# The Multiple Choice Test Test

One day, the logic professor announces a pop quiz. "All questions in this test are about the test itself. In fact, the test is open note since you have a copy of the test itself. Good luck!"

1. What is the 1st question whose answer is B?

a. 1 b. 2 c. 3 d. 4 e. 5

2. The only 2 consecutive questions with identical answers are

a. 6, 7 b. 7, 8 c. 8, 9 d. 9, 10 e. 10, 11

3. The number of questions with answer E is

a. 0 b. 1 c. 2 d. 3 e. 4

4. The number of questions with answer A is

a. 4 b. 5 c. 6 d. 7 e. 8

5. This question and what question have the same answer?

a. 1 b. 2 c. 3 d. 4 e. 5

6. What is the answer to #17?

a. C b. D c. E d. None of the above e. All of the above

7. The answer to this question and the following question are how far apart alphabetically?

a. 4 b. 3 c. 2 d. 1 e. 0

8. How many questions have vowels as an answer?

a. 4 b. 5 c. 6 d. 7 e. 8

9. The next question with the same answer as this one is

a. 10 b. 11 c. 12 d. 13 e. 14

10. The answer to #16 is

a. D b. A c. E d. B e. C

11. The number of earlier (1-10) questions with answer B is

a. 0 b. 1 c. 2 d. 3 e. 4

12. The number of questions with consonants as an answer is

a. an even number b. an odd number c. a perfect square d. prime e. divisible by 5

13. The only odd-numbered problem with answer A is

a. 9 b. 11 c. 13 d. 15 e. 17

14. The number of questions with answer D is

a. 6 b. 7 c. 8 d. 9 e. 10

15. The answer to #12 is

a. A b. B c. C d. D e. E

16. The answer to #10 is

a. D b. C c. B d. A e. E

17. The answer to #6 is

a. C b. D c. E d. none of the above e. all of the above

18. the number of questions with answer A is also the number of questions with answer

a. B b. C c. D d. E e. none of the above

19. The answer to this question is

a. A b. B c. C d. D e. E

20. Of the following, which are great logicians?

a. Boole b. De Morgan c. Russell d. Whitehead e. All of the above

Hint:

use the fact that each question has only 1 answer

Note: this puzzle is taken from the Mu Alpha Theta math competition. However, it earlier appeared in a slightly different form at http://faculty.uml.edu/jpropp/srat-Q.txt.

• I'm putting this on hold because it's an unacknowledged copy of someone else's work and we need to know whether OP got its creator's permission to post it here. Dec 22, 2018 at 23:34
• This is James Propp's Self-Referential Aptitude Test. @Display name, do you have Dr Propp's permission to post it here? (Even if you do, the question needs to give proper credit to its original author.) Dec 22, 2018 at 23:35
• ... if you don't have the permission of the original creator then you're probably breaking copyright law. It is not the job of PSE moderators to be copyright police, and we will not remove things merely because they're copyright infringements (and there are actually legal reasons why we shouldn't start trying) -- but purely informally I'd request that other people's stuff not be copied here without their permission. Dec 22, 2018 at 23:57
• I don't think we know that Propp is the original author, but on the page I linked to (and the other pages linked therefrom) he is clearly claiming to be, and unless he's outright falsifying things there is plenty of evidence there that his claim to authorship goes back at least to the year 2000. Is the competition you mention older than that? Dec 22, 2018 at 23:59
• Once again, the actual creator is (unless he's lying about it) James Propp, so it doesn't matter who was on the Mu Alpha Theta problem committee. Anyway, this is now suitably attributed so I'll reopen it again. Dec 23, 2018 at 1:41

Question 5 must match the answer of question 5, since there is only one answer that is correct for each question. So 5 is E and 1-4 can not be E.

Thus

Question 10 is linked to Question 16, and the only one that matches both ways is 10 - A and 16 - D

Thus

Question 17 is linked to Question 6, the only way that can be is if one is B and the other is D. Since 16 is D and 16 and 17 can't have the same answer (from Q2), then 6 is D and 17 is B.

Thus

Question 13 says only one odd question can have answer A. It can't be A-13 (which would mean 13 and 9 do), B-11 because question 1 says there is a B in the first 5 questions and so 11 cant be A, C-13 because then 13 would not be A, E-17 from previous result. So 13 is D and 15 is A. Which means 12 is A.

Thus

Question 20 should be E

So

Question 3 is the number of Es. It can not be E-4 because then D-3 would also be true. We already have two Es (20 and 5), so 3 is C-2 or D-3

So

Question 1 is the first B answer. This can not be A-1 nor B-2 because these are contradictions. Also, we know from Q5, that it can't be E-5. and C-3 is wrong because Q3 is C or D. So Q1 is D-4. This means that Q4 is B.

