Number Games A Lesson in the Math Fun

Number Games A Lesson in the “Math + Fun!” Series 1111111 1111 11111 111 11111 111111111 Apr. 2005 22222 222 222 222222222 222 222 222 3333 33333 333333333333 333 3333 333 333 33 333 333 Number Games 444444444 444444444 444 444444444 5555555 555 5555555 5555 55555 55555 1

About This Presentation This presentation is part of the “Math + Fun!” series devised by Behrooz Parhami, Professor of Computer Engineering at University of California, Santa Barbara. It was first prepared for special lessons in mathematics at Goleta Family School during the 2003 -04 and 2004 -05 school years. The slides can be used freely in teaching and in other educational settings. Unauthorized uses are strictly prohibited. © Behrooz Parhami Apr. 2005 Edition Released First Apr. 2005 Revised Number Games Revised 2

What Number Did You Choose? 1. Pick a number from this list: 1 2 3 4 5 6 7 8 9 10 2. Multiply the number by 9. If the result has two digits, add the two digits; otherwise leave the number alone. 3. Subtract 5 from the number that you got in step 2. 4. Pick the letter that corresponds to the number you got in step 3. 1 = A, 2 = B, 3 = C, 4 = D, 5 = E, 6 = F, 7 = G, 8 = H, etc. 5. Pick a country in Europe whose name begins with your letter. A = Austria, . . . C = Croatia, . . . E = England, . . . B = Belgium, . . . D = Denmark, . . . F = France, . . . etc. 6. Use the last letter in your country’s name as the first letter of an animal. 7. Use the last letter of your animal’s name as the first letter of a color. Apr. 2005 Number Games 3

How Did the Game Work? 1. Pick a number from this list: 1 2 3 4 5 6 7 8 9 10 2. Multiply the number by 9. 9 18 27 36 45 54 63 72 81 90 If the result has two digits, add the two digits You get 9 in all cases! 3. Subtract 5 from the number that you got in step 2. You get 4 4. Pick the letter that corresponds to the number you got in step 3. 1 = A, 2 = B, 3 = C, 4 = D, 5 = E, 6 = F, 7 = G, 8 = H, etc. 5. Pick a country in Europe whose name begins with your letter. A = Austria, . . . C = Croatia, . . . E = England, . . . B = Belgium, . . . D = Denmark, . . . F = France, . . . Kangaroo 6. Use the last letter in your country’s name as the first letter of an animal. Orange 7. Use the last letter of your animal’s name as the first letter of a color. Apr. 2005 Number Games 4

More Mind-Reading Games 1. 2. 3. 4. 5. Think of a 2 -digit number (10 -99). Multiply your number by 5. Add 500 to the result of step 2. Add the number of your siblings. Double the result of step 4. You got a 4 -digit number that starts with 1, has your original 2 -digit number in the middle, and double the number of your siblings at the end. 1. Think of a number between 1 and 30. 2. Tell me in which of the five lists below the number appears. List A: 1 3 5 7 List B: 2 3 6 7 List C: 4 5 6 7 List D: 8 9 10 11 List E: 16 17 18 19 Apr. 2005 9 10 12 12 20 11 11 13 13 21 13 14 14 14 22 15 15 23 17 18 20 24 24 19 19 21 25 25 Number Games 21 22 22 26 26 23 23 23 27 27 25 26 28 28 28 27 27 29 29 30 30 A B C D E E 5

Activity 1: Mind-Reading Game Play the following mind-reading game with your friends and family (First, discover how to tell the number if you know which lists it is in) 1. Think of a number between 1 and 20. 2. Tell me in which of the five lists below the number appears. List A: 1 4 6 8 List B: 2 6 7 8 List C: 3 4 6 9 List D: 5 7 8 9 List E: 10 11 12 13 9 12 13 15 14 11 17 14 16 15 14 20 18 17 16 16 19 19 20 18 19 20 17 18 19 20 Challenge: Add a List F to the five lists given above, and complete the lists, so that you can play the game with numbers up to 40. Apr. 2005 Number Games 6

Activity 2: More Mind-Reading Games I know what your number is! 1. 2. 3. 4. 5. 6. 7. Pick any number and write it down Add 10 to your number Add 20 to the result Subtract 4 from the result Add 15 to the result Subtract 5 from the result Subtract your original number Your final result is 36, isn’t it? Explain how this game works. Change the numbers to make your own game. The Birthday Number Game 1. 2. 3. 4. 5. 6. 7. 8. Write down your BD month (1 -12) Multiply your number by 5 Add 6 to the result Multiply the result by 4 Add 9 to the result Multiply the result by 5 Add the day of your BD (1 -31) 8. Subtract 165 from your result Apr. 2005 Number Games Your final result shows the month and day of your birth Explain how this game works. 7

Euclid’s Game for Two Players A referee writes two numbers on the board The two players take turns writing the difference of two numbers already on the board, provided the difference is a new number The player who cannot write a new number loses Player 1 loses Referee 24 Player 1 18 6 Player 2 80 15 65 12 35 50 The total number of values written on the board is n = the larger number / greatest common divisor of the two numbers If n is odd, player 1 wins Apr. 2005 Number Games 8

Activity 3: The Game of Nim Write three numbers on three lines The two players take turns reducing one of the numbers The player who makes the last move loses Start 5 3 1 Player 1 2 3 1 Player 2 2 2 1 Player 1 2 2 0 Player 2 2 1 0 Player 2 0 0 0 Loses! Challenge: 1. Discover a winning strategy with (5, 3, 1), if you are the first player 2. Play the game with other numbers such as (7, 4, 2), (6, 5, 3), . . . Apr. 2005 Number Games 9

