? What amount is represented by each Roman numeral? 17. DCCXXXIV 18. CMCM 19. XLMMCDLVII 20. MMMXL What amount is represented by each traditional Chinese numeral 21. 四 24. 二百千二六十 22. 七千五百二 23. 九万五千三百六十 What amount is represented by each Ionic Greek numeral? 25. λζ 26. φ δ 27. εφνε 28. δκβ What amount is represented by each Babylonian numeral? page 12

13 What amount is represented by each Mayan numeral? 1.1 Ancient Systems of Numeration Explore 37. Represent the number of happy faces shown below in each ancient system of numeration. JJJJ JJJJ JJJJ JJJJ JJJJ JJJJ JJJJ JJJJ JJJJ JJJJ JJJJ JJJJ JJJJ JJJJ JJJ 38. Represent the number of eyes on the happy faces shown above in each ancient system of numeration. 39. Write your birth date in each ancient system of numeration described in this section. 40. Suppose ancient cultures had telephones. How would you write the phone number YEA-MATH in the ancient systems of numeration described in this section? page 13

14 Section 1.2 From Hindu-Arabic to Binary 1.2 From Hindu-Arabic to Binary The system of numeration used worldwide originated with the Hindus in India in about 150 b.c. and was further developed in Persia. Hence, it is called the Hindu-Arabic System. It is a place-value system based on 10 with a zero placeholder. By 900, this system had reached Spain. By 1210, it had been spread to the rest of Europe by traders on the Mediterranean Sea and scholars who attended universities in Spain. By 1479, the digits appeared in the form that is used today. The Hindu-Arabic system of numeration uses ten simple symbols and made computation a reasonable task. It is also called the decimal system, from the Latin deci, meaning tenth. The symbols used in the system are the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. The position a digit holds in a numeral gives it a value based on the powers of 10. The numeral 1,389,260,547 has place values as follows: Billions 1, In expanded form, Hundred millions Ten millions 9, 10 6 Millions Hundred thousands Ten thousands 0, 10 3 Thousands Tens Hundreds Ones ,389,260,547 = The position of a digit tells us what it really represents. The 9 represents nine millions (9,000,000), the 6 represents six ten thousands (60,000), the 4 represents four tens (40), and so on. Example 1 Write 4,175,280 in expanded form. Solution: Example 2 4,175,280 = 4,000, , , = In the numeral 576,239, what do the 5, 6, and 2 represent? page 14

15 1.2 From Hindu-Arabic to Binary Solution: 5 represents five hundred thousands (500,000). 6 represents six thousands (6000). 2 represents two hundreds (200). Numeration Systems with Other Bases The decimal system uses the powers of 10 to determine the place value of each digit used in a numeral. The base of 10 is probably the result of human beings having ten fingers. However, as we saw in Section 1.1, other place-value systems did not use a base of 10. The Babylonians had a system based on 60, whereas the Mayan system was based on 20. Primitive tribes have been discovered that had a system of numeration based on 5, the number of fingers on one hand. The Duodecimal Society of America in the 1960s advocated a change to a base of 12. Computers, on the other hand, use a base of 2. If animals could develop a system of numeration, a horse might use a base 4 system, an octopus a base 8, or an ant a base 6. In this section, we investigate how to write numbers in different bases and introduce the numbers used in technology. The first component of any place-value system is its base. The base of a numeration system is a whole number that is larger than 1. The whole number powers of the base give each position in a numeral its place value. For any base b, the place values are as shown below. # # # # # b b b b b The second component of a place-value system is its set of digits. The digits are symbols that represent the quantities from 0 to one less than the base. The base determines the number of symbols that are in the system. In the decimal system, the base is 10, and it has 10 digits (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9). The following is a summary of some numeration systems. It includes the base, the digits, and the place values. Notice that base 12 and base 16 require the use of letters to represent digits that are greater than 9. For example, the letter A represents the digit 10. System Base Digits Place Values binary 2 0, 1, 32, 16, 8, 4, 2, 1 quintary 5 0, 1, 2, 3, 4, 3125, 625, 125, 25, 5, 1 octal 8 0, 1, 2, 3, 4, 5, 6, 7, 32,768, 4096, 512, 64, 8, 1 duodecimal 12 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A(10), B(11), 248,832, 20,736, 1728, 144, 12, 1 hexadecimal 16 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A(10), B(11), C(12), D(13), E(14), F(15), 1,048,576, 65,536, 4096, 256, 16, 1 When you write a numeral in a base other than 10, the base is indicated as a subscript to the right of the number and each digit is read separately. For example, the numeral is read one two three, base five and not one hundred twenty-three, base five. The reason is that the concept of hundreds is part of the base 10 system and, therefore, should not be used in other bases. page 15

