Self-Referential Logic Puzzle

Logic Level 4

This logic puzzle was created by Mu Alpha Theta.

Each question in this puzzle has a unique answer. Good luck.

It is recommended that you set aside about a half hour to solve this.

If you want to warm up with another self-referential logic puzzle, try this one .

1. The first question whose answer is B is question
   (A) 1   (B) 2   (C) 3   (D) 4   (E) 5

2. The only two consecutive questions with identical answers 
   are questions
   (A) 6 and 7   
   (B) 7 and 8   
   (C) 8 and 9   
   (D) 9 and 10   
   (E) 10 and 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. The answer to this question is the same as the answer to 
   question
   (A) 1   (B) 2   (C) 3   (D) 4   (E) 5

6. The answer to question 17 is
   (A) C
   (B) D
   (C) E
   (D) None of these answers
   (E) All of these answers

7. Alphabetically, the answer to this question and the answer 
   to the following question are
   (A) 4 apart
   (B) 3 apart
   (C) 2 apart
   (D) 1 apart
   (E) The same

8. The number of questions whose answers are vowels is
   (A) 4   (B) 5   (C) 6   (D) 7   (E) 8

9. The next question with the same answer as this one is 
   question
   (A) 10   (B) 11   (C) 12   (D) 13   (E) 14

10. The answer to question 16 is
    (A) D   (B) A   (C) E   (D) B   (E) C

11. The number of questions preceding this one with the answer 
    B is
    (A) 0   (B) 1   (C) 2   (D) 3   (E) 4

12. The number of questions whose answer is a consonant is
    (A) Even
    (B) Odd
    (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 question 12 is
    (A) A   (B) B   (C) C   (D) D   (E) E

16. The answer to question 10 is
    (A) D   (B) C   (C) B   (D) A   (E) E

17. The answer to question 6 is
    (A) C
    (B) D
    (C) E
    (D) None of these answers
    (E) All of these answers

18. The number of questions with answer A equals the number of 
    questions with answer
    (A) B   (B) C   (C) D   (D) E   (E) None of these answers

19. The answer to this question is
    (A) A   (B) B   (C) C   (D) D   (E) E

20. What is 1+1 in base 1337?
    (A) 3   (B) .   (C) 1   (D) 4   (E) 2

Let $\mathcal{A}$ represent the sum of the numbers of the questions with answer $A;$ $\mathcal{B}$ for the sum of the numbers questions with answer $B;$ and so on, up to $\mathcal{E}$ .

Find $\mathcal{A}+2\mathcal{B}+3\mathcal{C}+4\mathcal{D}+5\mathcal{E}.$

Details and Assumptions :

If questions 3, 13, 15, and 16 have answer $Z,$ then $\mathcal{Z}=3+13+15+16=47$

The answer is 572.

2 solutions

Trevor B.
May 10, 2015

I will use the notation $Q_N$ where $Q$ is the question number and $N$ is the set of possible answer choices for question $Q.$ These will be shown after each step. Note: This is how I completed the puzzle. Your solution may be in a different order from mine.

$1_{ABCDE}2_{ABCDE}3_{ABCDE}4_{ABCDE}5_{ABCDE}$ $6_{ABCDE}7_{ABCDE}8_{ABCDE}9_{ABCDE}10_{ABCDE}$ $11_{ABCDE}12_{ABCDE}13_{ABCDE}14_{ABCDE}15_{ABCDE}$ $16_{ABCDE}17_{ABCDE}18_{ABCDE}19_{ABCDE}20_{ABCDE}$

Questions (5) and (20) are trivially E, making (3) not A or B.

(4), (6), and (19) are not E as a result of (2). Because (3) is not B, (1) is not C. $1_{ABDE}2_{ABCDE}3_{CDE}4_{ABCD}5_{E}$ $6_{ABCD}7_{ABCDE}8_{ABCDE}9_{ABCDE}10_{ABCDE}$ $11_{ABCDE}12_{ABCDE}13_{ABCDE}14_{ABCDE}15_{ABCDE}$ $16_{ABCDE}17_{ABCDE}18_{ABCDE}19_{ABCD}20_{E}$

We will now eliminate logical fallacies.

(1) is not A or B. (10) and (16) have answers that depend on each other. The only way the questions are satisfied is if (10) us A and (16) is D. Because of (2), (15) and (17) are not D.

A chain of linked answers in (6) and (17) means that one of the two is B and the other is D. Because (17) is not D, (17) is B and (6) is D. Because of (2), (18) is not B.

Because of the information in (1), (11) is not A. This means (2) is not E.

(12) has a unique solution. The number of questions with answers of consonants is either even or odd, so (12) can only be A or B. This means (9) is not C.

(13) is not A or C, as these lead to a contradiction. This means 9 is not A.

(15) implies that (12) and (15) have the same answer. Thus, (15) is not C or E. $1_{DE}2_{ABCD}3_{CDE}4_{ABCD}5_{E}$ $6_{D}7_{ABCDE}8_{ABCDE}9_{BDE}10_{A}$ $11_{BCDE}12_{AB}13_{BDE}14_{ABCDE}15_{AB}$ $16_{D}17_{B}18_{ACDE}19_{ABCD}20_{E}$

(7) and (19) are not A as a result of (13).

Casework on (7) yields that (8) is either A, C, or E.

(5) is not B, so (1) is not E. This means (1) is D, leading (4) to be B. Since there are exactly 5 A's, and so far we have found 2 E's, (8) is not A or C, making it E.

