I'm going to kick off 2014 by analyzing mana bases. The creation of a solid mana base is one of the most important aspects of deckbuilding, and today I'll try to answer questions like, “how many sources of black mana do I need to be able to consistently cast Thoughtseize on turn one?” and, “how many sources of white mana do I need to be able to consistently cast Supreme Verdict on turn four?” These questions are relevant if, for example, you are building an Esper deck with 27 lands, but you are still wondering how many Temples to play, how many basic lands to play, which ones, and whether or not you can fit in Mutavault. This article will provide guidelines for those kinds of challenges.

Brace Yourselves: Methodological Remarks Incoming

Before I can do any analysis, I need to define what it means to be able to “consistently cast” a certain spell on a certain turn. It is important that I don't make this definition too strict. After all, if I want to be 100% guaranteed to be able to cast Thoughtseize on turn one in my 60-card deck, then I would need 54 black mana sources, which is obviously not a path to success. Instead, I'll propose that it is sufficient to have access to the required mana 90% of the time. This number is mostly a product of experience and intuition, so it is up for debate, but I believe that it is a reasonable interpretation of “consistency.”

Now suppose that I want to have a 90% guarantee to be able to cast, say, a turn one Thoughtseize on the play, and that I'm trying to figure out how many black sources my 60-card deck needs. I could translate this in mathematical terms by saying that I want to find the smallest integer number of black sources, denoted by B, such that the probability of having at least one black source in a random 7-card opening hand is no smaller than 90%. Since this probability is 1-{(60-B)!/(60-B-7)!}/{60!/53!}, I am looking for the smallest integer B for which 1-{(60-B)!/(60-B-7)!}/{60!/53!}≥0.9. Straightforward calculation reveals that the solution is to play 16 black sources. If you have experience with mana bases, then this number may seem overly large, and you would be right. There is a problem with this approach. Indeed, under the same logic, the number of red sources required to cast a third-turn Boros Reckoner 90% of the time turns out to be 29, which is clearly excessive.

The main problem is that the 90% should be based on keepable opening hands only. Indeed, the above calculation doesn't account for mulligans at all, despite their ability to dig for lands if needed. An opening hand without a black source is very likely to contain no lands whatsoever, which in all likelihood would result in a mulligan that yields the desired black source after all. However, this opening hand would be counted as a hand without a black source in the method I described above. This over-inflates the number of required mana sources. Moreover, if your red/green deck is stuck with a Boros Reckoner in hand because the only lands you drew were two Mountains, then that is caused by not drawing enough lands—not because the Forest/Mountain ratio was off.

To remedy this, I will instead consider the probability of being able to cast, say, a turn one Thoughtseize on the play, after mulligans have been resolved, conditional on having at least one land. Similarly, I will consider the probability of being able to cast, say, a Boros Reckoner on turn three on the play, after mulligans have been resolved, conditional on having at least three lands by turn three. Hence, the analysis of the mana base becomes intertwined with the mulligan strategy and the numbers of lands, and I'll have to make some assumptions to obtain anything.

To obtain useful numbers, I will make the following assumptions:

1. A 40-card deck contains 17 lands, a 60-card deck contains 24 lands, and a 99-card deck contains 40 lands.

2. The only colored mana sources are lands, i.e., there is no Birds of Paradise or Springleaf Drum.

3. We mulligan any 7-card hand with 0, 1, 6, or 7 lands, any 6-card hand with 0,1, 5, or 6 lands, any 5-card hand with 0 or 5 lands, and we keep all other hands.

These assumptions will allow me to derive useful optimal numbers of colored sources, and I expect that these optimal numbers will remain valid as solid rules of thumb for decks that are merely close to satisfying these assumptions (say, decks with 20-28 lands and 0-4 non-land mana sources).

Brace Yourselves: Complex Math Incoming

To calculate everything exactly, I initially wanted to use the hypergeometric distribution and conditional probability. This worked, but the required expressions become quite complex, and it got quite tedious and error-prone. For the mathematically inclined amongst you, here's an example:

hyper

To make things easier on myself, I coded a simulation, for which the code is given here, and verified that it worked as intended by comparing a few of the simulated outcomes to the exact probabilities.

The Numbers

Okay, now on to the results! The following tables will provide the minimum number of colored mana sources needed in a 40-card, 60-card, or 99-card deck to play a card with either one, two, or three colored mana in its cost by a certain turn with 90% probability, taking into account mulligans and other assumptions as described above.

