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Multiple Choice Test (Posted on 2003-06-07) Difficulty: 4 of 5
Select the correct answer to each question.

1. The correct answer to Question 2 is
A. B
B. C
C. D
D. A

2. The correct answer to Question 4 is
A. D
B. C
C. A
D. B

3. The first question for which C is the correct answer is
A. 4
B. 2
C. 3
D. 1

4. The number of questions for which B is the correct answer is
A. 1
B. 2
C. 3
D. 4

5. The number of even-numbered questions for which either B or C is the correct answer is
A. 1
B. 3
C. 0
D. 2

6. The last question for which A is the correct answer is
A. 6
B. 7
C. 5
D. 8

7. The number of instances where D is the correct answer for two consecutive questions is
A. 2
B. 0
C. 1
D. 3

8. The only letter which is the correct answer to exactly three questions is
A. C
B. D
C. B
D. A

See The Solution Submitted by Bryan    
Rating: 4.4091 (22 votes)

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Solution Solution | Comment 9 of 19 |
C D D B D A C B

Finishing what I started earlier, the answer to 1 cannot be 'A' since we end up with more than one answer with exactly three choices, defying problem 8.

First, something I failed to do earlier, I will make a few observations. One of the first four answers must be a 'C' (from clue 3). Similarly, one of questions 5-8 must be answered with an 'A' (6). There is only one letter which is the correct answer to exactly three questions.

Assume, then, that the answer to 1 is 'B.'
B _ _ _ _ _ _ _
From this, the answer to 2 is 'C.'
B C _ _ _ _ _ _
Given that, number two is the first answer of 'C' and number 3 is 'B.'
B C B _ _ _ _ _
Also, from number two, four must be 'A.'
B C B A _ _ _ _
That means that only one question has an answer of 'B.' Since we already have two Bs, this entire train of thought is disproved.
B C B A _ _ _ _

Suppose, then, that the answer to #1 is 'C.'
C _ _ _ _ _ _ _
Obviously, then, it is the first 'C,' and number three is 'D.'
C _ D _ _ _ _ _
Also, number two must be 'D.'
C D D _ _ _ _ _
From that, number 4 is 'B.'
C D D B _ _ _ _
That means that there are two questions with 'B' as their answer, one of which is this question, and one of which is in the last four questions.

Suppose, first, that the fifth answer is 'B.'
C D D B B _ _ _
That means that three of the four even-numbered questions are answered with either 'B' or 'C'. Number two is not, and four is, so numbers six and eight must both be either 'B' or 'C.'
However, since we have that there are only two Bs altogether, which are already placed, six and eight must both be 'C.'
C D D B B C _ C
An answer of 'C' to question six means that the answer to #5 is 'A,' but that has already been penned as 'B.' So, the last assumption must be false.
C D D B B C _ C

Suppose, then, that number six is 'B.'
C D D B _ B _ _
That answer stipulates that number seven is 'A.'
C D D B _ B A _
An answer of 'A' there says that there are two instances of consecutive 'D' answers. There is already one, but with the currently assumed answers, there is no way to fit in another match.
So, the last assumption, again, must be false.
C D D B _ B A _

Suppose that the answer to number seven is 'B.'
C D D B _ _ B _
That says that nowhere in the problem are two consecutive answers of 'D,' but we already have one instance.
Therefore, this cannot be the case.
C D D B _ _ B _

Finally, for the previous assumption to be valid, the answer to number eight must be 'B.'
C D D B _ _ _ B
That answer says that there are exactly three Ds, and not three of any other answer in the set of problems. Two have been placed already as answers to numbers two and three, and the final 'D' must answer one of the fifth, sixth, or seventh questions.
Further, we know (given these assumptions) that there is exactly one more 'D,' and therefore no way to make another pair of consecutive Ds other than the pair already made.
So, the answer to number seven must be 'C.'
C D D B _ _ C B

Now, there are two answers left, one of which is 'A' (from the general assumptions based on question 6) and the other which is 'D' (from the current answer to question eight.
Looking at number six, we already have answered questions seven and eight. So, the 'A' must be number five or six, and the answer to question six is 'A' or 'C.'
However, we know that it must be one of 'A' or 'D' (for reasons stated above).
Therefore, it must be 'A.'
C D D B _ A C B
Finally, the third 'D' must answer question five.
C D D B D A C B
That answer implies that two of the four even-numbered questions in the problem are answered with either 'B' or 'C.'
Inspecint these answers, that is indeed the case (they are D, B, A, B).
Double checking the other answers with their questions:
The answer to #2 is 'D' (1)
The answer to #4 is 'B' (2)
The first answer of 'C' is to question #1 (3)
'B' correctly answers exactly two questions (4)
There are 2 even-numbered answers of 'B' or 'C' (5)
The last (and only) answer of 'A' in the solution set is to question six (6)
Two consecutive Ds appear only once in the solution set (7)
There are exactly three Ds and not three of any other respose letter (8)

Everything checked so [a] valid solution is
C D D B D A C B
11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111S
Lastly, to perform a thorouth check, what happens if we assume number one to be 'D'?
D _ _ _ _ _ _ _
If that is the case, number two is 'A.'
D A _ _ _ _ _ _
From that, number four must be 'D.'
D A _ D _ _ _
Also, since (as stated in the introduction) no Cs have been placed, but one of the first four must be a 'C,' which in this case must he answer to number three.
D A C D _ _ _ _

From the answer of 'D' to number 4, there are four Bs in the problem. None have been placed yet in the fire time,
Therefore, the remaining four blanks must all be Bs.
D A C D B B B B
This last assumption is quite easily disproven by any of the latter four questions.
D A C D B B B B

So, the only valid possibility for the answers is as follows:

  1. C

  2. D

  3. D

  4. B






  Posted by DJ on 2003-06-08 00:47:42
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