16 Converting to a Decimal Numeral 1.2 From Hindu-Arabic to Binary To understand what number or amount is being represented by a numeral in a base other than 10, we need to convert it to the system we are familiar with, the decimal system. For example, the numeral 23,014 in base 5, written 23,014 5, represents an amount. To understand what amount that is, we will convert it to a decimal numeral. Example 3 Write 23,014 5 as a decimal numeral. Solution: Use the powers of 5 for the place value for each digit in the numeral = 4 1 = = 1 5 = = 0 25 = = = = = Thus, = Example 4 Write 17A6 12 as a decimal numeral. Solution: Use the powers of 12 for the place value of each digit in the numeral. 1 7 A = 6 1 = 6 + A 12 1 = = = = = = Thus, 17A6 12 = Example 5 Write as a decimal numeral. Solution: Use the powers of 2 for the place values of each digit in the numeral. page 16

17 1.2 From Hindu-Arabic to Binary Thus, = = 1 1 = = 0 2 = = 1 4 = = 1 8 = = 0 16 = = 1 32 = Converting Decimal Numerals to Other Bases To convert a decimal numeral to another base, we need to find how many groups of each appropriate place value are contained in the decimal numeral. The division scheme shown in the examples that follow will allow you to convert a decimal numeral into another base. Example 6 Write 89 in base 5. Solution: The powers of 5 that are less than 89 are 25, 5, 1. We need to find how many groups of each of those place values are contained in 89. We can do that by using the division scheme shown below. Example ) 514 ) ) The number 89 contains 3 groups of 25, 2 groups of 5, and 4 groups of = = = Write 10,000 in base 8. Solution: The powers of 8 that are less than 10,000 are 4096, 512, 64, 8, and 1. Use the division scheme of previous example ) ) ) 816 ) ) page 17

18 1.2 From Hindu-Arabic to Binary Example 8 10,000 = = = 23,420 8 Write the decimal number 40,600 in base 16. Solution: The powers of 16 that are less than 40,600 are 4096, 256, 16, and 1. Use the division scheme again ) (E) ) 16) 152 ) ,600 = E = E = 9E98 16 Numbers and Technology You probably are wondering why numbers with a different base could possibly be important. Our technological world would not exist without base 2 binary numbers. Computers read data as electric pulses having low or high voltage levels. Disks and tapes store information using magnetic fields pointing forward or backward. Compact disks store data using tiny pits and non-pits on the surface of the disk. Recordable CDs store audio signals as dark marks and nondark marks on the CD. In all these instances, the two states are associated with the digits of the binary system of numeration. The digit 1 is used for one state and the digit 0 is used for the other. Each 1 or 0 is called a bit. The number of bits that can be processed at one time varies from device to device. For example, an audio compact disk uses 16-bits to digitize the music. An inkjet printer uses 8-bits for each symbol on the keyboard. Present day 32-bit computers process data as sequences of thirty-two 1 s or 0 s. Computers and their peripheral devices use binary digits to read, store, and write information. Each letter of the alphabet, digit, and special character, such as a period, comma, or space, has a binary code. This code is called ASCII, American Standard Code for Information Interchange. When a computer reads or sends information, it is processed in ASCII code. When it sends data to a device such as a printer, the printer decodes the binary information and converts it to the corresponding character. The ASCII codes for the upper case letters of the alphabet are shown below page 18

19 1.2 From Hindu-Arabic to Binary Letter ASCII Code Letter ASCII Code Letter ASCII Code A J S B K T C L U D M V E N W F O X G P Y H Q Z I R space Example 9 Write the ASCII code for the word, MATH. Example 10 Solution: From the ASCII Chart we get the following. M = A = T = H = Thus, the ASCII equivalent of MATH is The codes on eight 1970s punch cards are given below. What does it say? , , , , , , , Solution: Converting the ASCII code into letters, we have: = H = I = space = T = H = E = R = E Thus, the eight cards have a message of, HI THERE. Since sequences of binary numerals are difficult to read, write, and remember, codes are usually written as base 8 (octal) or base 16 numerals (hexadecimal). If you continue learning more about computers, you see how octal and hexadecimal are used and how computers calculate and process information with such numbers. page 19

