In ninth grade, I was in a play. It was the week of the performance, and I was staying late every night because we were doing full run-throughs. The play was a courtroom drama. I played the coroner. I was the very first witness, but because the play was all one long scene, once I returned to my seat in the courtroom, I was on stage for the whole play. For most rehearsals, I didn't need to be on stage, because after my part was done, I wasn't doing anything. But this was the full run-through, so we had to do it like we would for the performance, meaning I had to be on stage the whole time. That week, we would do the play once, have dinner, and then do the play a second time. We would end the rehearsal late at night.

I bring this up because that week I didn't do all my homework. On other weeks, I could do it backstage while we were running through the many scenes I wasn't participating in, but I wasn't ever backstage because we were doing the full run-through. Normally, I was a pretty good student, so not getting my homework done was atypical for me and I was a little sensitive about it. When my math teacher confronted me about it in class, rather than simply explain why I couldn't do it, I instead made a snide remark (which was also atypical for me). I said, "I'm not going to need math."

My math teacher just smiled. She said, "No one thinks they're going to need math, but I'll let you in on a secret. Everyone needs math."

I didn't say this next part aloud, but inside my head, I said, "No, I won't. I'm going to be a writer. If I can count my salary and count page numbers, I'll be fine."

If this was a movie, that scene would have started with "43 years earlier." Then we would cut to me at my current job as head designer for Magic: The Gathering with "today" on the bottom of the screen. I would be holding my head in one hand as I stare at my computer screen. I would let out with a sigh and ask, "Why is there so much math?"

It turns out my math teacher was correct. Not only would I need math, I would need a lot of math. In fact, way more than I ever expected. That's what today's article is about: all the various ways that a Magic designer uses math. It's a cautionary tale to all my young readers out there. Math is probably going to matter more than you expect, so do your math homework.

As-Fan

Magic is a trading card game. When people buy our game, they're most likely buying a booster pack. That booster pack will contain a certain number of cards. Those cards will be of various rarities. We don't control what specific cards players get, but we can control the play environment. This gets tricky because a popular way to play the game is Limited, where players play solely with cards from sealed boosters.

Each person who buys a board game like Monopoly gets the same pieces. They get the same board, properties, Chance and Community Chest cards, plastic houses and hotels, and player pieces. The game designer for Monopoly knows what to expect of the game because all players always have access to the same game pieces. In Magic, we don't know exactly what cards the players are going to open.

That doesn't mean we have no control over what players open. We control the likelihood that players encounter certain game elements. For example, let's say we have a new creature mechanic called awesomeness. I won't know if you get a creature with awesomeness, but I will know, on average, how many creatures with awesomeness you'll open in a Play Booster. The more of those creatures we put in Play Boosters, the higher the percentage chance is that you'll open one. Rarity also controls how often certain cards show up. A common creature with awesomeness, for example, will just show up in more Play Boosters than an uncommon.

So, how do we figure this out? How do we know what percentage a certain game aspect will show up in inside booster packs? It's math. It's something R&D calls "as-fan," short for "as-fanned." As-fan represents how many times the effect we're tracking shows up in a product. Basically, we use a formula where we plug in how many cards are in a set with a given quality and their rarities, then account for the number of cards of each rarity. This gives us a number.

If awesomeness has an as-fan of 1.5 in Play Boosters, that means the average Play Booster will have 1.5 cards that are a creature with awesomeness. How can we have half a card? Percentages aren't telling us exactly what any given Play Booster will have, but what the average is over many Play Boosters. For instance, if I had two Play Boosters, one with an awesomeness card and another with two awesomeness cards, I would have an average of 1.5.

That as-fan number is important because it allows us to track how often we can expect players to interact with a certain game component. Because we've made a lot of Magic cards and we track as-fan in each set, we can compare the current set against what other sets did to get a sense of whether we're at the right level. For those interested in calculating as-fan, I went through how to calculate it in part two of this year's "Nuts & Bolts" article. We will still have to playtest to double-check, but it helps us get closer to the correct number faster.

In addition to "as-fan," we have a statistic we look at called "as-played." This is like as-fan, but cards are weighted based on how much play we expect them to see. For example, a weak card with awesomeness might increase the mechanic's as-fan in a Play Booster, but that doesn't necessarily increase the as-play for the mechanic.

