Please Solve

Question

Three siblings, Alice, Bob and Caro truthfully reported their grades to their parents as follows:

 

Alice: If I passed math, then so did Bob

I passed English if and only if Carol did

 

Bob: If I passed math, then so did Alice

Alice did not pass history

 

Carol: Either Alice passed history or I did not pass it.

 

If Bob did not pass English, then neither did Alice. If each of the three passed at least one subject and each subject was passed by at least one of the three, and if Carol did not pass the same number of subjects as either of her siblings, which subjects did they each pass

Answer 1

Three subjects, three students. Each student passed at least one subject, and each subject was passed at least once.

 

Let's look at history first:

Bob states that (Alice did not pass).

Carol states that either (Alice passed) or (Carol did not pass)

Since we know (Alice did not pass), then (Carol did not pass).

Since neither Alice or Carol passed history, (Bob passed).

 

English:

Alice states that (Alice passed) if and only if (Carol passed). So either they both pass, or they both fail.

The last paragraph states that if (Bob did not pass), then (Alice did not pass.) This would then imply from the if and only if that (Carol did not pass.) However, this would mean that none of them passed English, which cannot be true.

Therefore (Bob did not pass) cannot be true, and it must be true that (Bob passed).

We also know that Carol passed a different number of classes as either sibling.

If both Alice and Carol do not pass English, then they must both pass Math. This would mean that Alice and Carol each passed one class, which contradicts Carol's condition. Therefore, (Alice passed) and (Carol passed) English.

 

Math:

Alice and Bob's first statements imply that they either both pass, or both don't pass.

Because of Carol's condition, we know that both Alice and Carol cannot both pass Math, but at least one of them must, since they are both passing one class so far.

-If (Carol passes) and (Alice does not pass), then (Bob does not pass.) This would mean that Bob and Carol both passed two classes, which contradicts Carol's condition. Therefore, (Carol did not pass) Math.

This implies that (Alice passed) and (Bob passed) Math.