Magic Math – Oath of the Gatewatch

Another set, another opportunity to analyze the new cards from a mathematical perspective. Let’s get to it!

It’s Hard to Miss with Oath of Nissa

Oath of Nissa provides green Standard decks with cheap card selection as well as protection from Dromoka’s Command. But what are the chances of hitting in your top 3? Here are the numbers.

Table 1

The way to read this table is as follows: The first column is the number of creatures, lands, and planeswalkers in your 60-card deck. (I left out the unlikely values below 30 and above 55 for brevity.) The second column gives the percentage probability of seeing zero of those cards with Oath of Nissa. The third column gives the percentage probability of seeing one of those cards, and so on.

As a methodological note: These are basic hypergeometric probabilities derived by assuming a deck size of 59 cards. The interpretation is that Oath of Nissa is removed from the deck and put on the stack before a game starts. You could remove a Forest from the deck as well and run the numbers with 58 remaining cards, but I don’t like the resulting interpretation because in a real game, you are likely to have one or more green sources in your opening hand anyway. As a result, the probabilities I provided are close to the ones you’d get with a turn-1 Oath of Nissa conditional on having kept or mulliganed into a hand that can play it on turn 1.

To illustrate, suppose your deck has 50 creatures, lands, or planeswalkers. That means you have 10 instants, sorceries, enchantments, or artifacts, which seems like a reasonable number. For such a deck, you will almost always be able to pick up at least one card with Oath of Nissa. You will only miss completely in 0.3% of the cases—you will see only one card in 5.5% of the cases—and you’ll get a choice between multiple cards in most of the cases: 33.9% for 2 cards and 60.3% for 3 cards. I like these numbers.

Nevertheless, these numbers can be a bit deceptive because they don’t take into account how often you’ll get nothing but lands. Seeing land, land, sorcery is not always going to make you happy. So let’s take a look at how often you’ll see a certain number of nonland cards.

Table 2

This table is based on a deck with 24 lands. That’s slightly less than most Standard decks used to run, but you can shave a land if you add multiple Oath of Nissas to your deck.

With 50 hittable cards and thus 26 creatures or planeswalkers, you’ll be forced to take a land in 16.8% of the cases. That’s not negligible, and it makes the card closer to Sleight of Hand than to Ponder, but it’s still acceptable to me. On the bright side, you get a choice between 2 nonland cards in 33% of the cases and between 3 nonland cards in 8% of the cases. So an ideal Oath of Nissa in the late game will happen frequently enough.

So far, we’ve seen that Oath of Nissa with 50 hits is okay. But how low can you go on your creature/land/planeswalker count? This may not be the ultimate question of life, the universe, and everything, but I think it’s an interesting question nonetheless.

My answer would be 42. That is, I would accept having Oath of Nissa in my deck with 42 hittable cards, but not lower than that. This way, you miss completely with probability 2.1%, which is tolerable to me, and you see a nonland card in a 24-land deck with probability 67.2%, which is not great, but satisfactory if your deck has some mana sinks for lands in the late game and your strategy is based around several key creatures or planeswalkers.

To clarify, this choice of 42 is subjective, and I view it as a bare minimum, not a target value. Ideally you’re at 50 or higher, but I won’t chastise anyone for adding a few Oath of Nissas—it doesn’t have to be a 4-of because there are diminishing returns and they eat into the noncreature, nonplaneswalker slots—to smooth out their deck as long as they have 42 hittable cards or more. As always, the more the better.

This Is Not a Pipe

I believe this is the first time that the sentence “Repeat this process once” has appeared on a Magic card. It could be considered unclear what the word “process” refers to. Does it refer to the first sentence of the card, or to both? If it refers to both sentences, then you would get stuck in an infinite loop!

To be fair, I think everyone grasps what the card’s intended effect is, and the language is arguably clear enough. But was I alone in thinking that its possible self-referential ambiguity reminded me of Gödel’s (first) incompleteness theorem?