20 1.2 From Hindu-Arabic to Binary 1.2 Explain Apply Explore Explain 1. How are exponents used in the Hindu-Arabic system of numeration? 2. What is the expanded form of a Hindu-Arabic numeral? 3. How many symbols would be needed to create numerals in base 9? What are they? 4. What is the common belief as to why modern cultures use a base 10 system? 5. Why should the number be read one three five base seven, rather than one hundred thirty-five base seven? 6. What are the place values for a three digit numeral in base 32? 7. What do the binary digits 1 and 0 represent to a computer? 8. What is a bit? What is a 16-bit machine? 9. What is ASCII? What is it used for? Apply Write each of the following in expanded form , ,345, ,400,500,000 Write each of the following as a decimal numeral A ABBA 12 Write the following decimal numerals in the specified base. 19. Write 186 as a numeral in base Write 8888 as a numeral in base Write 7777 as a numeral in base Write 1860 as a numeral in base Write 16,016 as a numeral in base Write 222 as a numeral in base 2. Write the ASCII code for each saying. 25. GO FOR IT 26. I LOVE YOU 27. JUST DO IT page 20

21 Explore 1.2 From Hindu-Arabic to Binary Represent the number of days in a leap year (366) in each of the following systems of numeration. 28. Quintary 29. Binary 30. Octal Represent the number of pounds in a ton (2000) in each of the following systems of numeration. 31. Duodecimal 32. Hexadecimal 33. Base 6 DAD can be considered as a word in the English language or as a numeral in a base larger than What is the ASCII code for each letter? 35. What is DAD 14 as a decimal numeral? 36. What is DAD 16 as a decimal numeral? 37. What is DAD 32 as a decimal numeral? In each pair of numerals, determine which one has a larger value or 12, C1 16 or or page 21

22 1.3 Magical Number Patterns Section 1.3 Magical Number Patterns With the establishment of systems of numeration came a fascination with number patterns. One of the most famous patterns is a square array of numbers, called a magic square, {1, 2, 3,...}, that has the same sum horizontally, vertically, and diagonally. Some ancient cultures believed that these magic squares showed a harmony of numbers and contained mystical powers. Interest in magic squares dates back to the Lo Shu square, c b.c., where knots on strings drawn on a back of a tortoise shell gave a sum of 15 for all rows, columns, and diagonals. The Lo Shu Magic Square Magic squares have also been found of larger sizes in other cultures though out history. Japanese mathematicians in the 1600s determined rules for creating magic squares of various dimensions and found magic squares with up to 19 rows and 19 columns. Example 1 Verify that the 4 4 array of numbers is a magic square. Solution: Add the numbers in each row, column, and diagonal to determine if the sum is the same. Rows Columns Diagonals = = = = = = = = = = 34 Thus, the 4 4 array of numbers is a magic square In a San Francisco Bay Area science fair project in 1982, Kevin Staszkow, a son of one the authors, used an Apple II computer to generate a magic cube. In his cube, the integers from 1 to 27 were arranged in three layers of nine numbers where the sum of the numbers page 22

23 1.3 Magical Number Patterns in each row and column were 42 and the diagonals of the cube were also 42. Years later, Kevin was astonished to discover that the Japanese mathematician Tanaka Kurushima (1662) found a magic cube and Kurushima Gita (1757) determined a magic cube without the use of calculators or computers. Example 2 Verify that the diagonals of the magic cube found by Kevin Staszkow do have a sum of Solution: The diagonals of a cube go from one corner of the cube through the center of the cube to the opposite corner. In the cube, the numbers in the diagonals have the following sums: = = = = 42 Besides squares and cubes other arrays of numbers can have equal sums. We will now examine some of these magical number patterns. Magic Triangles A magic triangle is a triangle in which an equal number of counting numbers, {1, 2, 3, }, is placed on its sides so that the sum on the numbers on each side is the same. Example 3 Place the whole numbers from 1 to 6 in the circles to create a magic triangle. Solution: Since there are only six numbers to place in the circles, you could use trial and error: place numbers, check the sums, make adjustments, and arrive at a solution. However, using a little logic might make it easier. Here are some factors to consider. 1. Three numbers are odd, and three numbers are even. Maybe putting either the odd numbers or the even numbers at the vertex points and then using trial and error to place the other three numbers might work. Doing that, we get two solutions: a sum of 10 and a sum of 11. page 23

