I don't recall how to move 'stacks of cards'. We don't do that, so I'm not sure on that one...
However, we set up the row of cards as suggested in the rules. Then we start at the left-most pair of cards. If one of the cards at the TOP of one of the 9 fans equals the sum of those two cards, we move it to that stack. Then We move to the next stack and do the same thing.
If there isn't a card on the top of the fans that equals the two cards in the particular stack we're on, we skip it, and move to the next one. If we can't make a play at all, we shuffle all the cards in the fans and rearrange them (but you're only allowed one shuffle per game). Keep going until the fans are gone!
Hope that helps.