### 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?"

----------------------------------------

Can you help Arnold solve Carlos's problem?

--------------------------------------------

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?"

----------------------------------------

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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