24 1.3 Magical Number Patterns Odd numbers at vertex points. 1 Even numbers at vertex points Sum = 10 Sum = You could also split the numbers into two groups: the smaller numbers (1, 2, 3) and the larger numbers (4, 5, 6). Putting either group at the vertex points and using trial and error to place the other three numbers, we get a sum of 9 and 12. Smaller numbers at vertex points. 1 Larger numbers at vertex points Sum = 9 Sum = 12 Example 4 In the boxes on the letter A, place the numbers, {1, 2, 3, 4, 5}, so that the sum of the numbers on each line segment is the same. Solution: Since the number placed at the top of the A is used twice, lets put the middle number, 3, into that box. The other four numbers can then be paired to have sums of six. Using trial and error and checking the sums of each segment, we get a solution Sum = 9 Did we find the only solution? Maybe selecting a different number for the top of the A and working with two pairs of numbers that have the same sum might give other solutions. If you select the 1 for the top of the A, the four other numbers can be paired to have a sum of 7. This will give another solution Sum = 8 Selecting any other number for the top of the A does not lead to a solution. page 24

25 1.3 Magical Number Patterns Example 5 In the following, the boxes with the same sum are marked with 1, 2, or 3 lines. Solution: For this example, the sum is 11. Note that the numbers on the single line, the double line, and the triple line each add to Finding the solution to a magic number pattern can be challenging. However, if you think logically and stick with it, you can find it rewarding. The exercises that follow will allow you to have some fun with magical number patterns. 1.3 Explain Apply Explore Explain 1. What is a magic square? 2. What is a magic cube? 3. Why is the square array shown at right not a magic square? 4. What is a magic triangle? 5. Why is the triangle below not a magic triangle? Apply A magic cross can be created by placing consecutive numbers, {1, 2, 3, 4, 5, 6, 7, }, into the boxes on the cross so that the sum of the vertical boxes and horizontal boxes are the same. Find all possible ways to solve each magic cross page 25

26 1.3 Magical Number Patterns Magic letters are created by placing consecutive numbers, {1, 2, 3, 4, 5, 6, 7, }, into the boxes at the intersection and the end of each line segment so that the sum of the numbers on each segment is the same. Determine the placement of the numbers in the following magic letters Magic numbers are created by placing consecutive numbers, {1, 2, 3, 4, 5, 6, 7, }, into the boxes placed on the number figure so that the sum of the numbers on each segment or curve is the same. Determine the placement of the numbers in the following magic numbers. (Note: Numbers connected by single, double or triple lines have the same sum.) page 26

27 Explore 1.3 Magical Number Patterns A magic circle is created in the diagrams below by placing numbers, {1, 2, 3, 4, 5, 6, 7, 8, }, so that the sum of the numbers on any circle and the number in the center is the same as the sum of the numbers on each line through the center of the circle. 19. Find three of the five 20. Find three of the ten possible magic sums. possible magic sums. 21. Arrange the numbers {1, 2, 3, 4, 5, 6, 7, 8, 9} in the circles of the magic triangle in Problem 5 to make it a magic triangle. Find the three possible magic sums. 22. Complete this very magic square that was in an Albrecht Dürer painting (c. 1525) Verify the magic properties of this 4 4 of this square. (a) The numbers in 2 2 squares in each corner and the center have the magic sum. (b) The numbers in the four corners have the magic sum (c) The sum of the numbers in the two top rows have the same sum as the sum of the number in the two bottom rows. (d) The sum of the squares of the numbers in the first and third rows is the same as the sum of the squares in the second and fourth rows (e) The sum of the numbers on the diagonals is the same as the sum of the numbers not on the diagonals. (f) The sum of the squares of the numbers on the diagonals is the same as the sum of the squares of the numbers not on the diagonals. page 27

