# Unstable Magic Math: Red and Artifacts

After many years, we have another silver-bordered set! Unstable will be released on December 8, featuring questionable science, references to other games, and other weird experiments. The cards can’t be used in regular tournaments, but they’re awesome for casual play… and for over-analytical mathematicians.

Last week I analyzed several white, blue, and black cards. Today, I’m back with the red and artifact cards that I found most interesting from a mathematical perspective, and I’ll return later this week with the green and multicolor cards.

## Infinity Elemental

The first question that came to my mind when I saw this card was, “what kind of infinite are we talking about?” As Georg Cantor already figured out in the 19th century, there are infinite sets of different sizes. For example, the set of integers or natural numbers is countably infinite, while the infinite set of real numbers is uncountable.

Given that Magic, with few exceptions, only uses natural numbers, it would seem reasonable to assume that Infinity Elemental’s power is equal to the size of a countably infinite set. You could argue that it would have been more appropriate to use the corresponding Aleph-null symbol rather than the infinity symbol for Infinity Elemental’s power, but the card is confusing enough already.

The key confusion stems from the fact that a creature’s power is normally a number, whereas we can’t treat infinity as a number. Infinity is just a concept. As a result, we should treat Infinity Elemental’s power as a concept as well—it has no bound, it can’t change with a pump spell, and its damage output is higher than any creature, planeswalker, or player can take. For most purposes, this works as you would expect.

The craziness starts when we combine it with other cards. First, let’s try to re-create Hilbert’s hotel in Magic.

Imagine that your opponent controls Infinity Elemental. You attack with Broodhatch Nantuko, and since your opponent is at 1 life, they have no choice but to block, generating infinite tokens on your side of the battlefield. You then pass the turn, only to see your opponent play Fervor, summon Nacatl War-Pride, and swing in with both creatures. So now you’re being attacked by infinite tokens that must be blocked by exactly one creature if able. Can you survive?

An initial idea is to block their first token with your first token, their second token with your second token, and so on. This ties up all of the tokens, but it would leave Infinity Elemental unblocked. So do you just lose?

Nah. If you imagine a battlefield with Infinity Elemental on the far left, followed by an infinite row of blocks (each with 1 Nacatl War-Pride and 1 Saproling) to its right, then you can just move all of your Saprolings one step to the left. As a result, your 1st Saproling would be blocking Infinity Elemental, your 2nd Saproling would be blocking the 1st Nacatl War-Pride, your 3rd Saproling would be blocking the 2nd Nacatl War-Pride, and so on. Effectively, you’ve conjured an extra token out of thin air, but that’s what you get when you introduce infinities to Magic.

In fact, you can bring more Saprolings to the left, while still retaining an infinite number to block your opponent’s Nacatl War-Prides. You could put 5 Saprolings in front of Infinity Elemental, keep a million Saprolings behind, survive the attack, and swing back for lethal. Infinities are weird. But as long as you are able to associate every Nacatl War-Pride with a unique Saproling, then things work out well for you.

What if your opponent has 2 Nacatl War-Prides? Surely then they have more tokens than you? Nope, this is still not a problem—you can assign all of your odd-numbered Saprolings to block all tokens from the first Nacatl War-Pride and all even-numbered Saprolings to block all tokens from the second. In mathematics, strange as it may sound, the set of odd integers has the same size as the set of integers. Even if your opponent deployed a countably infinite number of Nacatl War-Prides pre-combat, that wouldn’t pose a problem. Search for Hilbert’s Hotel to learn more.

An uncountably infinite number of attackers (no relation to Un-sets) would be problematic, but that requires a lot of acrobatics. Let’s just move from infinite tokens to infinite life.

If you cast Swords to Plowshares or Huatli on your own Infinity Elemental to gain infinite life and then take infinite damage from an opponent’s Infinity Elemental, what would happen?

If you ask a mathematician, then the answer would be that this is undefined. Infinity subtracted from infinity is indeterminate, even if we’re using the extended real number line. So then how do we deal with undefined numbers in Magic? Well, according to rule 107.2 from Magic’s comprehensive rulebook: “If anything needs to use a number that can’t be determined, either as a result or in a calculation, it uses 0 instead.” Hence, according to Magic’s rules, infinity minus infinity should equal 0. In other words, after taking a hit from Infinity Elemental at infinite life, you’d lose the game.

