First, the instructions are massively useful in completing this test. Keep this in mind as we progress.

Second, this is just one possible walkthrough, and it’s the second time I have gone through the test. It was much harder the first time.

Easy Ones:

It’s my sincerest wish that identity will hold in this exam; the answer to #5 is E.

#20 gets special attention in the instructions, and by consulting reference material, we establish that answer is E

Warming Up:

Questions #10 and #16 refer to each other. The only correct set of answers possible means #10 is A and #16 is D.

Questions #6 and #17 refer to each other. When one considers that the answer to #17 cannot be D because of question #2, the only correct set of answers; #6 is D, and #17 is B.

The instructions state that the solution to the quiz is unique. I am not afraid to use shortcuts, so let’s do some analysis on #12. Uniqueness would imply that neither C,D, nor E could be the answer, as that would necessitate A or B also being a possible answer. Thus, not only can we eliminate C,D, and E as choices, we also establish that the number of consonants cannot be in the following:

  • 1, 4, 9, 16 (perfect squares)
  • 2, 3, 5, 7, 11, 13, 17, 19 (primes)
  • 10, 15, 20 (divisible by 5)

This means that the number of consonant answers can be either 6, 8, 12, 14, or 18. These are all even numbers, so the answer to #12 is A.

#15 directly references this answer, so #15 is also A. #15 is now established as the only odd-numbered question with the answer of A, which means the answer to #13 is D.

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

Let Me See…

#1 cannot possibly be A or B and remain internally consistent. There are already two questions answered E so #1 cannot be C. The answer to #5 is E, eliminating E as a possibility. The answer to #1, therefore, is D. This means the answer to #4 is B.

#4 states that there are 5 A’s, and we already have two E’s. This means that the only possible answer to #8 is E. There are eight vowels and twelve consonants. The answer to #3 is D, as 8 vowels – 5 A’s = 3 E’s. We have eight vowels and twelve consonants.

My Head Is Starting To Hurt A Little:

#9 is not A because of the answer to #13 indicates that #15 is the only odd-numbered question with the answer A. It cannot be C because the answer to #12 is not C. It cannot be E because we already have 3 E’s per #3. Suppose the answer to #9 was B. The answer to #11 would then necessarily be B—but that would be a contridiction, since the answers to #4 and #9 would be B. Hence the answer to #9 is D.

The answer to #11 cannot be A since #4 is B. Hence the answer to #2 must be A as the other answers have been established. This means the answer to #7 is D. We can now count to see that the answer to #11 is B.

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

Almost There:

Let’s do a little letter counting. A: 4 B: 3 C: 0 D: 7 E: 3

#18 cannot possibly be B, C, or D because of the letter counts and number of questions left. It cannot be E because of #3. #18 must be A, satisfying the A requirement of 5.

#14 and #19 must therefore both be B to make 5 answers B. We see that the count for letter D in question #14 is correct.

Give yourself a pat on the back for asking the indomitable #19! Note also that the answer spells DAD BEDDED A BAD BAD BABE, which the author recognizes as a Fatal Attraction reference. What a great test!