Bagels, Pico, Fermi. Bob Albrecht & George Firedrake Copyright (c) 2004 by Bob Albrecht

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1 ,, Bob Albrecht & George Firedrake Copyright (c) 2004 by Bob Albrecht is a number-guessing game that is great exercise for your wonderful problem-solving mind. We've played it on blackboards, whiteboards, paper and pencil, and other media with multitudes of students and teachers. You can play anywhere with a friend. One person is the gamemaster and the other is the player. The gamemaster thinks of a secret number and the player tries to guess the number. After each guess, the gamemaster supplies clues with strange names such as "," "," and "." We'll tell you how to play a 2-digit beginner's game and a more challenging 3-digit game. Beginner's Game with 2-Digit Numbers The secret number is a 2-digit number from 10 to 98 that has two different digits. tens digit: any decimal digit 1, 2, 3, 4, 5, 6, 7, 8, or 9, but not 0. ones digit: any decimal digit different than the tens digit. Secret number = 73. The tens digit is 7. The ones digit is 3. The number can be 10 or 23 or 64, but it can't be 05 or 11 or 22. Can it be 37? Yes. Can it be 11? No. Can it be 44? No. Can it be 99? No. Your Turn #1. Please answer this question, then visit the Answers to Your Turn section way down yonder to see our answer. 1. How many different secret numbers are possible? Remember: the secret number is a 2-digit number from 10 to 98 and the digits are different from each other. Your answer:. Guesses and Clues for 2-Digit The player tries to guess the secret number. After each guess, the gamemaster gives a clue. Here are the possible clues for a 2-digit game: Clue What the clue means No digit of the guess is in the secret number. One digit of the guess is in the secret number, but is in the wrong place. Both digits of the guess are in the secret number, but are in the wrong places. One digit of the guess is in the secret number and is in the correct place. Oops! This is not possible. This clue cannot occur in the 2-digit game. Got it! Both digits of the guess are in the secret number in the correct places.,, 1 11/20/2004

2 Sample game. The secret number is 73. Here are guesses and the clues returned by the gamemaster. Secret Guess Clue What the clue means number No digit of the guess is in the secret number One digit, wrong place One digit, correct place No digit of the guess is in the secret number Two digits in the wrong places Got it! Two digits in the correct places. Sample game. It is helpful to make a list of the decimal digits and erase or cross out digits that can't be in the secret number. Suppose that the secret number is 82. Laran made a list of decimal digits () and tried to guess the secret number. Whenever she figured out that a digit couldn't be in the secret number, she erased that digit. If she found a digit that might be in the secret number, she highlighted it. For example: 2. Secret number Guess Clue Laran's list of decimal digits That's it! The secret number is 82. Aha! When Laran guessed 27, she got one. So 2 or 7, but not both, are in the secret number. If 2 is in the secret number, it must be in the ones place. If 7 is in the secret number, it must be in the tens place. Laran hasn't used the digits 8 or 9. "Aha," thought Laran "I'll try 78." Laran's guess of 78 got one, so 7 can't be in the tens place. Clever Laran deduces that 7 is not in the secret number and erases it from her list of digits. Since 7 isn't in the secret number, the for 78 means that 8 is the tens digit of the number. The for the previous guess of 27 must be the 2, so 2 must be in the ones place. The secret number is 82.,, 2 11/20/2004

3 Your Turn #2. Now you and a friend can play 2-digit. Before you do, try these puzzlements. Don't peek now, but you can find our answers in the Answers to Your Turn section. When you discover that a digit is not in the secret number, cross it out in the digit list. Highlight or circle digits that might be in the secret number. Guess Clue Secret number? Decimal digits In the next two puzzlements, you can use the clues to reduce the possible values of the secret number to a few possibilities. In each case, what would be your next guess? Guess Clue What numbers are possible? Decimal digits ,, 3 11/20/2004

4 Three-Digit The three-digit game is more challenging than the two-digit game. The secret number has three digits and no two digits are the same. The three digits of the secret number are different from one another. The secret number is a 3-digit number from 102 to 987 that has three different digits. No two digits are the same. hundreds digit: 1, 2, 3, 4, 5, 6, 7, 8, or 9, but not 0. tens digit: any decimal digit different than the hundreds digit and the ones digit. ones digit: any decimal digit different than the hundreds digit and the tens digit. The secret number is 259. The hundreds digit is 2. The tens digit is 5. The ones digit is 9. The numbers 123, 407, and 796 are A-OK, but the secret number cannot be 012 or 007 or 363 or 888. Your Turn #3. Answer this question, then visit the Answers to Your Turn section to see our answer. 1. How many secret numbers are possible? Remember: the secret number is a 3-digit number from 102 to 987 that has three different digits. No two digits are the same. Your answer:. Guesses and Clues for 3-Digit The player tries to guess the secret number. After each guess, the gamemaster gives a clue. If the clue has both s and s, the gamemaster gives s first, then s. Here are the possible clues for a 3-digit game: Clue What the clue means No digit of the guess is in the secret number. One digit is in the secret number, but is in the wrong place. Two digits are in the secret number, but both are in the wrong places. Three digits are in the secret number, but all three are in the wrong places. One digit is in the secret number in the correct place. One digit is in the correct place and one digit is in the wrong place. One digit is in the correct place and two digits are in the wrong places. Two digits are in the correct places. Oops! This is not possible. This clue cannot occur in the 3-digit game. Got it! Three digits of the guess are in the secret number in the correct places.,, 4 11/20/2004