So

Q8 counts the number of A and E answers. from Q4, we know there are 5 As and Q3 says there are 2 or 3 Es, so Q8 is D-7 or E-8.

So

Q8 must be E-8 because Q-7 is B,C,D,or E and since Q8 can't be A or B or C from previous answers, Q8 is E in each situation. So Q3 must be D-3.

So

Because Q4 is B-5, we know there are 5 As. 3 are known (10, 15, 12). The only two left that can be A (because of contradictions or because 9,10 can't both be A because 2 can't be D because 1 is D) are 2 and 18, so Q2 is A and Q18 is A.

So

Because Q2 is A, Q6 and Q7 are the same so Q7 is D.

So

Because Q7 is D, Q8 must be E or C, but it can't be C so Q8 is E. This means Q3 must be D (there are 3 Es).

So

Q11 must be 1 or 2 so must be B or C. But if Q9 is B, Q11 is B and False. So Q9 must be D and Q11 is B.

So

Q14 can only be B or C so the number of Ds is 7 or 8. The maximum number of Ds is 7 (So Q14 is B) and that means that Q19 is B which makes Q18 A.

1-D
2-A
3-D
4-B
5-E
6-D
7-D
8-E
9-D
10-A
11-B
12-A
13-D
14-B
15-A
16-D
17-B
18-A
19-B
20-E

• You say rot13(Dhrfgvba 20 fubhyq or R) in your solution, then arrive at a different letter at the end. Check your solution for mistakes. Dec 22, 2018 at 23:09
• Oops. so I did. Typo in the solution list is fixed now. Thanks @Displayname Dec 23, 2018 at 0:32
• Actually, I never noticed it until now, but your answer list purports that there are rot13(6 Q'f, juvyr lbhe nafjre gb dhrfgvba 14 pynvzf gung gurer ner 7). Dec 23, 2018 at 0:34
• @Displayname found my mistake I think. An incorrect assumption on Q3. Corrected i hope. Dec 23, 2018 at 1:21

From Q1 we know that Q1, Q2, Q3, Q4 or Q5 is B. We know it cannot be Q1 because that would mean the answer would have to be A and B. Similarly, it cannot be Q2, because then Q1 would be the first B. It can also not be Q5 as this would cause Q2 to become the first B. This leaves us with C (Q3) and D (Q4), from which I cannot continue.

Next, I'll consider Q10 and Q16. These will cause a paradox unless Q10 is A and Q16 is D. Two questions down, eighteen to go.

Q13 provides a pleasant surprise. This rules out A in Q3, Q5, Q7, Q19, and finally Q13 because it being A would cause Q9 to also be A, but there is only one odd-numbered question with answer A. We can also rule out Q13 being answered by

Similarly to Q10 and Q16, Q6 and Q17 will cause a paradox unless their answers are B and D. However, unlike Q10 and Q16, we do not know which letter answers which question.

Q20 is easy enough. Since there are logicians named George Boole, Augustus De Morgan, Bertrand Russell and Alfred North Whitehead, the only feasible answer is E.

Q12 tells us the amount of answers with consonants (B, C or D) is either: odd, even, square, prime or divisible by 5. Since a number cannot be neither odd or even, C, D or E must all be incorrect. The only numbers between one and twenty that aren't square, prime or divisible by 5 are 6, 8, 12, 14 and 18. From Q8 we can figure out that the amount of consonants must be 16, 15, 14, 13 and 12. Crossing this with our Q12 answers we discover there are only possibilities 12 and 14. We also know Q12 is A.

Q15 asks us for the answer to Q12 as its own. This means the answer to Q15 is A.

However, Q2 teels us there is only one situation in which two identical questions have the same answer. Since Q16 and Q17 are not one of the options for Q2, Q17 cannot be D and must therefore be B This also means Q6 is D.

Current Position:

Q1: C or D

Q2: ?

Q3: Not A

Q4: ?

Q5: Not A

Q6: D

Q7: Not A

Q8: C or E

Q9: ?

Q10: A

Q11: ?

Q12: A

Q13: Not A

Q14: ?

Q15: A

Q16: D

Q17: B

Q18: ?

Q19: Not A

Q20: E

Please post suggestions in the comments if you have any, it would be of great help.

• "The only numbers between one and twenty that aren't square, prime or divisible by 5 are 2, 3, 7, 11, 13, 17 and 19." This is wrong, so you'll have to go back to that point. Dec 22, 2018 at 21:37
• @a guy. Yes. Did you read the 2nd sentence of my comment? Dec 22, 2018 at 21:39
• There we go. Fixed. Dec 22, 2018 at 21:47