A game of Magic: The Gathering begins well before players lay down their first card. As a collectible card game, Magic requires competitive players to select the optimal deck of cards based on how they think it will function against hypothetical opponents with many different strategies—then the game itself offers proof or disproof of the player’s predictive powers. Because about 30,000 different cards are available today—though they’re likely not all owned by a single individual—there are many degrees of variation.
This abundance of possibilities has sparked plenty of questions and ideas. Some players have wondered how complicated the game really is. For example, does it involve enough complexity to perform calculations, as you would with a computer? To this end, software engineer Alex Churchill and two other Magic players created a game situation in which the cards act as a universal computer—as a Turing machine. They posted their work to the preprint server arXiv.org in 2019.
Their computer model sealed the deal: Magic is the most complex type of game, they concluded. Theoretically, any kind of calculation that a computer can perform, a particular Magic game can do the same. Ever since I learned this, the game has held a certain fascination for me.
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But in practice, of course, using a Magic deck for its calculating prowess is not particularly helpful. Coding such a Turing machine alone is extremely time-consuming. And who has the time to go one step further and go through the billions of different card combinations necessary to solve a math problem with Magic cards? The quicker option would be to type the problem into a computer through some elegant Python code (or another programming language).
As it turns out, people are quite willing to give their time to such “Magical” endeavors. For example, in 2024 Churchill and mathematician Howe Choong Yin developed a Magic programming language that used Magic moves to code elementary calculations such as addition, multiplication or division. Say you wanted to calculate 3 + 5. All you would need are a few cards (such as Vaevictis Asmadi, the Dire), Churchill and Howe’s instructions, and a little patience. Forget supercomputers, quantum computing and all that fancy stuff: the future of computing lies in Magic cards, right?
Probably not—even solving a division problem with Magic cards is cumbersome, and tackling more complex problems in this way proves to be nearly impossible, especially when it comes to dealing with open questions in mathematics. That hasn’t stopped others from trying, however.
Gameplay with Twin Primes
In the fall of 2024 Reddit user its-summer-somewhere posted a combination of 14 moves that use about two dozen Magic cards and could potentially deal infinite damage. The outcome of the game depends on the answer to a mathematical puzzle that is almost 180 years old: Are there an infinite number of prime number twins? Prime numbers, such as 2, 3, 5, 7, 11, and so on, are divisible only by 1 and themselves. Twin primes are pairs of prime numbers that differ by only two, such as 3 and 5, 5 and 7, 11 and 13, and 17 and 19.
Mathematicians have previously proven that there are an infinite number of primes. But their number decreases with increasing size: the further you progress in the number line, the less often prime numbers appear. This is even more true for prime number twins. The question that mathematicians have been asking themselves for centuries is: Are there also an infinite number of twin primes? Or will this parade end at some point?
In 1849 French mathematician Alphonse de Polignac put forward the now famous twin prime conjecture: there are an infinite number of prime number twins. But despite numerous attempts, the assumption has so far neither been proven nor disproven. The largest known twin prime pair is 2,996,863,034,895 x 21,290,000 + 1 and 2,996,863,034,895 x 21,290,000 – 1. Is it perhaps the last?
A Mathematical Magic Card
Interest in prime numbers among Magic players increased with the introduction of the new card set Duskmourn: House of Horror on September 27, 2024. The deck contains, among other things, the card Zimone, All-Questioning. Its description reads: “At the beginning of your end step, if a land entered the battlefield under your control this turn and you control a prime number of lands, create Primo, the Indivisible, a legendary 0/0 green and blue Fractal creature token, then put that many +1/+1 counters on it. (2, 3, 5, 7, 11, 13, 17, 19, 23, 29, and 31 are prime numbers.)”
That sounds cryptic, at least for Magic-inexperienced people like me. But the action of the card depends on the number of currency-generating cards called “lands” that a player controls—specifically on whether that number is prime.
After its-summer-somewhere posted their complicated, and not particularly realistic, game situation, the outcome of which depends on whether there are an infinite number of prime twins, another Redditor promptly commented, “Somehow I knew that introducing the concept of prime numbers into the game would be a bad idea. Good to know I’m not wrong.” (“To be fair,” responded a third user, “prime numbers have always been in the game, as have non-prime numbers. [This set] just introduced the concept of that mattering.”)
The idea, its-summer-somewhere wrote, is to create situations in which certain cards called “creatures” can be copied as often as desired using a particular card combination. Another card ensures that the copied creatures function as lands. If the number of lands controlled is not prime, a certain combination of cards creates two more lands. As soon as the number of lands corresponds to a prime number p, however, Zimone comes into play: It then creates two new Primo creatures, which in turn automatically also become lands. This means that you now have p + 2 lands. If p + 2 is also a prime, Zimone’s ability gets triggered again, leaving four Primo creatures on the battlefield. At that point you can use three of them to cause damage to the enemy. Thus, the opponent can only be harmed if Zimone is triggered twice in a row—in other words, only if the number of lands corresponds to a prime number twin. You can then repeat certain steps to increase your number of lands to the number of the next largest twin prime. The maximum damage that can be inflicted depends on the number of all existing twin primes: “Our maximum damage is infinite, if and only if the twin primes conjecture is true,” its-summer-somewhere wrote.
Does this now bring humanity closer to a solution to the prime twin conjecture? Probably not. Sure, you could sit two people down and have them play Magic for ages. But ultimately the gameplay is based on knowing whether numbers are prime twins instead of explicitly proving the conjecture.
Regardless, the imagined game is always entertaining and bizarre—and apparently it tempts nonmathematicians to deal with problems related to number theory. It may also have the opposite effect: As a math fan, I’ve been looking for a new hobby for a long time. Maybe I should give Magic a try.
This article originally appeared in Spektrum der Wissenschaft and was reproduced with permission.
