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I realized that if there were 9 players, that would leave the deck with 7 cards after first deal, and then it wouldn't work to swap out cards for players without difficulty.

Is there a term for this situation, like "the Cincinnati overdraw" or something?
It seems stupid to even start a game like that, so I'm sure it never got as far as 9 players, but is there a classic point where the line was drawn, and if so, did it have a cute term to refer to it?

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Interesting question, I never thought about this situation. It depends on what type of game is being played. In NLHE (no limit hold-em) and PLO(pot limit omaha) this is impossible with 10 players at the table. For other types of games like draw and stud based games, this is entirely possible however unlikely. Stud games have a maximum table size of 8 (i believe) making it less likely that the dealer runs out of cards. I found an interesting article that explains what happens when the dealer runs out of cards in different types of games:

https://learn.pokernews.com/news/2013/12/a-look-at-house-rules-running-out-of-cards-2643.htm

As far as the term used for running out of cards, I am not aware of one that exists. Most games have limits to maximum table size to prevent the game from running out of cards (or make it extremely unlikely). Players tend to gravitate towards games with smaller table sizes anyways because the action is faster and the game plays looser.

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