When Algorithm 1 processes each card in turn

What on earth are you talking about

I am a little reluctant to give up this game

Rummy is a classic cardgame where the objective is to be the first to get rid of all your cards, by creating melds, which can either be sets, three or four cards of the same rank, e.g. ♥8♠8♦8, or runs, which are three or more cards of the same suit in a sequence, e.g. ♥A♥2♥3. Aces are low, and sequences can not wrap around. There are many, many variations of Rummy that exist, this particular implementation is Basic Rummy, or Traditional Rummy.