5 The secret number is 207. Here are some guesses and the clues returned by the gamemaster. Secret Guess Clue What the clue means number No digit is in the number One digit in the wrong place One digit in the correct place Two digits in the wrong places Three digits in the wrong places One digit in correct place, two digits in wrong places Got it! Three digits in the correct places. Your Turn #4. Now you can play 3-digit with a friend or try these puzzlements. Guess Clue Secret number? Decimal digits In the next two puzzlements, you can use the clues to reduce the possible values of the secret number to a few possibilities. In each case, what would be your next guess? Guesses Clues Possible numbers? Decimal digits ,, 5 11/20/2004

6 Answers to Your Turn Your Turn #1. 1. The tens digit can be 1, 2, 3, 4, 5, 6, 7, 8, or 9, but not 0, so there are nine ways to pick the tens digit. For each choice of the tens digit, there are nine ways to pick a different ones digit, so there are 9 9 = 81 possible secret numbers. Your Turn #2. In the notes below, the letter T represents the tens digit and the letter U represents the ones (units) digit of a 2-digit number. For example: T3 can be any of these numbers: 13, 23, 43, 53, 63, 73, 83, or 93. 9U can be any of these numbers: 90, 91, 92, 93, 94, 95, 96, 97, or 98. Guess Clue Secret Notes number and 3 are in the number in the wrong places The number is 2U or T7. (T = tens digit, U = ones digit) No, so 2 is not in the number. 7 was the in the previous guess. The is 3 in the wrong place. The number is T3 or 9U. (T = tens digit, U = ones digit) 3 is not in the number, so 4 must be the ones digit. The number is T1 or 2U. (T = tens digit, U = ones digit) 2 is not in the number, so the number is T1. 2 is not in the number, so 5 is the. The number is T3 or 1U. (T = tens digit, U = ones digit) The number is 23 or is not in the number, so it must be 18. Neither 1 nor 2 are in the secret number. Neither 3 nor 4 are in the secret number. Neither 5 nor 6 are in the secret number. Either 7 or 8 is in the secret number in the correct place. The must be the 7 in the tens place. The only digit left for the ones place is 0. Guess Clue Possible numbers? or 58 Notes The number is T7 or 5U ,, 6 11/20/2004

7 , 25, or Your Turn #3 1. The hundreds digit may be 1, 2, 3, 4, 5, 6, 7, 8, or 9, but not 0, so there are nine ways to pick the hundreds digit. For each choice of the hundreds digit, there are nine ways to pick the tens digit, so there are 9 9 = 81 ways to pick these two digits. Now two different digits have been selected. That leaves eight digits from which to pick the ones digit. There are = 648 possible secret numbers. Your Turn #4. Guess Clue Secret number? Notes Must be 378 or 783. Not 378, so must be Guess Clue Possible numbers? Notes 132, 213, or 321 Not 132, so must be 213 or or ,, 7 11/20/2004

8 Play Sheet Use this play sheet to play 2-digit or 3-digit. 2-digit : The secret number is a number from 10 to 98 that has two different digits in place values 10 and 1. 3-digit : The secret number is a number from 102 to 987 that has three different digits in place values 100, 10, and 1. No two digits are the same. After each guess, you'll receive one or more of these clues: no digit of the guess is in the secret number. one digit of the guess is in the secret number, but is in the wrong place value. one digit of the guess is in the secret number and is in the correct place value. Guess Clue(s) Digit List and Notes Place value ,, 8 11/20/2004

9 Prime Time. Use only prime numbers as secret numbers. 2-digit prime numbers. A 2-digit secret number must have 2 different digits, so we crossed out 11: 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 3-digit prime numbers. A 3-digit secret number must have three different digits, so we crossed out primes that have 2 digits the same (101, 113, 131, and so on): 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997 For more prime numbers, go to The Prime Pages ( Composite Number. Use only composite numbers as secret numbers. 2-digit composite numbers. A secret number must have different digits, so we crossed out composite numbers that have two digits the same (22, 33, et cetera): 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99 3-digit composite numbers. We leave this as an exercise for you, the reader. Alpha. Play with 2-letter words. The word must be an entry in a designated dictionary. You decide whether to allow abbreviations we usually don't. The secret 2-letter word has two different letters. The secret 2-letter word is in the designated dictionary. A guess doesn't have to be a dictionary word. It can be any two different letters. means that no letter of the guess is in the secret word. means that a letter of the guess is in the secret word, but in the wrong place. means that a letter of the guess is in the secret word, and is in the correct place. When the player guesses the secret word, ask for a definition. Alpha. Play with 3-letter words. The word must be an entry in a designated dictionary. You decide whether to allow abbreviations we usually don't. The secret 3-letter word has three different letters. No two letters are the same. The secret 3-letter word is in the designated dictionary. A guess doesn't have to be a dictionary word. It can be any three different letters. means that no letter of the guess is in the secret word. means that a letter of the guess is in the secret word, but is in the wrong place.,, 9 11/20/2004

10 means that a letter of the guess is in the secret word, and is in the correct place. When the player guesses the secret word, ask for a definition. Here's a handy play sheet for playing Alpha. Guess # Guess Clue(s) A B C D E F G H I J K L M N O P Q R S T U V W X Y Z May dragons of good fortune be your companions as you play together, learn together. Bob & George MathBackpacks@aol.com,, 10 11/20/2004