Proving Rules on Cards

You’ll have to flip over the cards “A” and “3”.

Your original intuition might be to flip the cards “A” and “6”, but this isn’t the case. Let’s consider the following scenarios:

Letter	Number	Statement valid?
Consonant	Even	Could be
Consonant	Odd	Could be
Vowel	Even	Could be
Vowel	Odd	No

If we see a card with a consonant on one side, knowing what’s on the other side doesn’t help us see if the statement is true or false. This means we can leave the “D” card alone.

If we encounter a card with a vowel on one side, we should check if the number on the other side is even. If it is odd, we know that the statement isn’t true. However, if it is odd, we know that the statement could be true, we just need to check the other cards. So we’ll have to flip over the “A” card.

If we see a card with an even number on one side, notice from the table that it doesn’t give us additional information whether or not the statement is invalid. After all, the statement doesn’t say that a card with a consonant must have an odd number on the other side. We can actually leave the “6” card alone.

If we see a card with an odd number on one side, notice from the table that if we see a vowel, we know that the statement is wrong. We’ll need to flip the “3” card over too.

In summary, flipping the cards “A” and “3” will tell us if the statement is true.