The Four 4 s Puzzle Use four 4 s and any number of math symbols (such as +, -, , /, , . . . ) to form as many different numbers as possible. 0=4– 4+4– 4 Examples 1 = 4 – 4 + 4/4 2 = 4 / 4 + 4 / 4 or 4 – (4 + 4) / 4 3 = (4 + 4) / 4 4 = (4 + 4) / 4 + 4 5 = (4 4 + 4) / 4 6 = (4 + 4) / 4 7 = 4 + 4 – 4 / 4 or 44 / 4 – 4 8 = 4 (4 / 4) + 4 9 = 44 / 4 – 4 or 4 / 4 + 4 4 10 = (4 + 4 / 4) 4 11 = 44 / ( 4 + 4) 12 = 4 (4 – 4 / 4) Apr. 2005 13 = 44 / 4 + 4 14 = 4 ( 4 + 4) – 4 15 = 44 / 4 + 4 16 = 4 4 + 4 – 4 or 4 4 4 / 4 17 = 4 4 + 4 / 4 18 = 4 ( 4 + 4) + 4 19 = 4. 4 20 = 4 (4 + 4 / 4) 21 = 4. 4 4 + 4 22 = 4 4 + 4 23 = 4 (4 + 4) –. 4 24 = 4 ( 4 + 4) 25 = (4 + 4 / 4) 4 Number Games 10

Activity 4: The Five 5 s Puzzle Use five 5 s and any number of math symbols (such as +, -, , /, , . . . ) to form as many different numbers as possible. 0 = 5 (5 / 5 – 5 / 5) 1 = 55 / 5 – 5 2= 3= 4= 5= 6= 7= 8= 9= 10 = 11 = 12 = Apr. 2005 Examples 13 = 14 = 15 = 16 = 17 = 18 = 19 = 20 = 21 = 22 = 23 = 24 = 25 = Number Games 11

Activity 5: Birth Year Puzzle Use the digits in your year of birth, in order, plus any number of math symbols (+, –, . . . ) to form the numbers from 0 to 10. Examples for 1993 0 = 1 (9 – 9) 3 1 = 1 + (9 – 9) 3 2 = – 1 + 9 – 9 + 3 3 = (1 + 9 – 9) 3 4= 5= 6 = (1+ 9 / 9) 3 7= 8= 9= 10 = Apr. 2005 Examples for 1994 0 = 1 (9 – 9) 4 1 = 1 + (9 – 9) 4 2 = – 1 – 9 / 9 + 4 3 = – 1 + 9 – 9 + 4 4=1 9– 9+4 5=1+9– 9+4 6 = 1 + 9/9 + 4 7 = 1 (9 / 9) + 4 8 = (1 + 9 / 9) 4 9 = 1 + 9 – 4 10 = 1 9 – 9 + 4 Number Games Examples for 1995 0 = 1 (9 – 9) 5 1 = 1 + (9 – 9) 5 2= 3= 4= 5 = 1 – 9/9 + 5 6= 7= 8= 9= 10 = (1 + 9 / 9) 5 12

Activity 6: Your Phone Number Write down your phone number, with space between digits Example: 5 5 5 1 2 5 3 Now, try to make as many different numbers as you can by Putting + , - , , / signs between the digits. 0 1 2 3 4 5 6 7 8 Apr. 2005 = = = = = 5 5 + 5 + 5 5 5 5 + 5 5 5 1 + 1 1 1 1 Number Games 2 2 2 2 2 5 5 5 5 5 5 3 3 3 3 3 13

Another Game for Two Players Put some playing cards (or numbers) in a row Players take turns removing a card from either end of the row At the end, add up the numbers to see who wins Apr. 2005 Number Games 14

Game Tree Score for Player 1 Score for Player 2 Cards left 0 [1 2 5 4 8 6] 0 Player 1 1 [2 5 4 8 6] 0 6 [1 2 5 4 8] 0 Player 2 1 [5 4 8 6] 2 1 [2 5 4 8] 6 6 [2 5 4 8] 1 Player 1 6 [4 8 6] 2 7 [5 4 8] 2 3 [5 4 8] 6 6 [1 2 5 4] 8 9 [2 5 4] 6 Player 2 6 [8 6] 6 7 [4 8] 7 6 [4 8] 8 Player 1 1 1 3 [4 8] 11 7 [5 4] 10 1 2 9 [5 4] 8 3 [5 4] 14 2 2 9 [2 5] 10 1 1 Player who wins Apr. 2005 Number Games 15

Some Surprising Predictions Answer these questions quickly. If you pause, it won’t work. What is 2 + 2? What is 4 + 4? What is 1 + 1? What is 9 + 9? What is 2 + 2? What is the first vegetable that comes to your mind? What is 3 + 3? What is 4 + 4? What is 5 + 5? What is 8 + 8? What is 16 + 16? Quickly, pick a number between 12 and 5 What is 6 + 6? What is 7 + 7? No one knows why these guesses work for most people. What is 8 + 8? Apr. 2005 Number Games 16

Next Lesson Thursday, May 19, 2005 What is special about 40? (Hint: Forty) The only number with letters appearing in alphabetical order What is special about 6? (Hint: Divisible by 1, 2, 3) What is special about 38? (Hint: XXXVIII) Apr. 2005 Number Games The smallest perfect number The last Roman numeral in alphabetical order 17