I finally found time to work on the 8 questions puzzle.
I see in your December 11 update that others have solved
it, but here's my solution anyway: C-A-D-C-D-D-B-A, for
a score of 7.
The analysis actually wasn't too bad, once I managed to
drag myself out of a morass of bad paths. [:-)] First, I
tried finding a solution with all 8 correct. I use "1A"
to indicate "Q1 has answer A" and "1!A" to indicate "Q1
cannot have answer A".
2B and 2C are immediately self-contradictory.
2D => 1!D (else Q6 is false),
1!C (else 2D is false),
1!A (since 1A => 2A)
hence 1B
1B => 3B => 6A, but 6A/2D/3B is a contradiction.
So 2!D, therefore 2A.
2A => 1C => 4C and 3!C.
4C => 3 answers are D, so some answer occurs at most once.
So the answer to Q7 cannot appear anywhere else. Thus 7B,
and no other answers are B.
Now if 3A, then 6A, which doesn't leave enough answers to
make up the 3 Ds required by 4C. Thus 3D, hence 8A. But
if all the questions are correct, then 8A is false. So
there is no solution with all 8 correct.
Now, suppose there is a solution with 7 correct. Either 8B,
or 8 is the only wrong answer. Let's try the latter path.
Since it implies 1-7 are correct, all of the logic in the
8-correct analysis still holds, up to the point where 3D
implies 8A. We have: 1C 2A 3D 4C 5!B 6!B 7B 8A. By 4C we
need two more Ds, so 5D and 6D. This turns out to be a
consistent solution with Q8 being the only wrong answer.
-- Don Woods.
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Hi Ed,
I scored 7 in the test after arguing that 7 is the maximum one can score.
Here's my answer,
Q: 1 2 3 4 5 6 7 8
A: C A D C D D B A
Why 8 is not possible ?
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Consider Q2 which can have only two correct answers, A or D.
If its A then it forces the following answers
Q: 1 2 3 4 5 6 7 8
A: C A C D D D
and now none of the four answers for Q3 will be consistent with the
others!
If the answer for Q2 is D then
Q: 1 2 3 4 5 6 7 8
A: D C D
This leaves us with B as the only choice for Q1,
Q: 1 2 3 4 5 6 7 8
A: B D B C A D
But this cannot give a score of 8 since the answer for Q6 is wrong!
Thus a score of 8 is not possible. QED
An interesting problem it was! I thoroughly enjoyed solving it.
Thanks for this wonderful site.
Ram Prasath
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http://freezope1.nipltd.net/adept/Haskell has a program by
Dmitry Astapov.
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joykeller: CADCDDBA has 7 (the first 7) correct answers.
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1) c
2) a
3) d
4) c
5) d
6) d
7) b
8) a
7 answers correct (8 is the one that is wrong)
Colin Backhurst
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The unique highest scoring answer set is
C A D C D D B A
for a score of 7.
Thanks!
Daniel Reeves
PS: I used brute force -- here is my Mathematica code:
nthPos[l_,x_,n_:1]:= Module[{p=Position[l,x]},
If[Length[p]= bestScore,
bestAns= ans;
bestScore= s;
Print[i,": ",ans," -> ",s]]]
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I am quite certain that there is a unique highest score of 7
since I scored all possible combinations!
The number of combos of 8 questions with 4 answers each is
4 ^ 8 = 2 ^ 16 = 65536 which happens to be the maximum number
of rows in an MS-Excel spreadsheet (heh-heh)!
Bob Kraus
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