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Magic Math – Mana Bases with Battle Lands

The new dual lands from Battle for Zendikar (which I’ll refer to as battle lands) will change the way we build mana bases in Standard. Time to run the numbers.

In this article, I’m inspired by the following question: How many basic lands do you need to ensure that your battle lands will consistently enter untapped?

This is a good question, but it’s also a vague one, and the answer will depend on your interpretation of “consistently,” on how many fetchlands and other lands you run, on the turn you’re interested in, and on how you plan to sequence your lands. So there won’t be a clear-cut answer. What I will do instead is to determine various probabilities and generate insights under specific assumptions.

Assumptions Eat Mathematical Models for Breakfast

  • To obtain some numbers, I will make the following simplifying assumptions:
  • We have a 60-card deck consisting of a number of battle lands, fetchlands, basic lands, other lands (such as painlands, manlands, or tri-lands), and non-land cards.
  • We mulligan an opening hand if and only if it contains 0, 1, 5, 6, or 7 lands.
    In case of a mulligan, we don’t scry, for simplicity.
  • We are always on the play.
  • Our sequencing of lands is as follows:
    1. If there are at least 2 basics in play and we can play a battle land from hand, then do so
    2. Otherwise, if there are at least 2 basics in play and we can fetch a battle land, then do so
    3. Otherwise, if we can play a basic from hand, then do so
    4. Otherwise, if we can fetch a basic, then do so
    5. Otherwise, we play another land
    6. Otherwise, we play a battle land

The land sequencing disregards the possibility of playing or fetching a tapped battle land on turn one (which may be fine if you don’t have a 1-drop) but instead attempts to get us to an untapped battle land as quickly as possible, which is in line with the setting that I’m interested in. Overall, these assumptions need not capture fully optimal gameplay, but I consider them to be reasonable guidelines. More importantly, they are simply enough to allow us to simulate games and determine relevant probabilities in a straightforward manner. The code I used for that is available here.

Graphs Eat Numbers for Lunch

Let’s start by considering a Standard Abzan deck with 26 lands, of which 4 are Windswept Heath, 2 Canopy Vista, X basics, and 20-X other lands. We’ll vary the number of basics and determine the probability that we’ll have at least 2 basics in play at the start of turn 4 (or, equivalently, at the end of turn 3) for each deck configuration under the above-described assumptions. The results, based on 10 million simulations each, are shown in the following graph.

Even though this hypothetical Abzan deck is not too focused around Canopy Vista, it would be nice if it could enter the battlefield untapped at least 50% of time on turn 4, in time for Siege Rhino. (The 50% requirement is based on intuition and experience.) This means that I’d like at least 7 basic lands in the deck. (The total comes to 11 when you also include the Windswept Heaths.) The more, the better, of course, but there will likely be tri-lands, manlands, and painlands competing for the same spots.

Next, let’s do the same for a Standard Esper deck with 27 lands, of which 9 are fetchlands, 6 battle lands, X basics, and 12-X other lands. The results are shown in the following graph.

This Esper deck is much more reliant on its battle lands. As a result, it would be nice if they could enter the battlefield untapped at least 75% of time on turn 4. (Again an arbitrary number that I happen to find appealing.) This means that I’d like to have at least 7 basic lands in the deck. Or, if you count the 9 fetchlands as well, 16 basic lands in total. This number 16 coincides with the minimum number of colored sources I would be comfortable with for a 4-cost double-colored spell like Languish, so it makes sense to me when translated to the mana base context. Again, the more the better.

But why limit ourselves to these two sets of decks and to turn 4? In the remainder, I’ll consider five different decks (all with different mana bases) and provide wider descriptive statistics for each of them. My aim is to build some feeling for and insight into the new mana bases. I have also added creatures and spells to every mana base for illustrative purposes, but the lists are not meant to represent a well-tuned deck.

