Brain Teasers
Arnold and Carlos: Part 2
Arnold and Carlos love math puzzles. They are constantly making them up for each other to solve. One day, Carlos presents his friend Arnold with a problem. This problem has six clues. Here is the problem Carlos gave:
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I am thinking of a mystery number.
1. The mystery number is a 3-digit prime number with digits ABC.
2. Two 2-digit prime numbers occur in the mystery number (AB and BC).
3. A, B, and C are all distinct digits.
4. The mystery number does not contain a 9.
5. If A, B, and C are re-arranged from greatest to least, a 3-digit prime number is formed.
6. If A, B, and C are re-arranged from least to greatest, a 3-digit prime number is formed.
What is the mystery number?"
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Can you help Arnold solve Carlos's problem?
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I am thinking of a mystery number.
1. The mystery number is a 3-digit prime number with digits ABC.
2. Two 2-digit prime numbers occur in the mystery number (AB and BC).
3. A, B, and C are all distinct digits.
4. The mystery number does not contain a 9.
5. If A, B, and C are re-arranged from greatest to least, a 3-digit prime number is formed.
6. If A, B, and C are re-arranged from least to greatest, a 3-digit prime number is formed.
What is the mystery number?"
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Can you help Arnold solve Carlos's problem?
Hint
Make a list and organize your possibilities. It will lead down to one answer.The 3-digit prime numbers are:
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, and 997.
Answer
Answer: 617.Clue #1- "The mystery number is a 3-digit prime number with digits ABC." The 3-digit prime numbers are: 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, and 997
After Clue #1- we have 143 possibilities.
Clue #2- "Two 2-digit prime numbers occur in the mystery number (AB and BC)." The two digit prime numbers are 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97. Since the two prime numbers are AB and BC, they share one digit- B. The only combinations for ABC such that AB and BC are prime numbers is: 113, 117, 119, 131, 137, 171, 173, 179, 197, 231, 237, 297, 311, 313, 317, 319, 371, 373, 379, 411, 413, 417, 419, 431, 437, 471, 473, 479, 531, 537, 597, 611, 613, 617, 619, 671, 673, 679, 711, 713, 717, 719, 731, 737, 797, 831, 837, 897, 971, 973, and 979.Out of these 51 numbers, only 113, 131, 137, 173, 179, 197, 311, 313, 317, 373, 379, 419, 431, 479, 613, 617, 619, 673, 719, 797, and 971 are prime.
After Clue #2 we have 21 possibilities: 113, 131, 137, 173, 179, 197, 311, 313, 317, 373, 379, 419, 431, 479, 613, 617, 619, 673, 719, 797, and 971.
Clue #3- "A, B, and C are all distinct digits." Out of our 21 possibilities, 113, 131, 311, 313, 373, and 797 do not have distinct digits.
After Clue #3, we have 15 possibilities: 137, 173, 179, 197, 317, 379, 419, 431, 479, 613, 617, 619, 673, 719, and 971.
Clue #4- "The mystery number does not contain a 9."179, 197, 379, 419, 479, 619, 719, and 971 all contain a 9.
After Clue #4, we have 7 possibilities: 137, 173, 317, 431, 613, 617, and 673.
Clue #5- "If A, B, and C are re-arranged from greatest to least, a 3-digit prime number is formed." With our 7 possibilities (137, 173, 317, 431, 613, 617, and 673) we arrange their digits from greatest to least getting 731, 731, 731, 431, 631, 761, and 763 respectively. Of these new 7 numbers, only 431 (from 431), 631 (from 613), and 761 (from 617) are prime.
After Clue #5, we have 3 possibilities: 431, 613, and 617.
Clue #6- "If A, B, and C are re-arranged from least to greatest, a 3-digit prime number is formed." With our 3 possibilities (431, 613, and 617), we arrange their digits from least to greatest to get 134, 136, and 167 respectively. 134 (from 431) and 136 (from 631) are non-prime. 167 (from 617) is prime.
After Clue #6, we have one possibility: 617.
617 is the mystery number.
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Comments
Aug 18, 2006
317 is also a number that fits this scenario.
it was hard, but i eventually got it
it was hard, but i eventually got it
When clue 5 is applied to 317 it becomes 731, which is not prime.
Aug 18, 2006
317 becomes 713 which is prime
Aug 18, 2006
I'm sorry , I see my mistake now
This was an exercise, not a teaser!
The least you could have asked for was best procedure. How about -
Per clue #2 start with 2-digit primes but exclude 11 because its inclusion in ABC breaks clue #3. Also exclude all that involve 9's, per clue #4. : 13, 17, 23, 31, 37, 41, 43, 47, 53, 61, 67, 71, 73, and 83.
Make every combination except those that fail clue #3. : 137, 173, 237, 317, 371, 413, 417, 431, 437, 471, 473, 531, 537, 613, 617, 671, 673, 713, 731, 831, and 837.