28 1.4 Number Puzzles and Alphametics Section 1.4 Number Puzzles and Alphametics Mathematical puzzles have challenged students over the centuries. With the expanded use of the internet mathematical puzzles are quickly circulated through s. You may read them and be amazed at the seemingly magical results. This section will eliminate the mystery of such puzzles by showing you how to use mathematics to understand how such puzzles work. This section will also introduce other number puzzles in the hope of stimulating your mathematical creativity. The Magic Symbol Select any two digit number. Add the digits. Subtract that sum from the two digit number. Find the symbol next to the answer just obtained. What is that symbol? 99 b 89 J 79 R 69 L 59! 49 o 39 e 29 U 19 d 9 h 98 z 88 v 78 J 68 Ö 58 v 48 R 38 L 28 e 18 h 8 J 97! 87 U 77 d 67 o 57 M 47 U 37 a 27 h 17 a 7 M 96 % 86 L 76 b 66 J 56 % 46! 36 h 26 M 16 J 6 R 95 o 85 M 75 U 65 d 55 J 45 h 35 Ö 25 o 15 b 5 U 94 v 84 z 74 % 64 e 54 h 44 z 34 U 24 v 14 M 4 d 93 R 83 d 73 M 63 h 53 U 43 J 33 d 23! 13 e 3 z 92 U 82 e 72 h 62 v 52 d 42 v 32 J 22 d 12 o 2 L 91 J 81 h 71 L 61 a 51 R 41 d 31 M 21 J 11 Ö 1 % 90 h 80 Ö 70 z 60 U 50 d 40 L 30 b 20 R 10 U 0 v Try it again with any two digit number and the result will be the same. You are probably wondering, Is it magic? How does that work? With some algebra we can understand the magic symbol. Let The symbol is h. a = the tens digit of the number selected b = the units digit of the number selected Then, 10a + b = the number. a + b = the sum of the digits 10a + b (a + b) = 10a + b a b = 9a 9a is a multiple of 9 and all the multiples of 9 in the chart have the symbol, h. The result is not magic. It is mathematics. page 28

29 Example Number Puzzles and Alphametics Consider the number 15,873. Multiply it by any single digit counting number. Now multiply that result by seven. See all the numbers in the answer are the same as the single digit you selected. Why does this work? Solution: Let s see if this really works. Let s pick the counting number 4. Following the instructions, we get a result that was predicted With the help of some algebra, we will see that this is not magic. Example 2 Let x = the single digit selected Then, the problem becomes ( 15873x)( 7)= ( 15873)( 7) x= x. Thus, the single digit times gives the single digit repeating six times. Age Guess Magic for People Over 9 years old Have a person take their age and do the following: Multiply the tens digit of their age by 5. Add 6 to that result. Double that total. Add the units digit of their age to that. Tell you the answer. If you subtract 12 from the answer, it will be their age. How does this work? Solution: Let s see if it works using an age of 23. The first digit of 2 times = 10 Add 6 to that = 16 Double that = 32 Add second digit of age = 35 You subtract = 23 (Yes, it worked.) Let s now see if it works for any age. Following the steps here is what happens. Let x = the tens digit of the person y = units digit of the person page 29

30 Example 3 Multiply the first digit by 5. 5x Add 6 to that result. 5x + 6 Double that total. 2(5x + 6) = 10x + 12 Add the second digit of their age. 10x y Subtract 12 from the answer. 10x y 12 = 10x + y 10x + y is the age of the person What is the error in this classic proof that, 2 = 1? 1.4 Number Puzzles and Alphametics Step 1: Let a = b Step 2: Then, a 2 = ab (Multiply both sides by a.) Step 3: a 2 + a 2 = a 2 + ab (Add a 2 to both sides.) Step 4: 2a 2 = a 2 + ab (Combine like terms on the left.) Step 5: 2a 2 2ab = a 2 + ab 2ab (Subtract 2ab from both sides.) Step 6: 2a 2 2ab = a 2 ab (Combine like terms on the right.) Step 7: 2(a 2 ab) = 1(a 2 ab) (Rewrite the left and right.) Step 6: Thus, 2 = 1 (Divide both sides by a 2 ab.) Solution: You know that 2 does not equal 1. There must be an error. All the algebraic steps look fine. However, if you remember that a = b from Step 1, then a 2 ab = 0. That means in Step 6 both sides were divided by zero. Division by zero is not defined and that is what caused the impossible result. Alphametic Puzzles In an alphametic, short for alphabet arithmetic, each letter represents a unique whole number from 0 to 9. This means that no two letters are assigned the same digit value. In an Alphametic, the value assigned to a letter that starts a word cannot be a zero. The first published alphametic was created by the well-known puzzler H. E. Dudeney in His alphametic, SEND + MORE = MONEY, is given as an exercise. Example 4 Find the value of each letter in the given alphametic puzzle. A T D + S I X P N T S Solution: You could use trial and error to find the value assigned to each letter, or you could use some logic before you start. There are some underlying principles that are used in addition. page 30