A 5-Color Deck

The mana base yields 16 sources of every color, so a 5-color deck would technically be possible in the new Standard. I say “technically” because you have to make choices with your fetchlands—they are not actual quadruple-lands. Nevertheless, you can cast all of your spells with Forest, Mountain, Sunken Hollow, and Prairie Stream, which is not an overly difficult sequence. Maybe we should play more Evolving Wilds instead of a few Polluted Delta, as we are currently taking too much pain. Cutting one of the five colors is another option, in which case I’d expect a very similar mana base (or at least, a similar distribution of fetches, basics, and battle lands) to this five-color one.

As I was focusing on mana bases, the non-land part of the deck is currently a motley assortment of cards. With access to so many good cards in all colors, I frankly have no idea how to build this. Scythe Leopard, for instance, seems good in a deck with so many fetchlands, but you can’t play a turn-three Mantis Rider if you have a Forest on turn one, so perhaps one of the two should be cut. And Collected Company is powerful, but perhaps we should play Bring to Light, Exert Influence, or other spells instead. In that case, we may want to replace some of the aggressive creatures with Jace, Vryn’s Prodigy and consider Hangarback Walker as well. It will take a lot of testing before we know which cards you might want to include in a deck like this. It’s just cool to see that, in theory, this mana base contains enough sources for Mantis Rider, Savage Knuckleblade, Siege Rhino, and Crackling Doom.

For this mana base, my computer simulation yields the following performance measures.

  • Probability that you have to play a battle land on turn 1: 1.19%
  • Probability that you have to play a battle land on turn 2: 9.69%
  • Probability that a potential battle land on turn 3 would enter tapped: 10.38%
  • Probability that a potential battle land on turn 4 would enter tapped: 6.80%
  • Probability that a potential battle land on turn 5 would enter tapped: 4.41%

The first two probabilities have a different interpretation than the last three. For instance, the second probability states that you’ll have to play a battle land on turn 2 in 9.69% of the games because you simply didn’t have another land, in which case it would naturally enter the battlefield tapped. The third probability, in contrast, states that in 10.38% of the games you would have fewer than two basic lands in play at the start of turn 3 (or, equivalently, at the end of turn 2). In these games, if you were to play or fetch a battle land on turn 3, it would enter the battlefield tapped.

At the risk of bombarding you with a smokescreen of numbers and interpretations, I’ll add that if we would limit ourselves to only the set of games in which we played or fetched a battle land on turn three, then this land would enter the battlefield tapped in only 3.32% of the games. This probability is lower than 10.38% because it doesn’t count the games in which we were mana screwed and the games in which we played a basic land or fetchland, possibly a freshly drawn one, on turn three instead.

So it’s just a matter of which number you’re most interested in, but I think it’s fair to say that in this deck, the battle lands will enter the battlefield untapped reliably enough.

An Atarka Red Deck

Compared to the previous Standard, we can now play Atarka Red with 12 green sources rather than 10, which is a substantial leap in consistency. I always preferred Mono-Red in the previous Standard, but the 2 additional sources (and the lack of an Atarka’s Command replacement) help to sway me to the green side.

  • For this mana base, my computer simulation yields the following performance measures.
  • Probability that you have to play a battle land on turn 1: 1.49%
  • Probability that you have to play a battle land on turn 2: 13.51%
  • Probability that a potential battle land on turn 3 would enter tapped: 15.05%
  • Probability that a potential battle land on turn 4 would enter tapped: 11.04%
  • Probability that a potential battle land on turn 5 would enter tapped: 8.05%

I’m not in love with four lands that potentially enter the battlefield tapped in an aggro deck like this, even if they only enter tapped on turn 2 or 3 in roughly 1 in 7 games. You could reduce the turn-2 and turn-3 probabilities to 2.81% and 3.95%, respectively, if you cut 3 Cinder Glade for 2 Windswept Heath and 1 Forest, but that Forest is a bit awkward in a deck with 3 Lightning Berserkers. It’s a close trade-off.

A Black/Red Devoid Aggro Deck

The mana base showcases a slower two-color aggro deck with a higher curve and 24 lands, including two utility lands (Looming Spires and Blighted Fen) from Battle for Zendikar. There are 18 black sources and 18 red sources, which is more than enough. The nonland part of the deck is a quick black/red brew built to take advantage of Ghostfire Blade and Wasteland Strangler.

For this mana base, my computer simulation yields the following performance measures.

