### Brain Teasers

# Arnold and Carlos: Part 1

Arnold and Carlos love math puzzles. They are constantly making them up for each other to solve. One day, Arnold presents his friend Carlos with a problem. This problem has four clues. Here is the problem Arnold gave:

"I am thinking of a mystery number.

1. The mystery number is a 2-digit prime number.

2. When the digits of the mystery number are reversed, another 2-digit prime number is formed.

3. When the original number is added to the next highest prime number, the sum forms a square number.

4. When the mystery number is added to the 2nd prime number after it, the sum is evenly divisible by 4.

What is the mystery number?"

Can you help Carlos solve Arnold's problem?

"I am thinking of a mystery number.

1. The mystery number is a 2-digit prime number.

2. When the digits of the mystery number are reversed, another 2-digit prime number is formed.

3. When the original number is added to the next highest prime number, the sum forms a square number.

4. When the mystery number is added to the 2nd prime number after it, the sum is evenly divisible by 4.

What is the mystery number?"

Can you help Carlos solve Arnold's problem?

### Hint

Make a list and eliminate possibilities due to the clues provided. Eventually, it leads to 1 answer.### Answer

17.Clue #1 - All of the 2-digit numbers are 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97.

After Clue #1 we have 21 possibilities.

Clue #2 - When every number is reversed, the numbers are 11, 31, 71, 91, 32, 92, 13, 73, 14, 34, 74, 35, 95, 16, 76, 17, 37, 97, 38, 98, and 79. Of these, only 31, 71, 13, 73, 17, 37, 97, and 79 are different prime numbers (11 reversed is still 11).

After Clue #2 we have 8 possibilities: 13, 17, 31, 37, 71, 73, 79, and 97.

Clue#3 - Add the next prime number to form a perfect square. So, using our list from Clue #2, we know our ten base numbers. We must add 13 and 17, 17 and 19, 31 and 37, 37 and 41, 71 and 73, 73 and 79, 79 and 83, 97 and 101. These sums are 30, 36, 68, 78, 144, 152, 162, and 198 respectively. Of these sums-only 36 (which was formed using 17) and 144 (formed using 71) are perfect squares.

After Clue #3, we have 2 possibilities: 17 and 71.

Clue #4 - Add the 2nd prime number after to make a number evenly divisible by 4. This would mean we add 23 to 17, and 79 to 71. These sums are 40 and 150. Only 40 is evenly divisible by 4. So, only 17 fits.

After Clue #4, we have one possibility: 17.

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## Comments

Easy............

Very clever teaser... but I have to say it was NOT easy for me! My brain hurts!

TOO MUCH INFORMATION! Good, but one could speed up the process by skipping clue #2. Beyond 2, every prime is odd, so the sum of the two primes in clue #3 will be even. The average of the two primes must be an integer which is half of a square number (and falls exactly between the two primes.) Odd squares produce fractions when halved, so they can be ignored. One tests the prime numbers preceding 18, 32, 50, 72, and 98. (These are the half values of 36, 64, 100, etc.) The result has 17, 47, and 71 as possiblities. Clue #4 eliminates both 47 and 71, making clue #2 extraneous..

Nice teaser.

Fun teaser! Good job!

easy, i liked it, but it could have been more challenging

How can we include 23 in the list ?!!. Reverse 23 and we get 32 and it is not a prime number. This violates the second rule in the question.

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