There is exactly one solution to the following [ Editor: self-referential multiple-choice ] test.
logic/self.ref.test New
New entry, edited from material collected for the archive and not yet in a released version.
Problem
There is exactly one solution to the following [ Editor: self-referential
multiple-choice ] test. Can you find it?
Note: To avoid confusion, the word 'answer' refers to the index of the
answer, not the text. E.g., [ Editor: was "e.g. If" in the original ]
if the answer to question 3 were [ Editor: was "was" in the original ]
the first item 'a) d', the answer is called 'a', not 'd'.
1. The first question with answer 'c' is:
a) 1
b) 2
c) 3
d) 4
e) 5
2. The answer to question 1 is the same as the answer to question:
a) 12
b) 14
c) 16
d) 18
e) 20
3. The answer 'a' appears with the same frequency as the answer:
a) d
b) e
c) b
d) c
e) none of the above
4. The most frequent answer in this test is:
a) a
b) b
c) c
d) d
e) e
5. The number of questions before this one answered with a vowel is:
a) 0
b) 1
c) 2
d) 3
e) 4
6. The number of questions with answer 'd' is:
a) 2
b) 3
c) 4
d) 5
e) 6
7. The only even numbered question with answer b is:
a) 6
b) 8
c) 10
d) 12
e) 14
8. The answer to question 10 is:
a) b
b) e
c) d
d) c
e) a
9. The highest numbered question with answer 'a' is:
a) 12
b) 13
c) 14
d) 15
e) 16
10. The answer to question 12 is:
a) c
b) d
c) e
d) a
e) b
11. This is the first question with answer:
a) a
b) b
c) c
d) d
e) e
12. The answer to question 8 is:
a) c
b) a
c) d
d) e
e) b
13. The only two consecutive questions with the same answer are:
a) 3 and 4
b) 4 and 5
c) 5 and 6
d) 6 and 7
e) 7 and 8
14. The only false statement in the following list is:
a) The answers to questions 6 and 18 are the same.
b) The answer to question 14 is a vowel.
c) The answer to question 2 is a consonant.
d) The answer to question 16 is not 'a'.
e) The answer 'c' appears fewer times than the answer 'e'.
15. The next question with the same answer as this one is:
a) 17
b) 18
c) 19
d) 20
e) none of the above
16. The answer to the question whose number happens to be the same as
the number of questions answered with a consonant is:
a) c
b) e
c) d
d) a
e) b
17. The only true statement in the following list is:
a) The answer to question 4 is not a consonant.
b) The answer to question 7 is not 'a'.
c) The answer to question 13 is the same as the answer to this
question.
d) The answer to question 15 is a vowel.
e) The answer to question 19 is 'd'.
18. The sum of the numbers in the answers of the first two questions is:
a) 15
b) 17
c) 19
d) 21
e) 23
19. The only true fact about the number of questions answered with a
vowel is that it is:
a) not a prime number
b) a multiple of 4
c) an odd number
d) less than 6
e) 6
20. The answer to this question is:
a) a
b) b
c) c
d) d
e) e
Show solution
[ Editor: The test is by Trevor Janes (copyright 1996). ]
1 - e 5 - c 9 - a 13 - d 17 - b
2 - d 6 - e 10 - d 14 - b 18 - e
3 - e 7 - e 11 - b 15 - d 19 - c
4 - d 8 - c 12 - a 16 - c 20 - d
There are _way_ too many steps needed to arrive at this solution for me
to post here (and, in fact, to even mail on request) but you can check
that it satisfies all questions. I wasn't paying that much attention,
but it took me somewhere between two and three hours to find it.
-- Leonardo Blanco <Leo_B@msn.com> [ Editor: posted as zLeo_B@msn.com,
with the instruction to remove the "z" ]
[ Editor: Tim Firman and Glenn Rhoads had reported hitting a contradiction late
in their solutions; Firman: "I was getting 15 and 16 equal to the same
letter, violating question 13." ]
I also found a contradiction, and when checking the solution given
[ Editor: "given :" in the original ],
15 - d
has the problem, since 19(c) is the next answer with the same as 15(c).
But indeed, if the answer to 15 is d, then that too is correct, since
20 also has answer d.
Very good puzzle that makes you think that there is a fault with it!
