Math Girls Rock! Math Club for Young Women http://www.uvu.edu/csh/mathematics/mgr/ Project 1 Mathematical Card Tricks and Number Systems Project 1 Coordinator: Prof. Carolyn Hamilton Phone: (801) 863-8801 e-mail: hamiltca@uvu.edu Project 1 Assistants: Prof. Violeta Vasilevska (e-mail: Violeta.Vasilevska@uvu.edu) Prof. Meghan Dewitt (e-mail: dewitt@math.byu.edu ) Emily Stucki (e-mail: emilycstucki@gmail.com ) Sydney Butler (e-mail: 10555503@uvlink.uvu.edu) Kristen Smith (e-mail: smith.kristen.d@gmail.com Department of Mathematics Utah Valley University 800 W. University Parkway Orem, UT 84058 Fax: (801) 863-6254 October 3 / October 17, 2013
Content 1.1. Card Trick 1.2. Number Systems 1.3. Place Value Systems 1.4. Other Bases 1.5. The Binary Number System 1.6. Base Three 1.7. Card Trick Revisited 2
1.1. Card Trick Mathematics is used in many unexpected places even in magic shows! This project will demonstrate a common card trick and teach the mathematics behind the trick: The magician places twenty-seven cards face up on the table in three columns of nine cards each. A participant is chosen and asked for a favorite number between 1 and 27. The participant is also asked to concentrate on a card, divulging to the magician only which column the card is in. The cards are re-dealt, again face up, again into three columns of nine cards each. The participant tells the magician which pile now contains the card. The cards are re-dealt a third time, in the same manner, and the participant indicates which column contains the selected card. The magician collects all the cards and begins to count them, face down from the top of the deck, stopping on the participant s favorite number. The card is turned over, revealing the chosen card. How did the magician know which card was chosen? How did the card arrive in the desired position? Magic?? No! Mathematics!! 1.2. Number Systems Many civilizations developed different types of systems in an attempt to describe quantities for both whole numbers and for fractions. The decimal system we use today is the result of thousands of years of development. Roman Symbol Value I 1 V 5 X 10 L 50 C 100 D 500 M 1000 For instance, the Roman numeral system was developed using letters from the Latin alphabet to represent various quantities: Originally, the Roman numeral system was strictly additive with the value of a group of symbols being equal to the sum of the values of each of the symbols. Also, symbols representing larger quantities were written before symbols representing smaller quantities (reading left-to-right): Example Find the quantity represented by the Roman numeral XVII. Roman Decimal XVII X + V + I + I 17 10 + 5 + 1 + 1 3
Example Find the quantity represented by the Roman numeral XVII. Roman MMLV Decimal Question: What year is indicated by the Roman numerals found on the Admiralty Arch in London (shown below)? Answer: http://www.tripadvisor.com/locationphotos-g186338-d519586-admiralty_arch-london_england.html#1168618 Because it was difficult to distinguish nine (VIIII) from eight (VIII), or three (III) from four (IIII) without having to count out each symbol, the Romans developed a variation of their number system for quantities whose representations involved four repetitions of the same symbol, such as 400 = CCCC or 14 = XIIII. Instead of writing nine as VIIII (literally: four numbers after five), the Romans used IX (literally: one before ten ). Instead of representing forty by XXXX, the Romans used XL (ten before fifty). The variation of the number system followed two rules: Whenever a symbol representing a smaller quantity precedes a symbol representing a larger quantity, subtraction is implied. Whenever a symbol representing a smaller quantity follows a symbol representing a larger quantity, addition is implied. 4
Complete the following table: Decimal Roman 4 (one before five) IV 9 (one before ten) 14 (ten plus four) 19 (ten plus nine) 40 (ten before fifty) 90 (ten before 100) 400 (100 before 500) 900 (100 before 1,000) Example Find the value of the Roman numeral MCMLXIX. M C M L X I X 1,000 100 1,000 50 10 1 10 MCMLXIX = Question: What is the Roman numeral representation of the quantity 2013? Answer: Question: What quantity is represented by the Roman numeral MCMXLVI? Answer: Question: How would your birth year be written as a Roman numeral? Answer: 5
The Roman numeral system had some serious limitations: Question: How would 47,000 be represented using only the Roman numerals introduced so far? Answer: Question: Can you think of a way to change the Roman numeral system to more easily represent 47,000? Answer: 1.3. Place Value Systems In the Roman numeral system, additional symbols needed to be introduced in order to represent larger and larger quantities, since anytime an X was used in Roman numerals, for example, it always represented ten objects. Cultures such as the Mayans and the Babylonians invented number systems that used place value notation, sometimes also called positional notation. These number systems have two important properties: There is a positive integer (called a base) which is central to the system The same digit can represent different quantities depending on its placement (or position). The quantity represented by any digit depends on: o the digit itself (face value), and o the digit s position within a number (place value). The place value of any digit is a power of the base. One example of a place-value system is the system we use today. Our number system is a base ten system, also called a decimal system. Place values in the decimal number system are powers of ten: 10 0 = 1 ones 10 1 = 10 tens 10 2 = 100 hundreds 10 3 = 1,000 thousands 10 4 = 10,000 ten-thousands 10 5 = 100,000 hundred-thousands The place values are positioned from right to left beginning with the ones column. For large numbers, a comma is used to separate the place values into sets of three digits., 6