To figure out as-fan (and as-play), there are two other aspects we need to understand. Yes, both involve math.

Collation

How exactly do we make Magic cards? They're printed with printers from around the world. But how exactly do we print them? Well, a normal Magic card is way too small for these printers. Instead, we print large sheets that have many Magic cards on them. After printing, we cut them down to card size. The two most common printing sizes are what we call 110 and 121. The first is 10 x 11, and the second is 11 x 11. The first number is how many cards are on the sheet vertically, and the second number is how many cards are on the sheet horizontally.

Innistrad: Midnight Hunt Sheet

Normally, basic lands, commons, and uncommons each get their own sheet (and often multiple sheets) while rare and mythic rare cards share a sheet, with each rare appearing twice on a sheet versus each mythic rare. The number of cards we put into a set is based on this sheet math. For example, we tend to use 121 sheets these days. Most sets have 80 common cards because we can spread three groups of commons (80 x 3) across two sheets (120 x 2).

Some sets require components that need additional sheets. Double-faced cards are a classic example of this. Those can't be printed on the same sheet as single-faced cards (how we print the back is different than how we print the front), so we add an extra sheet to the set. Thus, those cards don't replace existing card slots but add more cards to the set. That's why sets with double-faced cards tend to have more overall cards.

And I haven't even gotten into foil cards or Booster Fun cards, which require additional sheets. Suffice to say it takes a lot of math to figure out what can and can't fit on our sheets to be able to print them optimally. The design lead of a set must understand the limitations, so you can know what is and isn't possible. I have an upcoming set, for instance, where I had to completely change how we were doing collation to be able to get a certain cool element into the set.

But wait, there's more math.

Booster Math

Once you have the sheets of printed cards, you're not done yet. You have to figure out how to get the cards into the set in the proper proportions. The way this is done is based on how they make a booster pack at the printer. Cards are put into card feeders, and some number of cards from each feeder is put into the booster. For instance, slots one through six could be common cards. That means the feeder is filled with the cut cards from a common sheet. You would then either have one feeder with commons spit out six cards or have multiple feeders with common cards spit out a number that adds up to six. Using multiple feeders helps with randomization. Each slot in a booster pack is allocated to a certain type of card that shows up on a certain sheet.

This gets trickier when some slots aren't dedicated to one specific sheet. For instance, Play Boosters have a wildcard slot that can pull from different rarities. The percentages are put into the computer for the feeders, which ensure that the correct sheet gets added to that slot the correct percentage of the time. This is important for figuring out as-fan as you need to know how many slots a certain rarity will have. This is how you end up with rarities having numbers in between whole numbers. For example, these six slots always have a common, but these other slots can occasionally get a common.

Design can also impact booster math by creating a dedicated slot for a particular theme. Innistrad had a double-faced card (DFC) slot. Dominaria had a legendary creature slot. March of the Machine had a battle slot. Dedicated slots are usually added to help a theme of the set come through. Locking it to a slot helps increase its as-fan and can guarantee its as-fan if the subset only comes from that sheet. DFCs in Innistrad had an as-fan of one, but also a guarantee of one, because the only way to open a DFC was from their dedicated slot.

When designing Magic sets, you have to be aware of your as-fans, collation, and booster math. They all interconnect and result in a lot of math.

Effects

We talked about using math to figure out larger structural decisions, but math also comes up in individual card design. For example, let's say you want to make a card that looks at the top N ("N" is R&D lingo for a number to be picked later) cards of your library for a certain card type and put it into your hand. How do we decide what N is? We use math. Let's say you're looking for lands. Okay, how many lands do we expect people to play? Normally, lands make up 40% of your deck. We can do the math to figure out the likelihood of getting a land across multiple values of N.

  • If N=1, you have a 40% chance.
  • If N=2, you have a 64% chance.
  • If N=3, you have a 78.4% chance.
  • If N=4, you have an 87.04% chance.
  • If N=5, you have a 92.22% chance.

This allows us to set N based on whatever rate (R&D speak for how powerful a card's effect is compared to its cost) we want the card to be at. Note this can get more complicated as we start talking about other card types. Land percentages are pretty consistent between decks, but other card types are not. So, it's not just about figuring out the average use but also what the power level will be if a deck dedicates itself toward meeting the goal of the card.

