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**