If the two moderates did not vote the same way as each other, then which one of the following could be true?

Irina on September 10, 2019

@farnoushsalimian,

The game involves 7 judges - 2 conservatives, 2 moderates and 3 liberate. Each of them votes either no or yes on the petition.

The following rules apply:

(1) If two conservatives and at least one liberal voted the same way as each other, then both moderates voted that way.

This rule allows us to infer that:

2 C + 1+ L = Y -> 2 M = Y
2 C + 1+L = N -> 2 M = N

(2) If the three liberals voted the same way as each other, then no conservative voted that way.

3 L =Y -> 2 C =N
3 L = N -> 2 C = Y

(3) At least two judges voted for and at least two voted against.

Combined with rule (1) we can infer that if two conservatives and 1 liberal vote the same way, then the remaining two liberals must vote the opposite way

2C +1L = Y -> 2M =Y + 2L = N
2C+1L = N-> 2M = N +2L = Y

(4) At least one conservative voted against.

Combined with rules (1) and (3), we can infer that only one of these scenarios is actually possible:

2C + 1 L = N - > 2M = N + 2 L = Y

It is impossible for two conservatives to vote the same way as each other if any of them votes Y because at least one of them has to vote N.

Combined with rule (2) we can also infer, that the only way all liberals could vote the same way as each other is if they vote Y, in that scenario all conservative vote N. It is impossible for all liberals to vote N because this would require all conservatives to vote Y, and we know that at least one must vote N.

3L =Y -> 2C = N

Overall, we can diagram all the inferences together as follows:

N
C C M M L L L

2+ Y
2+ N

2C + 1 L = N - > 2M = N + 2 L = Y
3L =Y -> 2C = N


The question asks us if two moderates did not vote the same way as each other which of the following could be true. Well, there are only two ways a judge can vote - Y or N. Since they did not vote the same way as each other we know that 1 voted Y and 1 voted N.

We also know that at least 1 C judge must vote N per rule 4

N Y N
C C M M L L L

From here we have a couple of options, we need at least one more judge to vote Y to satisfy rule (3). Let's say C judge votes Y, then L judges could either vote two Y and 1 N or 2 N and 1 Y. Liberal judges can never vote 3N because this would require conservative judges to vote 2N. And liberal judges cannot vote 3Y in this scenario because we assumed that one of the conservative judges voted Y.

Y N Y N Y Y N
C C M M L L L

or

Y N Y N Y N N
C C M M L L L


Another scenario would be to have both conservative judges vote N, if this is the case, all the liberal judges must vote Y, otherwise, per rule (1), both moderates must vote the same as each other and this is not the case in this question.

N N Y N Y Y Y
C C M M L L L

These are all the possible scenarios for this question. Let's look at the answer choices:

(A) No conservative and exactly 2 L voted Y.

Impossible. If no conservative voted Y, then all L have to vote Y.

(B) Exactly one conservative and exactly 1 liberal voted Y.

Correct. This is scenario 2 above, it is possible for 1 C and 1 L to vote Y and the rest of L and C judges vote no.

(C) Exactly one conservative and all three liberals voted Y.

Impossible. If all three L vote Y, then all C must vote N per rule (2).

(D) Exactly two conservatives and one liberal voted Y.

Impossible. At least one conservative must vote N per rule (4).

(E) Exactly two conservatives and exactly two liberals voted for.

Impossible. At least one conservative must vote N per rule (4).

Let me know if this makes sense and if you have any further questions.

farnoushsalimian on September 11, 2019

Yes. Thank you.