6. Five cards have the numbers $101,102,103,104$ and 105 on their fronts.
105
On the reverse, each card has a statement printed as follows:
101: The statement on card 102 is false
102: Exactly two of these cards have true statements
103: Four of these cards have false statements
104: The statement on card 101 is false
105: The statements on cards 102 and 104 are both false
What is the total of the numbers shown on the front of the cards with TRUE statements?
SolUtion
206

If card 101 is true then both card 102 is false and card 104 is false. So card 105 is true and in turn card 103 is false. This means we have exactly two true cards but that simultaneously card 102 is false. This is a contradiction.
So card 101 must be false. This means both card 104 is true and card 102 is true. This means card 105 is false and card 103 is false. This is two true cards.
The sum of the numbers on the two true cards is $102+104=206$.