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Magic Math – Battle for Zendikar

Battle for Zendikar contains several cards that are interesting from a mathematical perspective. Last week, I ran the numbers on Kiora’s -2 ability. Today, I’ll cover the other cards that I found interesting.

Oblivion Sower

Against a 25-land deck, Oblivion Sower will exile 1.67 lands on average. So it’s not quite Primeval Titan, but it’s at least similar. An upside is that Oblivion Sower can grab extra lands if your opponent had previously played a delve spell or if you exiled lands with a card like Fathom Feeder or Crumble to Dust. A downside is that you might hit fetchlands that you won’t be able to use (although this can be avoided to some extent by, for instance, adding Cinder Glade and Canopy Vista to your mono-green Oblivion Sower deck).

The figure of 1.67 lands is just an average. In reality, you might get lucky or unlucky. Against a 25-land deck, a straightforward application of the hypergeometric distribution reveals the following:

Table Oblivion Sower

So you’re more likely to hit 0 lands than 4 lands, but once in a while, you’ll hit the jackpot. Cherish those moments.

Landfall

Landfall has returned in slightly weaker form. The new early drops no longer get +2/+2 like Steppe Lynx and Plated Geopede did, but they still look good enough for Standard if you have enough fetchlands.

But how reliably can you expect to hit your land drops on the various turns of the game? I ran a simulation to determine the landfall probabilities on the early turns for various 60-card decks with different numbers of basic lands. I assumed that we mulligan a hand if it contains 0, 1, 5, 6, or 7 lands and that we’re equally likely to be on the play and the draw.

Landfall-Probability

As you would expect, you will almost always make your land drops early on. Later in the game, the probability of having landfall approaches the ratio of lands in your deck.

But the numbers for the midgame are the most insightful to me. If we focus on turn 4 with a 24-land deck, then under my mulligan strategy you will hit your turn-4 land drop in 77.2% of the games. This means that in slightly more than 1 in 5 games, you will have a small Scythe Leopard on turn 4 and you won’t be able to cast Siege Rhino on curve either. This is exactly why most Standard decks don’t run more than 10 cards of 4 mana or higher.

Endless One

Creatures have gotten much better over the years. For instance, if we compare Endless One to its old-school nephews Shifting Wall and Phyrexian Marauder, then the new variant is clearly better. It even supplants such hits as Mold Demon, Kasimir the Lone Wolf, and Torsten Von Ursus. May they rest in peace. (I’m sure the comments section will be filled with pathological corner cases to show that Endless One is not actually strictly better.)

While I like the simple and elegant design, I have a gripe with the flavor text. It claims that Endless One embodies all possible meanings of the word “infinite,” but from a mathematical perspective that’s not true. At best, it only embodies all possible meanings of countable infinities. But not uncountable infinities. If you want a rigorous explanation, then I’ll have to refer you to a course on set theory or mathematical analysis, but I’ll briefly describe the gist of the argument:

A set is called countably infinite if there exists a one-to-one correspondence or bijection from this set onto the natural numbers {1,2,3,…}. The set of possible powers of Endless One, for instance, is countably infinite. Even if you would combine it with any number of Doubling Seasons and Clone Legions, it’s still countably infinite. But as shown by Georg Cantor in 1874, there are also uncountably infinite sets that cannot be put into a one-to-one correspondence with the natural numbers. For example, the set of real numbers or the interval [0,1]. These sets can be said to be “larger” than the set of natural numbers, and it is impossible to find a one-to-one correspondence onto the set of possible powers of Endless One. Hence, there are “larger” infinities that are out of reach even for Endless One.

Void Winnower

Well, the Eldrazi certainly care about bizarre stuff. What will be next? If things continue like this, then I wouldn’t be surprised to see the following befuddling card in Oath of the Gatewatch.

Kozilek's Wits

On a more serious note, I’m glad that Void Winnower includes the reminder that zero is even. Let me explain why this is indeed the case. For this, we need an adequate definition of “even.” A customary one is that an integer p (i.e., a number in the set {…, -2, -1, 0, 1, 2, …}) is even if there exists an integer s such that p=2s. Such an integer indeed exists for zero: zero itself! After all, with p=0 and s=0 we get 0=2*0. Hence, zero is even.

Titan’s Presence

A reasonable deckbuilding question is how many colorless creature cards you need in a deck with Titan’s Presence. This question is similar to one that I answered for Orator of Ojutai. Based on my analysis there, I recommended 10 Dragons for a 60-card deck with Orator of Ojutai because that provided an 80% probability of being able to reveal a Dragon on turn 3 on the play. But for Titan’s Presence, I would like to have more than 80% because it does absolutely nothing if you don’t have a proper colorless creature card in hand.

My recommendation would be 14 colorless creature cards with reasonable power (so don’t count Fathom Feeder or Hangarback Walker) in a Standard deck with Titan’s Presence. By my calculations, this gives you a 91% probability of having a colorless creature card in hand on turn 3 on the play, conditional on having drawn at least one Titan’s Presence. You can go a bit lower than 14 if you have several card drawing spells, but I wouldn’t go much lower because Titan’s Presence is still very situational.

In Limited, you can take more risks for a good removal spell, but I would still like to have at least 6 medium-to-large colorless creatures in my deck before including Titan’s Presence.

Processor Assault

This mostly looks like a Limited card. The mathematical analysis is similar to the one for Titan’s Presence, except that Processor Assault is still a good draw on turn 10 when you’ve played an exile or ingest card earlier. As a result, I would be okay with one fewer setup card than what I recommended for Titan’s Presence. That is, I would like to have at least 5 exile or ingest cards before I would add Processor Assault to my Limited deck.

Transgress the Mind

You might see a reasonable discard spell that is great against Deathmist Raptor and that can set up your Wasteland Strangler or Ulamog’s Nullifier. I see a nice opportunity to run some numbers.

Against a typical Standard deck with 25 lands, 16 two-drops, 9 three-drops, 6 four-drops, and 4 five-drops, I found the following probabilities of being able to exile a card from your opponent’s hand, depending on which turn you cast Transgress the Mind.

Transgress

To obtain these numbers, I assumed that the opponent will mulligan a hand if it contains 0, 1, 5, 6, or 7 lands. They are equally likely to be on the play as on the draw. During their turn, they play a land if they have one, then cast a 5-drop if possible, followed by as many 4-drops and then 3-drops as possible.

What the numbers in the chart indicate is that you will reliably exile a card on turn 2 and 3: you’ll only miss in roughly 1 in 20 games. On turn 4, you still have an 87.4% chance of hitting a card, but after that, it quickly tapers off.

Against an Esper Dragons deck with a top-heavier curve and counterspells, Transgress the Mind would be even more reliable, even it costs more mana than Duress. However, against a typical Atarka Red deck (with 21 lands, 28 one-drops or two-drops, and 11 three-drops) the hitting probabilities are much lower: 77.8% on turn 2 and 43.2% on turn 4.

All in all, I view Transgress the Mind as a Constructed playable card, especially in a deck with Eldrazi Processors. However, in a deck without Eldrazi Processors, I would prefer to leave it in my sideboard, depending on the metagame.

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