I recommend you come here sporting at least 500000 gil. Go to the rightmost house and convert all gil into coins.
10000 coins for AF??? SERENDIPITY FRAGMENT - Chaos Crystal (02/05) + 500CP !
9800 coins for AF??? SERENDIPITY FRAGMENT - Priceless Gift (03/05) + 800CP !
10000 coins for AF??? SERENDIPITY FRAGMENT - Setzer’s Dice (04/05) + 1500CP !
You also get 1 gil, just to spite you. Now save your game. For the finale enter the slot machine, tape down the L1 button and go watch a movie or something. In order to get the fragment, your combined winnings must exceed 7777 coins (you can quit and come back later, and losses are NOT taken into account). Once that happens, you score AF??? SERENDIPITY FRAGMENT - Lucky Coin (05/05) + 1000CP !