Example The number 804,149 contains two 4 digits. What is the face value of the red 4? What is the place value of the red 4? What is the face value of the blue 4? What is the place value of the blue 4? What is the place value of the 1? What is the place value of the 9? The place-value form of the number 804,149 is: Question: Why is it important to have a symbol like 0 to represent nothing? Answer: 1.4. Other Bases Quantities can also be represented using a place value system with a base other than ten. In a base five system, the place values are calculated using powers of five. Base blocks (shown below) can be used as an aid in visualizing quantities and their numerical representations in a base five system: block unit long flat Value: 7
Fill in the base five place values in the table below. super block blocks flats longs units fives ones Question: What quantity is represented by the numeral? (The subscript five after a numeral means base five) Fill in the place-value form for the number: = = Notice that five units equal one long, and that five longs equal one flat, etc. Quantities are always represented using the fewest possible number of each base block. For example, to represent 30 objects, it would be incorrect to use six longs. Instead, use one flat and one long. Question: How would the quantity 200 be represented as a base five numeral? Question: How would the quantity 24 be represented using base five? Question: How would the quantity 25 be represented using base five? Question: What quantity is represented by the numeral? Example Use base blocks to demonstrate the quantity. Question: Why is the expression incorrect? Question: How should the base five numeral be changed to correctly represent the quantity? 5 8
Example a. Write the largest possible 3-digit numeral in base 5: 5 b. Find the base five numeral that is one more than the numeral above: 5 c. What quantities do the numerals in parts a) and b) above represent? Notice that for a base five system, only five different digits are required: 0, 1, 2, 3, and 4. The number system the majority of the world uses today is referred to as the Hindu-Arabic system and was developed as early as 300 A.D. It is a base ten place-value system and therefore uses ten different symbols, namely, the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. In 1757, J. E. Montucla published Histoire de la Mathematique which documented the different symbols that had been used at different times for the decimal number system: http://en.wikipedia.org/wiki/history_of_the_hindu%e2%80%93arabic_numeral_system 9
5. The Binary Number System Binary is the name given to the base-two place value number system. There are only two digits needed in binary: 0 and 1. The quantity 18, for example, is represented in binary as 100102. The binary system is used by computers, cellular phones, digital display devices, and digital recording devices to encode and decode information. In fact, anything digital uses a base two number system in some way. For example, the surface of a CD contains one long spiral track of data. Along the track, there are flat reflective areas and non-reflective bumps. A flat reflective area represents a binary 1, while a non-reflective bump represents a binary 0. The CD drive shines a laser at the surface of the CD and can detect the reflective areas and the bumps by the amount of laser light they reflect. The drive converts the reflections into 1s and 0s to read digital data from the disc. http://electronics.howstuffworks.com/question287.htm 6. Base Three What are the place values in a base three number system? ones Question: What quantity is represented by? Answer: Question: What quantity is represented by? Answer: Question: How would the quantity 19 be represented in base three? Answer: 10
Count from 0 to 26 using base three: Decimal 0 1 2 3 4 5 6 7 8 Base Three Decimal 9 10 11 12 13 14 15 16 17 Base Three Decimal 18 19 20 21 22 23 24 25 26 Base Three 11
7. Card Trick Revisited Consider three piles of nine cards each. Suppose the piles are picked up from left to right, so that the leftmost pile is on top and the rightmost pile is on the bottom. In the dealer s hand, the cards are in sets of nine: The cards are then re-dealt into three new piles, with cards being dealt across the piles. Notice that the sets of nine cards from the original piles have been separated across the columns into sets of three cards. Suppose the cards are again picked up from left to right, so that the leftmost pile is on the top and the rightmost pile is on the bottom. In the dealer s hand, the cards are now in sets of three: Question: What do you think will happen to the arrangement of the cards after being re-dealt across the piles a second time? Answer: 12
After being re-dealt across the piles a second time, the cards are in sets of one: Remember that the trick started with 27 cards, in sets of 9, which became sets of 3, which then became sets of 1. Notice that 1, 3, 9 and 27 are all powers of 3. This card trick uses counting in base three! Example Suppose the person s favorite number was 22. The chosen card must somehow be made to appear in the 22 nd position, which means 21 cards must be in front of the chosen card. Rewrite the number 21 using base three: 21 = 3 Now write the digits in reverse order: These digits show how to make the chosen card appear below 21 other cards in the 22 nd position! How?? The first digit (in reverse order) is 0. Digit 0 = no other piles on top. The first time the cards are picked up, the target pile must be placed on top. The second digit (in reverse order) is 1. Digit 1 = one pile on top. The second time are cards are picked up, be certain the target pile is the middle pile. 13
The third digit (in reverse order) is 2. Digit 2 = two piles on top. The last time the cards are picked up, be certain the target pile is on the bottom. Notice that by picking up the piles and placing the target pile according to the digits 0, 1, 2, the chosen card will end up as the 22 nd card in the pile! Try it! 14