Next, per rules #5 and #6 exclude every combination where the middle value digit is in the middle (B) position. (Together these two clues indicate A, B, and C are neither in order nor in reverse order.) : 173, 317, 371, 413, 417, 437, 471, 473, 537, 613, 617, 671, 673, 713, and 837.
This is first point one needs to check for 3-digit primes. These 15 possibilities are winnowed down to 5. : 173, 317, 613, 617, and 673.
Per clue #6, 613 reformed as 136 is not prime. Per clue #5, both 173 and 317 reform as 731, not a prime. 673 becomes 367, not a prime. 761 is prime, which means only 617 is left, and reformed as 167, also a prime, it also passes the clue #6 test.
The least you could have asked for was best procedure. How about -
Per clue #2 start with 2-digit primes but exclude 11 because its inclusion in ABC breaks clue #3. Also exclude all that involve 9's, per clue #4. : 13, 17, 23, 31, 37, 41, 43, 47, 53, 61, 67, 71, 73, and 83.
Make every combination except those that fail clue #3. : 137, 173, 237, 317, 371, 413, 417, 431, 437, 471, 473, 531, 537, 613, 617, 671, 673, 713, 731, 831, and 837.
Next, per rules #5 and #6 exclude every combination where the middle value digit is in the middle (B) position. (Together these two clues indicate A, B, and C are neither in order nor in reverse order.) : 173, 317, 371, 413, 417, 437, 471, 473, 537, 613, 617, 671, 673, 713, and 837.
This is first point one needs to check for 3-digit primes. These 15 possibilities are winnowed down to 5. : 173, 317, 613, 617, and 673.
Per clue #6, 613 reformed as 136 is not prime. Per clue #5, both 173 and 317 reform as 731, not a prime. 673 becomes 367, not a prime. 761 is prime, which means only 617 is left, and reformed as 167, also a prime, it also passes the clue #6 test.
"673 becomes 367," should have said "673 becomes 763,"
This was awsome...I should show it to my math teacher.
WOW!!
I only considered eight possibilities and only had to determine the primality of fourteen 3-digit numbers. It reduced to these possibilities rather easily: the only thing I wrote down in solving this was these eight numbers.
First, since AB and BC are prime with no 9s, then both B and C are 1, 3 or 7. Every combination of these digits forms a two digit prime, so BC is either 13, 17, 31, 37, 71 or 73. If A is not the middle value, then A must also be 1, 3 or 7.
Since B can't be the middle, then (B < C) = (A < C). Also, neither A + B nor A + B + C can be a multiple of three or the number will be divisible by three.
Using these simple rules, I then wrote down all of the 3-digit numbers that were candidates for each possible value for BC:
13: [713]: 2+1+3=6, so A is not the middle value and can only be 7
17: [317, 617]: 2+1=3, 4+1+7=12, 5+1=6, leaving 317 and 617
31: []: 2+1+3=6 and A can't be greater than B
37: [437]: A can't be less than B, 5+3+7=15, 6+3=9, leaving 437
71: [371, 671]: 2+7=9, 4+7+1=12, 5+7=12, leaving 371 and 671
73: [473, 673]: 2+7=9, 5+7=12, leaving 473 and 673
Eliminating the non-primes leaves 317, 617 and 673, so one of the pairs of numbers (137, 731), (167, 761) or (367, 763) is two primes. The prime numbers from these six numbers are 137, 167, 761, and 367, so (167, 761) is the prime pair and 617 is the number.
First, since AB and BC are prime with no 9s, then both B and C are 1, 3 or 7. Every combination of these digits forms a two digit prime, so BC is either 13, 17, 31, 37, 71 or 73. If A is not the middle value, then A must also be 1, 3 or 7.
Since B can't be the middle, then (B < C) = (A < C). Also, neither A + B nor A + B + C can be a multiple of three or the number will be divisible by three.
Using these simple rules, I then wrote down all of the 3-digit numbers that were candidates for each possible value for BC:
13: [713]: 2+1+3=6, so A is not the middle value and can only be 7
17: [317, 617]: 2+1=3, 4+1+7=12, 5+1=6, leaving 317 and 617
31: []: 2+1+3=6 and A can't be greater than B
37: [437]: A can't be less than B, 5+3+7=15, 6+3=9, leaving 437
71: [371, 671]: 2+7=9, 4+7+1=12, 5+7=12, leaving 371 and 671
73: [473, 673]: 2+7=9, 5+7=12, leaving 473 and 673
Eliminating the non-primes leaves 317, 617 and 673, so one of the pairs of numbers (137, 731), (167, 761) or (367, 763) is two primes. The prime numbers from these six numbers are 137, 167, 761, and 367, so (167, 761) is the prime pair and 617 is the number.
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