(9) and (10) cannot have the same answer so (2) is not D. $1_{D}2_{ABC}3_{CDE}4_{B}5_{E}$ $6_{D}7_{BCDE}8_{E}9_{BDE}10_{A}$ $11_{BCDE}12_{AB}13_{BDE}14_{ABCDE}15_{AB}$ $16_{D}17_{B}18_{ACDE}19_{BCD}20_{E}$

There are 8 questions with answers being vowels and 5 questions with answer A, so (3) is D. No other questions can have answer choice E.

(7) is not E, making (2) not B. (9) is not E, so (2) is not C. This means (2) is A, making (7) D. Also, (13) must be D because no other odd-numbered question can be A.

(18) is not E, so one of the letter choices must be the answer exactly 5 times. E has been shown to appear exactly 3 times, so (18) is not D. By (14), D occurs more than 5 times, so (18) is not C. Therefore, (18) is A.

There are 8 questions with answer being a vowel, so there are 12 questions whose answers are consonants. Thus, (12) is A. This means (15) is A. By (2), (14) is not A. $1_{D}2_{A}3_{D}4_{B}5_{E}$ $6_{D}7_{D}8_{E}9_{BD}10_{A}$ $11_{BCD}12_{A}13_{D}14_{BCD}15_{A}$ $16_{D}17_{B}18_{A}19_{BCD}20_{E}$

There are at maximum 2 questions preceding (11) with answer B, so (11) cannot be D.

Since the locations of the 5 A's have been found, (14) cannot be A. This means at least 1 more question has answer D. 5 questions have answer B. 2 of those questions have been determined, so out of the 4 remaining questions, 3 have answer B. This means there is exactly 1 more question with answer D, making (14) B.

We now know how many times each letter is the answer: 5 A's, 5 B's, 7 D's, and 3 E's. Thus, no other questions have answer C. This means (11) is B.

Since there are no other questions with answer B preceding (11), (9) is D.

Finally, since we have found the locations of the 5 D's, (19) is B and the puzzle is solved. $1_{D}2_{A}3_{D}4_{B}5_{E}$ $6_{D}7_{D}8_{E}9_{D}10_{A}$ $11_{B}12_{A}13_{D}14_{B}15_{A}$ $16_{D}17_{B}18_{A}19_{B}20_{E}$

We can now make a chart of which question has which answer.

$\begin{array}{|c|l|} \hline A & 2, 10, 12, 15, 18\\ \hline B & 4, 11, 14, 17, 19\\ \hline C & \\ \hline D & 1, 3, 6, 7, 9, 13, 16\\ \hline E & 5, 8, 20\\ \hline \end{array}$

$\begin{array}{|c|c|} \hline \mathcal{A} & 57\\ \hline \mathcal{B} & 65\\ \hline \mathcal{C} & 0\\ \hline \mathcal{D} & 55\\ \hline \mathcal{E} & 33\\ \hline \end{array}$

Thus, $\mathcal{A}+2\mathcal{B}+3\mathcal{C}+4\mathcal{D}+5\mathcal{E}=\boxed{572}$

One of the rule statement is: Each question in this puzzle has a unique answer.

So how could be, question number nine could be have more than one unique answer. In this case, question number 9, having possibilities answer of A and D. If 9, is A, then 13 also could be an A, where this lead to different statement of number 13 itself.

The rule break its own rule, that's why after found un-unique answer of question 9 and 13, I just stop my effort to solve this.

Aryell Caesar - 5 years, 9 months ago

Q13 cannot be A, since it will make consecutive As with Q12, when Q2 already specified "The only two consecutive questions" and answer {12 and 13} is not listed as possible answers.

Saya Suka - 1 year, 6 months ago

You start saying 5 is E but your reasoning is off. If 2 was B, 5B would be a perfectly correct answer. To put it differently: If you think every question has only one correct answer, how can you answer question 19?

Andreas Vestermo Nesje - 5 years, 7 months ago

Q2 cannot be B as you can't force Q1 to accept the answer 2B. Q19 accepts an answer that would validate the truthfulness of all the answers as a whole, including itself.

Saya Suka - 1 year, 6 months ago

you could eliminate E from 1 to 4 in the 2nd step itself as 5 says that the answer of 5 is E and the rest do not satisfy this so 1 to 4 cannot have E as an answer

Ajinkya Shivashankar - 4 years, 8 months ago

What a question! Hats off to the efforts.

Vijayant Patel - 1 year, 8 months ago

Saya Suka
May 7, 2021

Reading up to Q5, Q1 = { C, D } , Q2 ≠ B ≠ Q5.

{ Q6, Q17 } = { B <=> D }
Q8 = { A, C, E }
Q10 = A & Q16 = D
Q17 = B & Q6 = D
Q5 = { C, E }
By Q8, Q12 = { 12, 14, 16 } = { A, C }
By Q12, Q15 = { A, C }
Q20 = E

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

By Q8 & Q1, Q5 = E
Also, { Q3, Q4 } = { B <=> D }
Furthermore, Q8 = E, Q12 = A = Q15

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

From the above halfway filled table, we get :
Q1 = D = Q3 & Q4 = B
Q2 ≠ D, so Q2 = { A, C }
Q9 ≠ A ≠ Q11

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

As	Value	Qs	Sum	Product
A	1 , [ 5 ]	2, 10, 12, 15, 18	57	57
B	2 , [ 5 ]	4, 11, 14, 17, 19	65	130
C	3 , [ 0 ]	—	0	0
D	4 , [ 7 ]	1, 3, 6, 7, 9, 13, 16	55	220
E	5 , [ 3 ]	5, 8, 20	33	165

Input answer
= 57 + 130 + 0 + 220 + 165
= 572

Self-Referential Logic Puzzle

The answer is 572.

2 solutions

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