But Mark Rosewater, the rules manager for silver-bordered sets, decided on something different: “If you’re at infinite life and get hit with an opposing Infinity Elemental, you’d still be at infinite life.”

Personally, I find Maro’s ruling counterintuitive. It’s not consistent with the combination of mathematics and the rulebook. But silver-bordered sets don’t need perfect logic, so we’ll just have to roll with it.

## Just Desserts

We’re getting close to building Euler’s identity in Magic! In gameplay terms, Just Desserts will act like a Lightning Strike that can only target creatures, except that 8 copies of this spell deal 25 rather than 24 damage. Not sure when that would be relevant, but hey, you never know.

The big question is: What if you combine Just Desserts with Stuffy Doll? Well, according to Mark Rosewater: “It does damage like normal, but will create fractional amounts. If you ever have to calculate Just Desserts, you have my permission to use 3.14.”

All right. But how about Soul-Scar Mage or Druid’s Call? I’m not quite sure how to represent a fractional number of -1/-1 counters or a fractional number of Squirrel tokens. For example, can I chump block with a fractional token? I honestly have no clue. I guess it would function as kind of a Schrödinger’s token, simultaneously there and not there with probability 0.14159… but that would also be quite weird.

To make matters even more difficult, the word “fractional” is actually misleading because, as Flaky from the flavor text would surely be able to tell us, pi is an irrational number. In other words, it cannot be expressed as a fraction of two integers. This means that no matter how many copies you make of that token, you would never be able to assemble a complete Squirrel. Or at least, you’d always have squirrel parts left over. It’s lugubrious, really.

## Sword of Dungeons & Dragons

A 1 in 20 chance to repeat the process is not a lot, but this is a good card if you’re feeling lucky. Once every 400 times, you get to create at least 3 Dragons, and you can keep going from there.

A fun exercise is to figure out how many Dragons a trigger would net in expectation. Let’s call this X. Then X is equal to 1 plus 1/20 multiplied by the expected number of Dragons you get after repeating the process… which is X again! So X=1+X/20, which if we round to two decimals solves to X=1.05. The extra Dragons for the super-lucky are literally a rounding error.

## Boomflinger, Hard Hat Area, Gift Horse, Thud-for-Duds

The difference between the results of two six-sided dice has the following distribution.

 Outcome Probability 0 6/36 (16.7%) 1 10/36 (27.8%) 2 8/36 (22.2%) 3 6/36 (16.7%) 4 4/36 (11.1%) 5 2/36 (5.6%)

The mean outcome is 1.9, which is less than 3.5, the expected value of rolling a six-sided die. Another difference is that with the Contraptions, zero is now a possibility. In other words, sometimes you don’t get any effect at all.

Weirdly, though, I wouldn’t consider these Contraptions to be high-variance cards. The variance of a random variable is a mathematical way to describe how far the possible outcomes are spread out from the average outcome. While the roll of a single six-sided die has a variance of 2.92, the difference between two such die rolls has a variance of only 2.05.

An intuitive explanation for this relatively low variance is that unlike a six-sided die, not all outcomes are equally likely—a difference of 1 is the most likely result, closely followed by a difference of 2. In contrast, an outcome of 4 or 5 is pretty rare. What this does mean is that when you do roll a 5, it’s all the more exciting, and these big swings may create more drama than a balanced six-sided die.

## The Grand Calcutron

I follow RoboRosewater on Twitter to marvel at the cards designed by its neural network. They can range from funny to incomprehensible, but automatically generated cards with a lot of text on them are rarely coherent.

One day, I saw RoboRosewater tweet out The Grand Calcutron, and I thought, “there’s no way that an artificial intelligence could have designed this card.” A short while later, I noticed the set symbol, and a broad smile appeared on my face. RoboRosewater was the perfect preview outlet for this card!

In gameplay terms, this card is a callback to the mechanics of RoboRally—a game that was also designed by Magic’s creator Richard Garfield. The game had a mechanic that tasked you with placing your cards in order, forcing you to think in new ways. I enjoyed playing it back in the day, and I expect that The Grand Calcutron will be a lot of fun as well.