31 1.4 Number Puzzles and Alphametics 1. The effect of carrying: If the sum of digits in a column is less than 10, you do not carry a one to the next column. If the sum of the digits is more than 9, you carry a one to the next column. A B + C A E B + C must be less than 10, no carrying. A B + C D E B + C must be greater than 9, carrying occurred, D = A The same letter in a column: If the same letter appears in the ones column, the value of the letter is 0. A B + C B B must be 0. D B If the same letter appears in another column, the value of the letter is 0 or 9. E A B + F A C G A D A is 0 if B + C is less than 10. A is 9 if B + C is greater than The same letter in top and bottom of a column: If the same letter appears in the top and bottom of a column, the value of the letter in the middle is 0 or 9. C A + B D T E A D must be 0. (no carrying) C A + D R T C E D must be 9. (carrying from the ones column) Armed with those principles, let s solve the Alphametic. A T D + S I X P N T S a) P = 1 (One more digit in the answer indicates carrying.) b) I = 0 (Numbers in the top and bottom are the same in the tens column.) Realizing that each letter represents a different digit, we logically place digits, check sums, and make adjustments with the remaining digits to get a solution. page 31

32 1.4 Number Puzzles and Alphametics There are other possible solutions to this Alphametic puzzle. The exercises that follow will give you the opportunity to solve math puzzles and Alphametics. Have fun! 1.4 Explain Apply Explore Explain 1. If a and b are the first and second digits of a two digit number, why is the number 10a + b? 2. If a, b, and c are the first, second and third digits of a three digit number, why is the number 100a + 10b + c? 3. What is an alphametic puzzle? 4. In the alphametic below, why is the value of B = 1 and E = 0? G O T Apply + T H E B E A T 5. Your Shoe Size and Age Write down your age at the end of 2006 and your shoe size (nearest whole number). Double your shoe size. Add five to that result. Multiply that answer by 50. Add 1756 to that result. Subtract the year of your birth. The last two digits give your age at the end of 2006 and to the left of that your shoe size. a) Use your shoe size and age at the end of 2006 to see if this works. b) Use algebra to show why this works for any shoe size or age. 6. Your Birth Date Write down your birth date, i.e. 11/08/1987. Double the month. Multiply by 50. Subtract 7. Add the day of your birth. Multiply by 100. Add Add the year of your birth. Subtract 11,111. page 32

33 The answer is your birth date in standard form, i.e. 11/08/87. a) See if this works with your birth date. b) Use algebra to show why this works for any birth date. 7. A Proof That All Numbers Equal Zero Let x = y 2 x = xy (Multiply both sides by x.) 2 x y = xy y (Subtract y from both sides.) ( x+ y)(x y) = yx ( y) (Factor both sides.) x+ y= y (Divide both sides by x y.) x = 0 (Subtract y from both sides.) 1.4 Number Puzzles and Alphametics Since x could represent any number, all numbers equal zero. Since we know all numbers do not equal zero, what is wrong with the proof? 8. I Can Guess Your Age (Age must be between 12 and 111 years old, inclusive) Take your age. Add my lucky number 88 to your age. Cross out the first digit of that answer and add the crossed out digit to the two remaining digits. Tell me that result. If I add 11 to that number, it will be your age. a) Use your age to see if this works. b) Use algebra to show why this works. (Note: this is a good one to impress your friends with you magic math abilities.) In Problems 9 18, find at least one solution for the following alphametic puzzles. 9. M O O + M O O C O W S 10. B O W + W O W D O G S 11. B I G + B A D W O L F S T E A L P A S S S C O R E 13. F U N I N S U N + O N O A H U 14. R O W R O W + R O W B O A T page 33

34 1.4 Number Puzzles and Alphametics 15. H I P + H O P R O C K 16. R U N + F O R G O L D 17. G O L F + R A I N N O F U N 18. G O T + T H E B E A T Explore 19. It s Always 222 Choose three digits from 1 to 9. Write all the possible three digit numbers using those digits. (There are six of them.) Add the six numbers. Divide that answer by the sum of the three digits. The answer is 222. a) See if this works with your three selected digits. b) Use algebra to show why this works. 20. Problem 5 works for your age at the end of How would you change it to make it work at the end of 2007? 2008? 2009? 21. In Problem 8, how could you make the magic math work for each lucky number from 89 to 99? Find the range of ages for each of these lucky numbers. 22. Your telephone number. Multiply the first three digits of your telephone number (not the area code) by 80. Add 1 to the result. Multiply that result by 250. Add the last four digits of your telephone number to that result. Add the last four digits of your telephone number again. Subtract 250. Divide the result by 2. a) Do you recognize the result? b) Use algebra to show why this works. 23. Solve H. E. Dudeney s 1924 alphametic. S E N D + M O R E M O N E Y page 34