  • Probability that you have to play a battle land on turn 1: 0.93%
  • Probability that you have to play a battle land on turn 2: 9.24%
  • Probability that a potential battle land on turn 3 would enter tapped: 17.09%
  • Probability that a potential battle land on turn 4 would enter tapped: 12.24%
  • Probability that a potential battle land on turn 5 would enter tapped: 8.67%

A reason why the numbers for turns 3-5 are larger than those for Atarka Red is that this deck has 2 utility lands where Atarka Red had spells. This deck will sometimes keep 2-landers with such a utility land (which doesn’t help for the battle lands) whereas Atarka Red would have mulliganed those hands.

An Abzan Midrange Deck

With or without Deathmist Raptor, a deck like this will remain strong in Standard. The mana base seems reasonable, with 16 white sources, 16 black sources, and 18 green sources. There are 8 tapped lands plus 2 Canopy Vista, and there are 8 basic lands (including Evolving Wilds). I considered adding a Wooded Foothills and Smoldering Marsh over a Swamp and an Evolving Wilds, which is likely worth testing, but I didn’t want to increase the battle land count while decreasing the basic count at this point.

For this mana base, my computer simulation yields the following performance measures.

  • Probability that you have to play a battle land on turn 1: 0.11%
  • Probability that you have to play a battle land on turn 2: 3.02%
  • Probability that a potential battle land on turn 3 would enter tapped: 49.96%
  • Probability that a potential battle land on turn 4 would enter tapped: 42.36%
  • Probability that a potential battle land on turn 5 would enter tapped: 35.64%

Note that you can find the turn-4 tapped probability of 42.36% (or rather, the complementary probability of 57.64% for having enough basics) in the first graph of this article: look for 8 basics, including the Evolving Wilds.

An Esper Dragons Deck

The mana is very kind to the Grixis, Jund, Naya, Bant, and Esper shards. We have 20 blue sources, 18 black sources, and an overkill of 15 white sources without even trying all that hard. Life is easy when 8 of your fetchlands can get all three of your colors. Dismal Backwater may be better than Bloodstained Mire, but that would lower the effective basic land count, which I dislike. I considered adding extra colorless utility lands like Blighted Cataract or Blighted Fen, but I don’t love these lands when you need double-blue on turn 2 and double-black on turn 3. In a two-color control deck, however, these colorless utility lands may be good enough.

For this mana base, my computer simulation yields the following performance measures.

  • Probability that you have to play a battle land on turn 1: 1.62%
  • Probability that you have to play a battle land on turn 2: 11.00%
  • Probability that a potential battle land on turn 3 would enter tapped: 30.78%
  • Probability that a potential battle land on turn 4 would enter tapped: 23.70%
  • Probability that a potential battle land on turn 5 would enter tapped: 18.03%

Note that this deck has the 7 basics plus 9 fetchland mana base that I suggested at the beginning.

Some Final Thoughts

Hopefully these numbers and mana base can help you for the new Standard. For now, I’d aim for at least 11 basic lands (including all fetchlands and Evolving Wilds) in a deck with 1-2 battle lands, and I’d strive for 16 or more in a deck with 4-6 battle lands.

I’ll close out with a few final, tangentially related recommendations that may help you construct a reliable mana base in Standard:

  • Don’t play more than 9 taplands (where the battle lands count as a half or quarter tapland, depending on the probabilities)
  • Don’t play more than 6 painlands
  • If you have an aggressive Grixis, Jund, Naya, Bant, or Esper deck with fetchlands instead of tri-lands, then you may want to play more 1-drops and 2-drops than you might be used to. After all, you can often curve two basic lands into a stream of (fetched) battle lands, in contrast to playing Temples on turns 1 and 2.
  • Play at least 14 sources for a single-color 1-drop or 2-drop (such as Jace)
  • Play at least 12-13 sources for a single-color 3-drop or 4-drop (such as Nissa)
  • Play at least 18 sources for cheap double-colored cards (such as Silumgar’s Scorn)
  • Play at least 16-17 sources for more expensive double-colored cards (such as Gideon, Ally of Zendikar)
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