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Solution to: Ten Questions

We denote the answers to the ten questions as A1, A2, A3, ..., A10. There are two possible cases:

  1. A10 is an integer. If A10 is a number, it must be an integer because all other numbers in the answers are integers, and the total of these numbers (A1) is also an integer.
  2. A10 is not a number. In this case, only the answer "true" affects the other answers of the puzzle (due to question 2: how many questions in this puzzle have the answer "true"?).

First, we consider the case where A10 is an integer.

We can note the following:

  1. A6 (the number of questions with the same answer as this question, including this question) is at least 1 and at most 6, because only A1, A2, A4, A6, A8, and A10 are numbers.
  2. A2 (the number of questions with the answer "true") cannot be 0, because A6 is at least 1 according to (i), and A6 would then be guaranteed to be greater than A2. This would mean that A7 (the answer to question 6 is greater than the answer to question 2) would be "true," which would require A2 to be at least 1. Furthermore, A2 is at most 4, because only A3, A5, A7, and A9 can be "true". Thus, A2 is at least 1 and at most 4.
  3. For A4 (the answer to question 1 divided by the answer to this question), the following holds: A4 = A1 : A4. This means that A4 = √A1 or A4 = -√A1.
  4. From (iii), it follows that A1 is a square and thus at least 0.
  5. Since 6 of the answers are numbers, the following holds for A8 (the average of all the numbers in the answers): A8 = A1 : 6.
  6. From (v), it follows that A1 is a multiple of 6.
  7. From (iv) and (vi) together, it follows that A1 (the total of all the numbers in the answers) is both a square and a multiple of 6. Combined with (iii), we can determine several possible values for A1, A4, and A8 (of course, there are infinitely many possible values, but below are only the smallest four possibilities for A1):
    A1A8A4
    totalA1 : 6+/-√A1
    000
    3666
    366-6
    1442412
    14424-12
    3245418
    32454-18
  8. For A1 (the total of all numbers in the answers), the following applies: A1 = A1 + A2 + A4 + A6 + A8 + A10. This means that A2 + A4 + A6 + A8 + A10 = 0. Therefore, for A10, the following applies: A10 = - (A2 + A4 + A6 + A8).
  9. We already know from (ii) and (i) that both A2 and A6 are at least 1, and from (vii) it can be inferred that A4 + A8 is at least 0. Thus, the sum of A2, A4, A6, and A8 is greater than 0. Combined with (viii), it follows that A10 must be negative.
  10. Since A10 is negative according to (ix), A3 (all numbers in the answers are greater than 0) is "false".
  11. Since A10 is negative according to (ix), it can be inferred in combination with (i) that A6 (the number of questions with the same answer as this question, including this question) is at most 5. As seen in the table from (vii), A1, A4, and A8 are also not equal to A6. This means that A6 can only be equal to itself and A2, and thus can only be 1 or 2. Moreover, A6 can only be 2 if A2 is also 2.
  12. Since A3 is "false" according to (x), A2 (the number of questions with the answer "true") can be at most 3 (if the answers A5, A7, and A9 are all "true"). However, A2 can only be 3 if A7 (the answer to question 6 is greater than the answer to question 2) is "true," which would mean that A6 must then be at least 4, which is not possible according to (xi). Therefore, A2 is either 1 or 2.
  13. It is impossible for both A2 and A6 to be 1. If A6 (the number of questions with the same answer as this question, including this question) is 1, then none of the other answers can be 1.
  14. From (xi), (xii), and (xiii), it follows that there are only two possible combinations of A2 and A6 remaining:
    • A2 = 2 and A6 = 1
    • A2 = 2 and A6 = 2
  15. From (xiv), it follows that A7 (the answer to question 6 is greater than the answer to question 2) is "false".

Of the two possible combinations for A2 and A6, as determined in (xiv), we first consider the case A2 = 2 and A6 = 1. We combine these values with the smallest possibilities for A1, A4, and A8 from (vii) and the calculation of A10 from (viii). We now need to check whether A2 corresponds to the number of answers that are "true." Based on (x) and (xv), it is already known that A3 and A7 are "false." The question is whether A5 and A9 are both "true," so that the value of A2 is correct.

A2A6A1A8A4A10A3A7A5A9
number "true"number equal to A6totalA1 : 6+/-√A1- (A2 + A4 + A6 + A8) all numbers >0?A6>A2?A1 the largest?A6 - A2 - (A6 × A4) A8 = A6 - A2 - (A6 × A4)?
21000-3falsefalsefalse-1false
213666-15falsefalsetrue-7false
21366-6-3falsefalsetrue5false
211442412-39falsefalsetrue-13false
2114424-12-15falsefalsetrue11false
213245418-75falsefalsetrue-19false
2132454-18-39falsefalsetrue17false

The above table shows that in none of the cases is A9 "true," and it is clear that this is also the case for larger values of A1. Therefore, with these values for A2 and A6, there is no solution.

Now we consider the case where A2 = 2 and A6 = 2. Again, both A5 and A9 must be "true" for the value of A2 to be correct.

A2A6A1A8A4A10A3A7A5A9
number "true"number equal to A6totalA1 : 6+/-√A1- (A2 + A4 + A6 + A8) all numbers >0?A6>A2?A1 the largest?A6 - A2 - (A6 × A4) A8 = A6 - A2 - (A6 × A4)?
22000-4falsefalsefalse0true
223666-16falsefalsetrue-12false
22366-6-4falsefalsetrue12false
221442412-40falsefalsetrue-24false
2214424-12-16falsefalsetrue24true
223245418-76falsefalsetrue-36false
2232454-18-40falsefalsetrue36false

The table shows that there is a solution in which all answers have a correct value. Moreover, it is clear that this is the only solution, as for larger values of A1, A9 will never be "true."

In a similar manner, it can be deduced that there is no possible solution for the case where A10 is not a number.

Conclusion: the answers to the ten questions are as follows:


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