-- Herman De Wael <hermandw@innet.be>
I suspect that these people fell into an intentional trap concerning
Q15. [ Editor: Here is my solution in detail. (Condensed from Janes'
introduction: "I am posting my own solution in detailed explanation
for anyone interested.") ]
Short Solution:
1e 2d 3e 4d 5c 6e 7e 8c 9a 10d 11b 12a 13d 14b 15d 16c 17b 18e 19c 20d
Long Solution:
Note: "Qx" means "question x"
"Ax" means "answer to question x"
"(x)" means "step x of the solution"
(1) From Q1: A1 cannot be 'a' or 'c'.
(2a) From Q7: A4 cannot be 'b',
(2b) From Q7: A18 cannot be 'b',
(3) From Q10+Q12: A8 cannot be 'a', 'b', 'd' or 'e'.
(4) From (3): A8 is 'c'.
(5) From (4): A10 is 'd'.
(6) From (5): A12 is 'a'.
(7) From A8+A10: A11 cannot be 'c' or 'd'.
(8) From Q13+A8+A10: A9 cannot be 'c' or 'd',
(9a) From Q13+A12: A11 cannot be 'a',
(9b) From Q13+A12: A13 cannot be 'a'.
(10) From Q18: A1 cannot be 'b' or 'd'.
(11) From (1)+(10): A1 is 'e'.
(12a)From (11): A5 is 'c'.
(12b)From (11): A11 cannot be 'e'.
(13) From (7)+(9a)+(12b): A11 is 'b'.
(14) From A8+A10+A12: A7 cannot be 'b', 'c', or 'd'.
(15) From A8+(14): A13 cannot be 'e'.
(16) From (13): A6 cannot be 'b',
(17) From (16): A7 cannot be 'a'.
(18) From (14)+(17): A7 is 'e'.
(19) From (18): A14 is 'b'.
(20) From A14 [ Editor: was "(A14)" in the original ]: A6 = A18.
(21) From A14 [ Editor: was "(A14)" in the original ]: A4 cannot be 'c'.
(22) From (21) [ Editor: was "(22)" in the original ]+A5: A13 cannot be 'b'.
(23) From A1+A14: A2 cannot be 'b'.
(24) From A1+(23): A18 cannot be 'c'.
(25) From (20)+(24): A6 cannot be 'c'.
(26) From A5+(25): A13 cannot be 'c'.
(27) From (9b)+(15) [ Editor: was "(a5)" in the original ]+(22)+(26):
A13 is 'd'.
(28) From A7+A13: A6 is 'e'.
(29) From (20)+A6: A18 is 'e'.
(30) From A1+A18: A2 is 'd'.
(31) From A7: A17 is 'b'.
(32) From A17: A4 cannot be 'a' or 'e'.
(33) From (2a)+(21)+(32): A4 is 'd'.
(34) From Q5+A1+A2+A4: A3 cannot be 'b' or 'c' or 'd'.
(35) From A4: A3 cannot be 'a'.
(36) From (34)+(35): A3 is 'e'.
(37) From A13: A9 cannot be 'b'.
(38) From A14: A16 cannot be 'a'.
(39) From (38): A9 cannot be 'e'.
(40) From (8)+(37)+(39): A9 is 'a'.
Note: Solving the four remaining questions (15, 16, 19 and 20)
requires more complex logic than has been used up until now.
Some people might prefer to get the last part with 'brute force'
(i.e. try all combinations), but real fans of logic puzzles will
want to solve the entire thing by deduction. So here goes...
(41) From counting the answers so far, 9 are consonants.
(42) From counting the answers so far, 7 are vowels.
(43) From counting the answers so far, 4 are 'd'.
(44) From Q19: A19 cannot be 'a' or 'e'.
(45) From [ Editor: ">From" in the original ] (42): A19 cannot be 'd'.
(46) From A6+(43): there are exactly 2 more 'd' answers.
(47) From (41)+(45)+(46) there are either 12 or 13 consonant answers.
(48) From (47): there are either 7 or 8 vowel answers.
(49) From Q19+(48): A19 cannot be 'b'.
(50) From (44)+(45)+(49): A19 is 'c'.
(Note: At this point, many people conclude that A15 must also be 'c',
but that is a trap for the unwary!)
(51) From A19: there are 7 vowel answers.
(52) From (51): there are 13 consonant answers.
(53) From (52)+A13: A16 is 'c'.
(54) From A16+Q13: A15 cannot be 'c'.
(55) From A14+Q13: A15 cannot be 'b'.
(56) From A17: A15 cannot be 'a' or 'e'.
(57) From (54)+(55)+(56): A15 is 'd'.
(58) From A15: A20 is 'd'.
As far as I know, this is the only solution for which all the questions
remain true.
-- Trevor Janes <tjanes@newbridge.com>