In loose layman’s terms, that theorem says that in certain non-contradictory axiomatic systems capable of elementary arithmetic, there are statements which are true, but not provable. Yeah, that was my attempt at “loose layman’s terms.” It’s complicated and mind-boggling stuff, but Gödel described a rigorous way to construct self-referential statements like “this statement is not provable” in a formal mathematical system. And then we really cannot prove it. It may have been a stretch, but my mind leapt to these self-referential statements after reading Remorseless Punishment.

Gödel’s theorems impose fascinating limitations. I’m not an expert on the subject and definitely won’t attempt to go into detail here. If you want to learn more, then you could check out thick, semi-accessible books like Gödel, Escher, Bach: An Eternal Golden Braid by Douglas Hofstadter. In the meantime, I’ll eagerly await the following card in the next set.

createcard

With 10 Spells, a Turn-5 Goblin Dark-Dwellers Is About 80% Reliable

This is an excellent card that can generate a lot of value by flashing back Crackling Doom, Kolaghan’s Command, Hordeling Outburst, and so on. Alternatively, a cheaper spell like Roast or Duress is also a valid target. It doesn’t pair well with Painful Truths, but I still expect that Goblin Dark-Dwellers will see plenty of play in Standard.

What I was wondering about is the following question: If I have a deck with 26 lands, 4 Goblin Dark-Dwellers, X spells of cost 3 or less, and 30-X other creatures, then how often will I be able to flash back an instant or sorcery when I cast Goblin Dark-Dwellers on turn 5? I’ll assume that you cast a spell over another creature whenever possible, that all lands enter the battlefield untapped, that you’re on the play and draw with equal probability, that there is no extra card draw, and that you mulligan any opening hand with 0, 1, 6, or 7 lands. Given those assumptions, I coded a simulation to approximate the probability of interest.

As a methodological note: I could just determine the probability of drawing at least one spell by turn 4, but that would likely result in a larger number than the probability I’m actually interested in. Specifically, I’m focusing on the situation where you cast a Goblin Dark-Dwellers on turn 5, which means that several “slots” for a cheap sorcery or instant are already taken by lands and the 4/4. I felt it was worth a little extra effort to take this aspect into account.

GoblinChart

You don’t need that many spells to get value reliably, but you still need to have Goblin Dark-Dwellers in mind when building your deck. I would aim for at least 10 sorceries or instants with cost 3 or less, with as many 3-cost ones as possible.

With 10 hits in the deck, you are over 80% to get value with Goblin Dark-Dwellers on turn 5. Of course, it only goes up as the game goes longer, and the fallback option of a 4/4 menace with no other ability is not terrible.

The configuration of many contemporary Jeskai or Mardu decks would already largely suffice, although they would have to flash back an unspectacular Fiery Impulse a little too often for my taste. Deck building is always a balancing act, but I would be inclined to add an extra Kolaghan’s Command and perhaps replace a Painful Truths by a Read the Bones if I were to add Goblin Dark-Dwellers to my Jeskai or Mardu deck.

Could We Get a Real Rampant Growth, Please?

How many Wastes do you need for this to work? I took the average of the hypergeometric probabilities for playing first and drawing first to obtain the following table.

Table 3

We don’t have to require 90%, but at least 70% seems reasonable. Otherwise, I’d rather just have Rattleclaw Mystic or Sylvan Scrying. Those cards have their downsides too—Rattleclaw Mystic gets killed by Fiery Impulse, and Sylvan Scrying’s target of Shrine of the Forsaken Gods won’t accelerate you until later in the game—but at least you know what you can expect.

So if I want to run Ruin in their Wake under my (subjective, experience-driven) 70% minimum, then that means at least 8 Wastes. Or 4 Evolving Wilds and 4 Wastes. You can get into actually consistent numbers with 4 Evolving Wilds, 4 Fertile Thicket, 4 Oath of Nissa, and 4 Wastes. I actually like the sound of that. I wouldn’t count other copies of Ruin in their Wake—they can turn on subsequent ones, but they won’t enable turn-2 ramp, which is what I am most interested in.

All those Wastes are a tall order, but it may not be impossible. Especially in a more midrange Eldrazi deck based around a turn-3 Thought-Knot Seer rather than a turn-6 Ulamog, having a Wastes or 2 next to a bunch of land-tutor effects might work out. I would be interested to see if a Standard deck with these cards will come together.

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