Multiple choice(johndcook.com)
johndcook.com
Multiple choice
http://www.johndcook.com/blog/2015/07/06/multiple-choice/comment-page-1/
24 comments
Yup, I agree! After figuring it out by hand I was having fun encoding it for sat via this horrible pipeline:
... find replace, find replace, ...
[gamma]
[epsilon]
edit: I think I am someone who interprets "one of the above" as "at least one of the above but possibly more" not "exactly one of the above". not that the interpretation makes a difference for this problem.
step 1 : write it out by hand in PBL [alpha]
step 2 : convert it to CNF format using [beta]
step 3 : hand-reformat it tediously to produce [gamma]
step 4 : plug machine readable CNF into [delta]
step 5 : receive a feasible solution! [epsilon]
Main_Exp : ((a <=> (b & c & d & e & f)) & (b <=> (~c & ~d & ~e & ~f)) & (c <=> (a & b)) & (d <=> (a | b | c)) & (e <=> (~a & ~b & ~c & ~d)) & (f <=> (~a & ~b & ~c & ~d & ~e)))
... find replace, find replace, ...
[gamma]
c multiple choice
p cnf 6 29
-1 2 0
-1 3 0
-1 4 0
-1 5 0
-1 6 0
-2 -3 -4 -5 -6 1 0
-2 -3 0
-2 -4 0
-2 -5 0
-2 -6 0
3 4 5 6 2 0
-3 1 0
-3 2 0
-1 -2 3 0
-4 1 2 3 0
-1 4 0
-2 4 0
-3 4 0
-5 -1 0
-5 -2 0
-5 -3 0
-5 -4 0
1 2 3 4 5 0
-6 -1 0
-6 -2 0
-6 -3 0
-6 -4 0
-6 -5 0
1 2 3 4 5 6 0
[epsilon]
This is MiniSat 2.0 beta
============================[ Problem Statistics ]=============================
| |
| Number of variables: 6 |
| Number of clauses: 29 |
| Parsing time: 0.00 s |
============================[ Search Statistics ]==============================
| Conflicts | ORIGINAL | LEARNT | Progress |
| | Vars Clauses Literals | Limit Clauses Lit/Cl | |
===============================================================================
| 0 | 6 29 75 | 9 0 nan | 0.000 % |
===============================================================================
Verified 29 original clauses.
restarts : 1
conflicts : 2 (2 /sec)
decisions : 5 (0.00 % random) (5 /sec)
propagations : 10 (10 /sec)
conflict literals : 5 (28.57 % deleted)
CPU time : 1 s
SATISFIABLE
v -1 -2 -3 -4 5 -6 0
edit: I think I am someone who interprets "one of the above" as "at least one of the above but possibly more" not "exactly one of the above". not that the interpretation makes a difference for this problem.
Simple solution:
* A and B can't both be true, which means that C is false.
* If C is false, A must be false.
* B and D have a paradoxical relationship, given the others - if B were true, D would be true, which would make B false, so B must be false.
* D is false, given A, B, C are false.
* E is true, given A, B, C, D are false.
* F is false because E is true (typo corrected as per comment)
* A and B can't both be true, which means that C is false.
* If C is false, A must be false.
* B and D have a paradoxical relationship, given the others - if B were true, D would be true, which would make B false, so B must be false.
* D is false, given A, B, C are false.
* E is true, given A, B, C, D are false.
* F is false because E is true (typo corrected as per comment)
F is false because D is true.
F is false because E is true.
F is false because E is true.
I just went down one by one and ruled them out, I'm surprised that people have harder ways to do it. My reasoning for ruling out each one:
a) b contradicts it.
b) cde encompass all the choices, so you can't have none of those.
c) a contradicts b.
d) abc have already been rejected.
e) works.
f) e already works.
a) b contradicts it.
b) cde encompass all the choices, so you can't have none of those.
c) a contradicts b.
d) abc have already been rejected.
e) works.
f) e already works.
Absolutely - thanks for sharing; you were far more concise than how I was going to explain my solution. :)
E A very interesting mental brute force process to work it out though. How did other people work it out, is there a more elegant method?
[deleted]
Figured I'd dust off my old constraint programming language (https://github.com/dbunker/SABR/blob/master/test/Real/Multip...)
with result (https://github.com/dbunker/SABR/blob/master/test/Real/Multip...). Fifth position being E.
A bit on the verbose side, but might be preferable to some since it explicitly enumerates all the possibilities for each rule.
with result (https://github.com/dbunker/SABR/blob/master/test/Real/Multip...). Fifth position being E.
A bit on the verbose side, but might be preferable to some since it explicitly enumerates all the possibilities for each rule.
That's a fun quick brainteaser! Reminds me of those old puzzles where you'd have a list of facts -- like there are 7 houses in a row, all different colors and with different species of pets in the yard, the blue house is next to the orange house, the green house has a pet with 2 legs, etc. etc., and they provide just enough info that you can logically chain together the order of colors and pets.
They're a bit like the word-problem version of working out sudoku (vs. guessing at it, which is a different game).
They're a bit like the word-problem version of working out sudoku (vs. guessing at it, which is a different game).
Either e or f, I can't think of a reason why it can't be f...
Because e == f, and if f is true then e is true, in which case f is false, hence a contradiction, hence it cannot be f.
>Because e == f, and if f is true then e is true, in which case f is false
e != f, because they produce different results. You could say e() == f(), each having the previous answer choices as arguments and each requiring them to be false. But since e() evaluates to true, f() can't be true.
e != f, because they produce different results. You could say e() == f(), each having the previous answer choices as arguments and each requiring them to be false. But since e() evaluates to true, f() can't be true.
A better way to phrase it would be that, f implies e, but e doesn't imply f. In fact e implies 'not f'.
Aaaaah, I understand now. Thanks :3
The answer is E.
For what it's worth, the answer is E. :)