Associate each letter with the correct number, given that:
- If A is 1, then B is not 3.
- If B is not 1, then D is 4.
- If B is 1, then C is 4.
- If C is 3, then D is not 2.
- If C is not 2, then D is 2.
- If D is 3, then A is not 4.
Lets number the statements.
1) If A is 1, then B is not 3.
2) If B is not 1, then D is 4.
3) If B is 1, then C is 4.
4) If C is 3, then D is not 2.
5) If C is not 2, then D is 2.
6) If D is 3, then A is not 4.
From statement 5, at least one of C or D is 2
From statements 2 and 3, one but not both of C or D is 4
Therefore, one of them is 2 and the other is 4.
(So statement 4 and 6 are extraneous, because their predicates are false)
I think the problem statement intends a 1 to 1 correspondence between the letters and numbers, which means that A and B in some order must be 1 and 3. But if A = 1, then statement 1 is a contradiction, so A = 3 and B = 1
And then, from statement 3, C = 4 so D must equal 2.
Only answer, if all numbers are accounted for, is [A,B,C,D] = [3,1,4,2]. I submit this as the solution.
But, since Jer brings it up, there are 15 solutions if numbers are allowed to duplicate. Basically, [A,B] can take any value except [1,3] and C and D depend directly on B. The 15 are:
A B C D
1 1 4 2
2 1 4 2
3 1 4 2 <== no duplicates
4 1 4 2
1 2 2 4
2 2 2 4
3 2 2 4
4 2 2 4
2 3 2 4
3 3 2 4
4 3 2 4
1 4 2 4
2 4 2 4
3 4 2 4
4 4 2 4
Edited on January 3, 2013, 3:42 pm
Full analytical solution (spoiler)