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So what does this mean for 60-card decks? Well, if you want to be able to consistently cast a 1-drop such as Soldier of the Pantheon or Boros Elite on turn one, then this table recommends 14 colored sources for your deck. Note that these should be untapped sources: for 1-drops, Plains is good but Selesnya Guildgate is not. The recommendation for 2-drops such as Pyroclasm or Mana Leak is 13 colored sources. And so on. If you're splashing a card with a single colored mana in its cost and you don't intend to cast it early on, such as a splashed Blood Baron of Vizkopa in Black Devotion or a splashed Assemble the Legion in Red Devotion, then 9 colored sources should be sufficient.

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So, for 60-card decks, if you want to be 90% sure you have WW on turn 4 for Supreme Verdict or RR on turn 4 for Splinter Twin, then 18 colored sourced is a good amount to aim for.

For 40-card decks, the requirements that I computed are rather high. Surprisingly high, I might say. They suggest that the classic 9-8 mana base is often not sufficient for a two-color deck with double-color requirements in both colors. So, decks with both Wingsteed Rider and Prescient Chimera are shaky, and I would discourage you from splashing a third color. I have always picked lands quite highly in draft because I value a stable mana base, but I may have to pick them even more highly. Next time I get a choice between Temple of Mystery and Nessian Asp, I'll grab the land.

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So, if you want RRR for Boros Reckoner or UUU for Nightveil Specter by turn 3, then 22 sources is the recommended amount. Many Standard decks play slightly less, which increases the likelihood of those dreaded Mutavault, Island, Island, Nightveil Specter draws. However, the benefit of being able to attack with Mutavault probably outweighs the downside of a less consistent mana base. As I mentioned before, my numbers should be mainly seen as useful guidelines.

How to Account for Fetchlands, Mana Creatures, and Card Draw Spells?

Basic lands are simple. A Plains counts as a white source, and a Forest counts as a green source. But what if (contrary to what I initially assumed) the deck contains non-land mana sources as well? The numbers in my tables should still act as good guidelines, but does a Birds of Paradise or Serum Visions count as a full colored mana source? A good rule of thumb is to count fragile mana producers (e.g., Birds of Paradise or Deathrite Shaman) as half a colored mana source, to count any cheap cantrip (e.g., Remand or Peek) as 0.25 colored sources, and to count any cheap scry card (e.g., Temples) as 0.2 colored sources.

Regarding fetchlands, I usually consider Verdant Catacombs, Terramorphic Expanse, and the like as a full mana source for any color that they might be able to fetch. However, they may count for slightly less than a full source if you have heavy color requirements in multiple colors. For example, an opening hand with Kitchen Finks, Liliana of the Veil, Chandra, Pyromaster, Forest, Swamp, Mountain, and Terramorphic Expanse is much worse than the opening hand in which Terramorphic Expanse is replaced by Savage Lands.

A Remark on Dual Lands and Gold Cards

Although dual lands can usually be counted as a source for both of their colors, you have to be careful with this if you want to cast gold cards. For example, if you want to cast Voice of Resurgence in a Bant deck, there is a substantial difference between a deck with 4 Temple Garden, 4 Selesnya Guildgate, 4 Sunpetal Grove, and 12 Islands, and a deck with 4 Hallowed Fountain, 4 Breeding Pool, 2 Simic Guildgate, 2 Azorius Guildgate, 6 Forest, and 6 Plains. Although both mana bases have 12 sources of each color, it is much easier to cast Voice of Resurgence with the second one. For a 2-drop like Voice of Resurgence, I would recommend to have at least 20 lands that tap for either green or white (cf. the table on 2 colored mana) along with 13 green sources and 13 white sources (cf. the table on 1 colored mana).

Examples with Recent Top Decks

The numbers in my tables are based on hardcore analysis. To investigate whether or not the results are reasonable, I will compare them to the actual numbers in recent decks that won a Pro Tour or finished in the Top 8 of Grand Prix. The examples also serve to illustrate the process of determining the required numbers of colored mana sources and the process of counting mana sources.

Reid Duke, Modern Jund - GP Detroit 2013

My tables recommend 19 black sources for Liliana of the Veil, 18 red sources for Chandra, Pyromaster, 13 green sources for Tarmogoyf, 14 turn-one black sources for Thoughtseize, and 14 turn-one red sources for Lightning Bolt.

This matches up well with the actual numbers in Reid's deck. Counting the 4 Deathrite Shaman as 2 sources for each color, Reid's deck contains 19 black sources, 19 red sources, and 18 green sources. That excess green mana will satisfy Scavenging Ooze. As Deathrite Shaman and Raging Ravine don't count as turn-one sources, Reid has 17 untapped black sources and 15 untapped red sources for Thoughtseize or Lightning Bolt on turn one. Excellent!

