Today’s article is a guest article by my friend Michael Lee, or as many of you might know him, Curby on MtG:Salvation or kirbysdl on The Source. The content is his, but I’ve done a little editing here and there for clarity. The math that follows is based on primarily on hypergeometric distribution .

The 60-Card Max Rule

It’s said that beginners don’t know the rules, skilled practitioners follow the rules, and true masters know when to break the rules. Let’s look at the “60-Card Max” rule we all follow:

Always build 60-card decks. –Everyone

Now consider not the rule itself, but the purpose behind it. Magic limits the effectiveness of powerful cards through the one-two combo of capping cards at 4 maximum copies and requiring at least 60 total cards. The purpose of “60-Max” is to maximize the chances of getting your most powerful cards. It’s a noble goal but not the only goal. If we consider all aspects of deck-building, we get something less quotable but much more useful:

Build decks to balance the chances of drawing the cards in the deck. Prioritize the chances of drawing your staple playsets, but don’t neglect the chances of drawing powerful 3-ofs and pairs. The goal is not to achieve any one number, but rather to build in a nuanced manner to achieve a cohesive whole. –No One Ever

So people cite “60-Max” because adding cards beyond 60 decreases the chances of seeing your playsets. Sure, but by how much? Don’t worry, I brought some math to this fight.

Adding a Creature

Let’s say a given Death and Taxes deck has room for a single Thalia, Heretic Cathar (THC). Based on metagaming considerations, the pilot would like to add a second THC. The right answer is often to replace a less relevant card with the new choice, but what if the player simply adds a THC to their 60-card deck?

60 cards

Chance to open on at least 1 of 4 Aether Vials: 39.9%

Chance to draw at least 1 of 3 Flickerwisps by turn three: 39.1%

Chance to draw at least 1 of 2 Serra Avengers by turn three: 28.0%

Chance to draw the single THC by turn three: 15.0%

61 cards

Chance to open on at least 1 of 4 Vials: 39.4%

Chance to draw at least 1 of 3 ‘wisps by turn three: 38.6%

Chance to draw at least 1 of 2 Avengers by turn three: 27.5%

Chance to draw at least 1 of 2 THCs by turn three: 27.5%

Having added that second THC, the pilot should expect to miss seeing Vial from just one more opening hand among hundreds of games. In other words, two players could take these similar decks through every round in several 2-day Legacy GPs and easily never see any difference in Vials. Similarly, the chances of drawing any triples and pairs does go down, but the difference is again merely a rounding error (≈0.5%). However, the goal of drawing THC more often is fully met: the chance of naturally drawing it by turn three nearly doubles.

Adding a Land

Recent printings of powerful 3-drops such as THC, Sanctum Prelate, and Recruiter of the Guard have increased the average mana costs of many D&T builds, requiring pilots to hit their first few land drops more consistently. Some of us have proposed adding an extra land to relieve the pressure. Let’s suppose our 60-card deck starts with a standard 23 land manabase including 10 Plains and 3 Karakas. If we add another Plains, the probabilities for spells follow the example above but what happens to the manabase?

The original deck had a 83.7% chance to open on one or more Plains or Karakas; the modified deck enjoys 85.6%. This is important to allow a first-turn Swords to Plowshares, for example. Another worry is being stuck with 3-drops and no way to cast them. The chances of opening on three or more uncastable 3-drops goes from 10.3% to 9.7% when we add a land (for details, see my chance-to-draw spreadsheet).

What about downsides? The original deck has a 24.6% chance of flooding with four or more land in the opener, but the modified deck sees 26.6%. It’s clear that modifying the manabase is a complex issue, and the effects are wide-ranging. However not everything changes: the relative chances of seeing our different spells remain the same. Adding a land allows us to get more mana without changing the ratios of our spells.

Other Examples

This analysis is not limited to Death and Taxes. Imagine a deck of any archetype where you’ve locked 57 maindeck cards, and are debating between two cards for the last slots. You could awkwardly fit a “2-1” combination, but what if they’re important in different, equally-common matchups? In this situation, I’d consider adding two of each. Doing so roughly doubles your chance of drawing the card you would otherwise only have one of, and only slightly reduces your chances of seeing your more numerous cards.

Alternately, imagine a deck with a small tutor component to find silver bullets against particular matchups. You’ve tightened the maindeck down, but you’re heading into a metagame with a common matchup where an additional tutor target would help. Instead of removing an existing staple and heavily reducing the chance of seeing it, it may make sense to add the new target as a 61st card. Doing so gives you game-1 access to the new card while only slightly affecting the card you might have otherwise cut.

Trade-offs

I feel so strongly about considering trade-offs that I put it right in my forum signature. You aren’t really forced to judge the relative power of cards until you start considering what to cut. However, cutting a card is not the only possible outcome. Instead of cutting one card and reducing the chance of seeing it by a lot, you can reduce the chances of seeing every other card by a tiny bit. It amazes me that one of those trade-offs is so commonly accepted, and the other so thoroughly vilified.

The Road to Mastery

The goal here isn’t to advocate 61+ cards in every deck, but rather to honestly consider when it might be an appropriate solution. If there’s a combo deck that completely centers around getting two different cards, any reduction in finding those cards is a prohibitive cost compared to cutting a less important utility card.

However, other decks are more well-rounded. Aggro decks field redundant threats. Control decks have a variety of overlapping control elements. While some cards in those decks are still more important than others, the deck can probably function perfectly well without drawing them in a given game. In those decks, slightly reducing the chances of seeing all cards may be a viable alternative to heavily reducing the chance of seeing one card.

Where Does it End?

A common counter-argument to someone considering a 61st card is, “Why stop at 61? Why not add a 62nd card? Why not add ten cards?” Well, it’s actually not a counter-argument but rather something to consider. Could 62 or more cards ever be ideal?

The more cards you add, the more you dilute the most powerful cards in the deck. We saw above that a 61st card drops the “Vial in opener” chance by ≈0.5%. The next several additional cards each reduce it further by ≈0.5%, so the cumulative effects will be increasingly obvious as you add cards. Furthermore the number of potential cuts goes up too, so additional candidates have to be exceptionally powerful to all justify inclusion. Eventually, you simply run out of cards whose power levels equally justify destabilizing the rest of the deck.

Having done that analysis, the answer to, “Why not 62?” is either that you can make the case for a given 62nd card, or that you can find a card to cut rather than further weakening the deck. In short, the answer depends on the situation.

Mastering “60-Max”

You can sometimes actually increase the consistency and effectiveness of a deck by adding a 61st card. You may not agree with all of my examples above, but the overall point still stands: “60-Max” is a useful guideline, not an unbreakable requirement.

A zero-tolerance viewpoint is usually the lazy way out: it is so much easier to simply decree that every 61-card deck is fundamentally broken than to examine whether or not that is actually true. Indeed, the nuances of any complex issue cannot be fully understood without considering when the rules should be broken.

Given that we’re here at Thraben University, I’ll end with a professor’s words: