Number Nightmare Trivia Quiz

Answer: 89

Since all the answers in this quiz are interrelated, rather than list individual answers I will step through a process that derives all of the answers.

First of all, the answer to question 9 has to be in the range of one to ten. The answer to to question 9 cannot be 10 because 10 is not prime, and all answers would have to be prime in order for an answer of 10 to be correct. Therefore, the answer to question 9 is in the range from 0-9, which means that question 9 is the question whose answer is not a two digit number. Every other
answer must be a two digit number.

Answer: 13

Since all answers are less than 100 (1 or 2 digits), the square root of any answer has to be less than the square root of 100. Therefore the maximum value that question 2 can have is the sum of two different integers less than 10. The maximum value that question 2 can have, therefore is 9+8 since 9+9 would be invalid because 9 squared would have to be the answer to two questions, violating the rule that no answer is repeated.

Can we refine the maximum value of question 2 even further? Well if 9+8 were a correct answer for question 2, then the minimum value for question 1 would be 81+64 (9 squared plus 8 squared). This is a three digit number. Therefore 17 is not a valid answer for question 2).

Using this technique further, we can show that 16 is not a valid answer for question 2 (8+8 and 9+7 each violate either the no answer repeated rule or make the answer to question 1 invalid). Similarly, 15 is not a valid answer because 9+6 and 7+8 are both invalid, and 14 is not a valid answer because 9+5, 8+6, and 7+7 are all invalid. Therefore, the answer to question 2 must be 13 or lower. The only two digit prime number less than 13 is 11, so the answer to question 6 must be 11, and the answer to question 2 is either 12 or 13.

Answer: 36

Now look at question 3. Possible values are 18, 27, 36, 45, 54, 63, 72, and 81. Note that the minimum value of question 4 would be the answer to question 3 plus 12 (the minimum value of question 2). Also, note that the answer to question 5 is really 2 times the answer to question 4.

Therefore, the answer to question 5 is even. The maximum value possible for question 5 is 98, meaning that the maximum value for question 4 is 49. This means that the only valid values for the answer to question 3 are 18, 27 or 36, because all others would cause question 4 to excede its maximum possible value.

Answer: 49

The product of any two digit numbers is a number with more than three digits. Therefore the answers to question 7 and
question 8 must be numbers multiplied by the answer to question 9.
The answer to question 9 must be at least 2 because
the answers to question 6 and to question 10 are prime.
The answer to question 9 is also 6 or less because at least four answers are
not prime (all possible answers to question 3 are divisible by 3, the answer to question 5 is divisible by 2, and the answers
to question 7 and question 8 can be factored by the answer to question 9. We now have the answer to question 9 being in
the range of 2 through 6.

Answer: 98

Getting back to question 2, possible combinations for an answer of 12 are 7+5, 8+4, and 9+3. Possible combinations for
an answer of 13 are 7+6, 8+5, and 9+4. The 9+3 combination can be eliminated because 3 is the square root of 9, and 9 cannot
be one of the answers because the only one digit answer is in the range of 2 through 6.
If no numbers that are the square of integers appear in the answers to questions 3, 4 and 5, then the squares must be
in questions 7 and 8 (the answer to question 1 is the sum of these numbers, the answer to question 2 is 12 or 13 which are
not the squares of integers, the answers to questions 6 and 10 are prime, and the answer to question 9 is too small to
be the square of any of the possible remaining numbers). Since both question 7 and question 8 the product of question 9,
the squares of the pair of numbers that add up to the answer to question 2 must have a common factor. This eliminates
49 and 25 (the 7+5 combination), 49 and 36 (the 7+6 combination), 64 and 25 (the 8+5 combination), and 81 and 16 (the 9+4
combination). The only remaining combination is 8+4, which means that the answers to queston 7 and 8 must be 64 and 16.
16 however, cannot be expressed as the product of a two digit number and a number in the range of 2 through 6, so that
combination must be elimated also. Therefore, at least one of the squares that makes up the answer to question 2 must
be in the answers to questions 3, 4, and 5.

Answer: 11

The answer to question 2 is either 12 or 13, the answer to question 3 is either 18, 27, or 36 (repeating previous
conclusions) Working through the six possible combinations of answers for questions 2 and 3, the only time any of the
answers to questions 3, 4, or 5 are the square of an integer occurs when the answer to question 3 is 36. Therefore,
the answer to question 3 is 36, and 6 must be one of the numbers that makes up the answer to question 2. Therefore,
the answer to question 2 cannot be 12, because this would cause the two numbers adding up to the answer to question 2 to
both be 6, which would mean that two answers in this quiz must be 36, which would violate the rule that all answers
are unique. Therefore the answer to question 2 must be 13. Adding 36 and 13 yields 49, the answer to question 4, and
adding 36, 13, and 49 yields 98, the answer to question 5. 49 is also the square of 7, so the answer to question 4 also
supplies the other square that is needed for the answer to question 2. Therefore the answer to question 1 is at least
36+49 = 85.

Answer: 92

At this point we know that the answers to questions 3, 4, 5, 7, and 8 are not prime, and that the answers to questions 2, 6, and 10 are
prime. If the answers to quesitons 2, 6, 10 were the only primes, then the answer to question 9 would be 3.
However 3 is also prime so this answer
is incorrect. It looks like we are stuck since the answer to question 1 so far is 85 which is not prime. But wait! If the answer
to question 9 were 4, then it also would be the square of an integer. This means that 4 should be added to the answer to question
1 which changes the answer to question 1 to be 89 which is prime. Thus, the answer to question 9 is 4, and the answers to questions
1, 2, 6, and 10 are the only questions with prime number answers. The answer to question 1 is 89 (36+49+4). Note that since the
only one digit answer has been used already to compute the answer to question 1, any other squares of an integer that can
possibly be added to question 1 must also be two digits long. The minumum square possible then is 16. If this were added to 89, you would
get 105 which is over two digits. Therefore no other numbers can be added to question 1, so 89 is the final answer. Also note
that 2, the square root of the answer to question 9, is too small to be used for any of the numbers used to answer question 2 (making
2 one of the numbers added in question 2 would cause the other number to be so large that it would have to be the square root of a
three digit number). Therefore, the answers question 2 remain unique, even though another answer is now also the square of an integer.

Answer: 52

At this point, questions 7, 8, and 10 are the only questions that remain to be answered.
Since the answer to question 1 is the third largest answer on the test and
the only answer larger than 89 is the answer to question 5 (98), one of the answers to question 7, 8, or 10 must be larger than 89.
If the answer to question 10 were larger than 89, it would have to be a prime number in the range of 90 to 99. The only prime number
in that range is 97. However 97 is one less than the answer to question 5, which violates the stamement made in question 5. Therefore,
the answer to question 10 is not greater than 89. Since the answer to question 8 is not greater than 89, by process of elimination
we get question 7 as having the other answer that is greater than 89.

Answer: 4

Given all the answers that we know so far, the only valid products of the answer to question 9 and the answer to any other question
are 44 (the answer to question 6 times the answer to question 9), and 52 (the answer to question 2 times the answer to question 9).
All other answers are too large. Therefore in order get an answer to question 7 that is greater than 89 and less than 100, 4 (the answer
to question 9) must be multiplied by either 23 or 24. The only available spot for this answer is question 10. Since this answer must be
prime, then the answer to question 10 must be 23, and the answer to question 7 is 4 times 23 or 92.

Answer: 23

This leaves question 8 as the only remaining unanswered question. We know that this number is either 44 or 52. If it were 44, then
the difference between the digits would be 0. Dividing any number by 0 would not result in a two digit integer. Therefore, the answer
to question 8 cannot be 44, and must be 52. Whew!