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The number of possible decks is incomprehensibly large. See this question for context.

However, the number of games that can be played is incomprehensibly larger—for every deck you have, there are about 60! ways to draw the cards in order (depending on the size of your deck). This doesn’t even take into account the way that you could die before drawing all of those cards, or the number of actions you can take during each round (or not take).

Given that every single card presents new possibilities to the set of possibilities from previous cards, is it even practical to try to calculate this?

I am here defining a game as a discrete set of actions that could be taken by two players while properly following the rules of Magic: The Gathering, and the sequence of collections of cards included in each players’ respective draw piles, graveyards, hands, and exile piles over time (for those familiar with the terminology). (Technically an unlimited number of players could play Magic: The Gathering against each other in a single game, but to avoid the trivial case of an infinite number of games being played, I’m restricting the rules to the typical case of two.)

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  • $\begingroup$ I'll point out that this question isn't yet a mathematical question, because the complete set of possibilities hasn't been described with mathematical precision. I suspect that rough estimates is the best one will be able to get, given the complicated nature of the Magic rules. One note at least: $60! \approx 10^{82}$ is barely a measurement error compared to the total number of decks.... $\endgroup$ Nov 26, 2022 at 18:25
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    $\begingroup$ Since the game is Turing-complete, you can in principle encode any TM as a state of an MtG game. Since there’s no upper limit to the size of a deck I think this means the number of games is infinite? $\endgroup$ Nov 26, 2022 at 18:30
  • $\begingroup$ @GregMartin Regarding 60!, I guess this is only in any way significant because the 60! possibilities apply to each and every possible deck, and they’re just the draw order (not even taking actions into account). $\endgroup$ Nov 26, 2022 at 18:38
  • $\begingroup$ @templatetypedef Perhaps. However, since this case is trivial in how large it is, I’d prefer to exclude it. Maybe just the number of non-Turing machine, 60-card-deck two-player games. $\endgroup$ Nov 26, 2022 at 18:55

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Uncountably infinite

  1. Both players each play 6 lands and a Leashling.
  2. For any given real number between 0 and 1, consider its binary representation.
  3. During each player's turn, that player returns their Leashling to their hand, returns one of two cards in their hand to the top of their library depending on the next binary digit, and then re-plays the Leashling. They then draw back that card at the beginning of their next turn.
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    $\begingroup$ More precisely, there are $\beth_1$ games: you've described that many, but it's also an upper bound as each game consists of countably many decisions, each with countably many options. (This cardinal may also be written as $c$, $\mathfrak{c}$ or $2^{\aleph_0}$.) Similar reasoning tells us there are $\aleph_0$ games of finite duration. $\endgroup$
    – J.G.
    Nov 27, 2022 at 10:24
  • $\begingroup$ Throughout all the different answers to this question (mostly in the comments), I realized that most of the answers lie, not in considering the different sets of events that could result from a given card, but whether a given card can cause looping behavior. Because there’s so much potential for looping behavior, the answer being that there are an infinite number of possible games that could be played comes up again and again. I wonder if, in conjunction with this card behavior, the Turing Machine capability of Magic: the Gathering produces different types of infinities for game possibilities. $\endgroup$ Nov 29, 2022 at 5:42

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