35 24. A Proof that Three = Two Let Then a+ b = c 1.4 Number Puzzles and Alphametics 3a+ 3b= 3 c (Multiply both sides of original by 3.) 2a 2b= 2c (Multiply both sides of original by 2.) 3a 2a+ 3b 2b = 3c 2c ( Add both equations.) + 2a + 2b + 2a + 2b (Add 2a and 2b to both sides.) 3a+ 3b = 3c+ 2a+ 2b 2c 3c 3c (Subtract 3c from both sides.) 3a+ 3b 3c= 2a + 2b 2c 3( a+ b c)= 2( a+ b c) (Factoring both sides.) 3= 2 (Dividing both sides by a + b c.) Since we know that three does not equal two, what is wrong with the proof? 25. Explain why this is not an alphametic. D O + T H E M A T H 26. Student Mindy Lindsay enrolled in Math 155, Math for the Associate of Arts at Ohlone College created this alphametic on St. Valentines Day, There are at least two solutions. L O V E + L O S T H U R T S page 35

36 1.5 Computation Shortcuts and Estimation Section 1.5 Computation Shortcuts and Estimations Sometimes it is necessary to make calculations in everyday life when we have no access to a calculator. We also, at times, need an approximate answer to a numerical problem. In this section, we will look at some shortcuts for performing various computations and look at some techniques for quickly estimating answers. Computation Shortcuts Zeros in Subtraction When subtracting, you may use the technique of borrowing from digits to the left of a particular place value. When the number to be subtracted from ends in zeros, the technique shown below will simplify that process =? Borrow 1 from 700 getting 699 and subtract digit by digit. 830,000 51,639 =? , , , Example 1 97,000 56,783 =? Solution: 40,217 1 Borrow 1 from getting and subtract digit by digit , , , Zeros in Multiplication Multiplying by numbers that end in zeros using normal methods may look like this. 52,000 4,000 =? page 36

37 1.5 Computation Shortcuts and Estimation , 0 0 0, Most of the computation was spent writing and aligning zeros. The result is more easily determined if you multiply the non-zero parts (52 4 = 208) and attach the total number of zeros (six zeros). Thus, 52,000 4,000 = 208,000,000. Example ,000,000 =? Solution: 27 5 = 135 and there are 8 zeros in the multipliers. Thus, zeros , 000, 000 = Zeros in Division = 13, 500, 000, 000 Dividing numbers that end in zeros using normal methods may look like this. 3,600, =? ) Again, most of the computation was spent writing and aligning zeros. The result is more easily determined if you eliminate the same number of zeros from both the divisor and dividend. 3, 600, , 600, = = = Eliminating those zeros is equivalent to dividing the numerator and denominator of a fraction by the same amount, page 37

38 Example 3 What is 7.5 trillion divided by 5 million? Solution: 7.5 trillion = 7,500,000,000,000 5 million = 5,000, Computation Shortcuts and Estimation 7, 500, 000, 000, 000 7,500,000,000,000 5,000,000 = 5, 000, 000 7, 500, 000 = = 1, 500, Vedic Multiplication I Vedic mathematics is the name given to a system from ancient India that was rediscovered by Sri Bharati Krsna Tirthaji between 1911 and A main focus of this system is to improve mental computation. We will see how multiplication of two-digit numbers can be simplified in this system using a vertically and crosswise formula. You will see that this technique accomplishes the same results as normal long multiplication methods. For example, what is 32 21? Long Multiplication Vedic Multiplication a) Vertically: 3 2 = = 2 b) Crosswise: = 7 This gives the middle digit. The vertically and crosswise formula works quickly with the multiplication of any two twodigit numbers. Example =? Solution: The Vedic method is set up as above. However, since the crosswise total is the two-digit number, 29, the 2 is carried to the left vertical and added to Crosswise: = = 3796 page 38