One game of Rummy followed from the deal to the finish, move by move, with the reason for each choice.
Three players — you, Ada and Bo — so everyone gets seven cards. Bo deals, and you, on Bo's left, go first. Your hand: ♥ 4 5 6, ♣ 9, ♦ 9, K♠ and 2♣. 9♠ is turned up to start the discard pile.
You already hold a run, the four, five and six of hearts. Everything else counts against you if somebody goes out: 9, 9, 10 and 2, for 30.
9♠ on the pile makes a set with your two nines, so you take it instead of drawing from the stock. Then you throw K♠: it is worth 10 and has nothing near it.
You now hold a run and a set, with only 2♣ left over. You could lay both melds down now. Instead you keep them in your hand: if you can go out later in one turn, having laid nothing down, the hand counts double.
Ada holds K♥ and K♦, so she takes your K♠ from the pile and lays the three kings down. She throws Q♣, her highest loose card, and is left with four cards: 3♦, 8♦, A♣ and 5♠.
Q♣ is no use to Bo, who draws 3♣ from the stock. He lays down a run of spades, 10♠ J♠ Q♠, and throws 8♥. He keeps 7♦, 6♣, 2♦ and 3♣. Neither of them knows you are collecting hearts.
8♥ is one card too far from your run, so you draw from the stock: 7♥. That is the card you were waiting for. You lay down the run ♥ 4 5 6 7 and the set of nines, and throw 2♣ as your last card.
Your whole hand went down in one turn, and nothing of yours was on the table before it. That is rummy.
Ada is caught holding 3♦, 8♦, A♣ and 5♠: 3 + 8 + 1 + 5 = 17. Bo is caught holding 7♦, 6♣, 2♦ and 3♣: 7 + 6 + 2 + 3 = 18. That is 35 between them, and going rummy doubles it, so you score 70. Ada and Bo score nothing.
Had you laid your melds down on the first turn and gone out now, you would have scored 35. Had Ada gone out first, though, you would have been caught with all seven cards, 44 points, and scored nothing.
Holding back was a gamble that paid, because your hand was one card from finished and your leftover card was cheap. Ada's kings show the other side: she laid them down at once, and they could never be counted against her.
The next hand is dealt by you, and Ada starts. Hands are played until somebody reaches 100 at the end of one, and the highest score wins.