There are a lot of cards that have variables that you have to account for. Sometimes you have whole mechanics. Figuring out how to properly adjust them takes math.

Another issue for card design is that the gameplay of Magic itself has a lot of math. Often, figuring out when you have a good attack relies on a mathematical analysis of the damage you'll do versus damage you'll receive. Because there are shifting factors (like the value of a creature attacking instead of staying behind the block), this math can get pretty complex. A lot of testing new cards involves playing with them, so playtesting itself involves a lot of math just in the gameplay alone. Understanding whether a new card or mechanic is working has a lot to do with understanding the math in gameplay.

Mana Curves and Costing

Another important structural part of design is making sure themes and archetypes are viable, both in Limited and Constructed formats. You must ensure there are things to do at every step of the game. This is what we call a mana curve. We want each deck to have things to do each turn. Decks tend to skew toward low-cost cards because you can cast multiple of them in the late game, but more expensive cards can get stuck in your hand in the early game.

This has to be balanced when we cost cards because we must ensure the mana cost is appropriate for the effect of the card. If we make it too cheap, the card can be broken, negatively impacting formats. If we make it too expensive, the card is useless and few people will play it. Trying to get the balance of each individual card along with the mana curve—which is the result of a suite of cards in the same color—can be tricky. Cards can have other numbers, including creatures, like power and toughness. Each of these numbers can be tweaked to find the proper mana cost. The whole system involves math.

Creating Magic cards relies on determining a lot of numbers and doing a significant amount of math.

Internal Data

Once you've made the cards, you have to test them. Each playtest provides you with data, but the data sets are small, so we collect the data, but it's not a statistically relevant as we'll get with external data (more on this in a second). The data we obtain in playtests is used more for what we call directional input. It can help us track larger things like how much certain colors are getting played, so we can figure out if colors are imbalanced. Each set goes through one or more play days where the entire department stops our normal work to play with a set to help stress-test it. Play days give us our largest pool of internal data.

We also do a series of polls where we ask what people think of specific cards (usually rare and mythic rare ones) or mechanics. Play days also allow us to collect information on people's impressions of all the game components. All this data is collected into spreadsheets where we can compare it with historical data we collected from older sets. This gives us a sense of first impressions and can often highlight things needing extra attention.

External Data

Once the set comes out, we get a whole stream of data. This is partly due to more people playing the set—millions of players as opposed to R&D's double digits—and partly because a lot of people play digital formats, which are great for collecting a lot of crunchy play data. A normal part of our process is a postmortem where we look at how our sets performed in various formats, especially Limited. This allows us to double-check our internal data to see how close our estimations were to the actual played environments.

Also, much like we do polling internally, we do external market research. We talk to players about all aspects of the set and get them to give us their opinion. We can take all this data and compare it against our historical data. If you've ever read one of my "Storm Scale" articles where I list how well a mechanic did, that's coming from this data.

But wait, there's even more data. We can measure sales. We can measure play numbers. We can measure trends on social media. There are significant amounts of data produced every time we put out a new set. All this data helps us get a better sense of how the set performed, which can teach us things to repeat, things to tweak, and things to avoid. Note that this contrasts with a lot of anecdotal feedback I get on social media and my blog. It's not that the individual pieces of feedback aren't important or valuable, but I'm talking math today, and those aren't statistically relevant from a mathematical perspective.

All The Rest

Today, I'm focusing on the math that designers use, but there's a lot more math required to make Magic sets. People in other departments have to do things like figure out print runs (or how much product we print for a set), determine how many printers we need to use, create logistical systems that get all the cards around the world for a global release day, calculate the percentages of premium and Booster Fun cards, and the list goes on. In fact, there's so much math, Magic has its own economists on staff and an entire team to crunch all the data we collect to help us understand it. Suffice to say, when making Magic, everyone needs math.

With that, it's time for me to say goodbye today. I hope this peek into this aspect of Magic design was entertaining and informative. If you have thoughts on today's article or any of the elements I shared, please feel free to email me feedback or contact me through social media (Bluesky, Tumblr, Instagram, TikTok, and Twitter)

Join me next week when Edge of Eternities previews begin.

Until then, do your math homework, kids.