Patrick Dickmann, Modern Twin - GP Antwerp 2013

My tables recommend 22 blue sources for Cryptic Command, 21 red sources for Kiki-Jiki, Mirror Breaker, 18 red sources for Splinter Twin, 19 blue sources for Vendilion Clique, 14 turn-one blue sources for Serum Visions, and 14 turn-one red sources for Lightning Bolt.

This matches up quite well with the actual numbers in Patrick's deck. Counting the entire collection of blue card draw spells as 4 sources for each color, Patrick's deck has 23 blue sources and 20 red sources. As Serum Visions and Sulfur Falls don't count as turn-one sources, Patrick has 15 untapped black sources and 12 untapped red sources to play Serum Visions or Lightning Bolt on turn one. Seems good overall!

Craig Wescoe, RTR Block Selesnya - Pro Tour Dragon's Maze 2013

My tables recommend 18 green sources for Advent of the Wurm, 14 turn-one green sources for Experiment One, and 14 turn-one white sources for Judge's Familiar.

Unfortunately, given the available card pool for Block Constructed, these requirements are impossible. At least, if you want to play 23 lands, which seems to be reasonable for the low mana curve of this deck. Cutting a Plains for a Selesnya Guildgate would have resulted in an additional green source for Advent of the Wurm, but would have reduced the white sources for a turn-one Judge's Familiar. I still think that switch would be worth it, but apart from that I can't really fault Craig's mana base, given the card pool available to him. In formats like Block Constructed, mana consistency sometimes has to take a back seat.

Tom Martell, Old Standard Aristocrats - Pro Tour Gatecrash 2013

My tables recommend 22 red or white sources for Boros Reckoner, 19 white sources for Silverblade Paladin, 13 black non-Cavern sources for Orzhov Charm, 11 red sources for Falkenrath Aristocrat, and 14 turn-one white sources for Champion of the Parish.

This matches up very well with the actual numbers in Tom's deck. Counting Cavern of Souls as a full source for creature spells in each color, Tom's deck has 23 red/white sources, 19 white sources, 12 black non-Cavern sources, and 12 red sources. As Clifftop Retreat and Isolated Chapel don't count as a turn-one sources, Tom has 14 untapped white sources to play Champion of the Parish on turn one. Excellent!

I'm going to round out my examples with two interesting decks from the recent Standard Grand Prix in Japan.

Shota Takao, Esper Humans - GP Shizuoka 2013

This deck looks crazy, but I like it a lot. The one-ofs are solid, the card quality is high, and Xathrid Necromancer plus Supreme Verdict is a great combo. But how about the mana base?

My tables recommend 18 black sources for Desecration Demon, 18 white sources for Supreme Verdict, 12 blue sources for Detention Sphere, and 14 turn-one white sources for Soldier of the Pantheon.

This matches up reasonably well with the actual numbers in Shota's deck. Counting the entire collection of Temples as 1 source for each color, Shota's deck has 17 black sources, 16 white sources, and 12 blue sources. So, you could consider cutting a Mutavault for an additional Plains, but it's fine overall. As Temple of Silence doesn't count as a turn-one source, Shota has 11 untapped white sources to play Soldier of the Pantheon on turn one. That's a little low, but not a big problem, as Temples are also acceptable turn-one plays.

Shota Yasooka, UB Control - GP Shizuoka 2013

Again, a work of art. Only a brilliant deck builder like Yaso-san could register this beauty for the Grand Prix. His creations are not for everyone, but I like how it attacks from unexpected angles.

Regarding the mana base, my tables recommend 19 blue sources for Dissolve and 19 black sources for Hero's Downfall.

Counting the entire collection of Temples and card selection spells as 1 source for each color, Shota's deck has 16 black sources and 18 blue sources. That comes fairly close to my requirements, and I believe the mana base is fine. Adding more sources would require cutting Mutavault (an important piece of the deck's game plan) or adding Dimir Guildgate (which slows you down considerably). In the end, consistency always has to be weighed against the pain it can cause.

Conclusion

My analysis has shed some light on the number of colored sources required for a solid mana base. I showed that, to consistently cast Thoughtseize on turn one, you need 14 black sources in your deck, and to consistently cast Supreme Verdict on turn four, you need 18 white sources in your deck. The numbers I derived turned out to correspond quite well to the mana bases seen in winning Standard and Modern decks, so they should act as useful guidelines for future deck building. Now, let the brewing commence!