Math and Magic

Magic has a lot of numbers, that is undeniable. We have mana values, power, toughness, life totals, cards in libraries, plus and minus counters. But numbers don’t stop at the game itself, we also have the deckbuilding aspect. Examples include optimizing lands and spells, choosing a minimum number of creatures for Collected Company, mana curves, or figuring out if Yorion, Sky Nomad is worth the percentage points it costs. On top of all that, we also have tournament Math. Players need to know how to navigate the world of tiebreakers to make an informed decision; whether they should ID (Intentional Draw) the last round to sneak in top 8.

Having said all that, you don’t need a degree in Math or a related field to be a good Magic player, but you get a significant advantage when using Math to your benefit; that is why I am here. The goal of this article is to give you tools to understand how to use Math to up your game, even if you are not a fan of Math.

Here’s an outline of what I will be talking about in this article: First, we are going to look at combat Math, looking at some board states to see if it’s a good idea to attack or to block. Next, we are going to look a little bit at the concept of clock. Then, we are going to move into tiebreaker Math where I intend to give you a very comprehensive guide on when to ID or not if you intend to make the top 8. Following that, we will talk about deckbuilding and figuring out the best numbers of specific cards to put in your deck. Finally, we will wrap everything up with a short conclusion.

 

Combat Math

Combat Math is the most fundamental intersection of Math and Magic. Combat happens in the vast majority of games and it is crucial to know what actions we should take instead of saying the phrase that gives me goosebumps: “Math is for blockers”. If you know how to optimally navigate attacks and blocks, you are on your way to becoming an excellent Magic player, because most players just don’t know this.

Should I attack?

Let’s look at a very simple example: say Ana controls six 1/1 vanilla tokens and Nestor controls a 5/5 creature. Should Ana attack with all her tokens?

Well, it depends of course. What are the life totals? How many cards does each player have in their hand? Let us assume that Ana has 20 life and Nestor has 8 life and both players are empty handed. With those assumptions, attacking makes a lot of sense. In this turn, Nestor goes from 8 to 3, Ana loses one token, leaving 5. Next turn, even through a topdeck blocker, Ana still has a lethal attack.

Let’s change the scenario. Now, Nestor has 12 life and 2 cards in hand and Ana has 20 life and no cards in hand still. An attack would mean putting Nestor to 7 and losing a token. Next turn, Nestor can play another blocker (with 2 cards in hand and a draw step, it is likely that Nestor will have something relevant) and stabilize the board. In this scenario, unless Ana has a lot of direct damage in her deck, an attack is not a good strategy.

Should I block?

Now, let’s flip the table, and since people say Math is for blockers, this scenario shall sound more realistic. We are playing MH2 limited and Ana, your opponent, controls Hell Mongrel, a Zombie Army token that is currently a 2/2, a Lazotep Chancellor, 4 untapped lands and has 5 cards in hand. Ana just looks at you dismissively and taps all her creatures for an attack. You control a beautiful and powerful, yet lonely, Svyelun of Sea and Sky and you are at 17 life. Should you block?

Unfortunately for our Merfolk friend, the answer is a resounding yes. The opponent is currently attacking for 7, however, for every card and mana available, represents 3 damage if both creatures are unblocked. Also, every card without mana can be converted into a point of damage through Hell Mongrel. With 5 cards in hand and 4 untapped lands, the opponent can deal a total of 7 + 4×3 + 1 = 20 damage, which some would consider lethal (and the others would just be wrong).

 

The Clock

Imagine a scenario where you have two creatures, a 2/2 and a 3/3. Both are vanilla creatures. You are playing against control, stock with wrath effects and you just attacked your opponent down to 15 life. Should you play another 2/2 creature? You shouldn’t cast the creature because that won’t change your clock. Right now, your clock is three turns (15 to 10 to 5 to 0) and if you cast the 2/2 your clock is still three turns (15 to 8 to 1 to 0). The new 2/2 won’t make you win the game faster. However, imagine that instead you got a 3/3 creature, your clock would change to two turns (15 to 7 to 0).

Now, let’s go to a different scenario. Say you and your opponent are empty handed, you have a 3/3 flier, your opponent has two 2/2 vanillas and it is your combat phase. Should you attack? It depends on context of course! What are life totals? If your opponent is at 10 and you are at 12, you are going to lose this race, because you are going to die first if you take the aggressive line. However, if your opponent is at 9, you are winning the race instead!

 

Tiebreaker Math

If you are a grinder in the competitive Magic scene or just a good MTGO player wanting to take a shot at paper Magic (cough*, the best version of Magic), it is very important to know if you still are live for prizes or if your time would be better spent playing side events or cubing with your scrub friends. Regardless, in the last round, sometimes people will ask you “do you wanna ID?” and to answer that, you really need to know how good your tiebreakers are and what number of people will get in with a record higher than yours.

Essentially, we want to know how many players will have an undefeated record, a record of x-1 (x wins and 1 loss) and x-2 in the last round. Reason is, if the tournament pays out to top 16, and it’s a 226-player event with 8 rounds, you have no chances of getting prizes after your third loss because there will be 32 players with a 6-2 record or better.

The Math concept behind this is very simple and it’s called the Pascal Triangle, it essentially gives you how many players will have a certain record after a given number of rounds. I won’t explain the Math behind it but if you want to dig deeper, just do a google search on the term and you will find lots of good information.

For example, for a three-round tournament with exactly 8 players and assuming no draws, we get exactly one person at 3-0, three people at 2-1, three others at 1-2 and finally one unlucky person at 0-3. Under the Swiss Pairings algorithm, it is mathematically impossible for more players to exist at an x-1 record than the second number of the respective line in Pascal’s Triangle.

Number of Players Number of Rounds
33-64 6
65-128 7
129-226 8
227-409 9
410+ 10

 

If I am x-1 going into the last round, can me and my x-1 opponent both ID into top 8? If the number of players is closer to the lower limit (for example, in a 34-player tournament, the number of players in the tournament is approximately 33, which is the lower limit for a 6-round tournament), then your chances of making it into top 8, regardless of your tiebreakers, are excellent. However, if the number of players is closer to the upper limit (for example, 126 players, near the upper limit for 7 rounds), you and your opponent can only ID if your tiebreakers are the best amongst all the x-1s, otherwise, you will both likely be out of top 8.

If I am x-2, should I drop? It depends, what else are you going to do with your time and what are your goals with the event? If you are expecting to get prizes, look at how far the payout is. If the tournament only pays to top 8, forget about it, you are out of contention (though, there is a chance the undefeated people can drop, be disqualified, have a heart attack, catch fire, etc.). If the payout is for top 16, look at how many players started the event and how many rounds it is. If the number of players is closer to the lower limit (for example, in an 8-round event, the lower limit would be 129), then, your chances for top 16 are very good. However, if the number of players is closer to the upper limit (226 for an 8-round event), your chances of top 16 are about 40% and entirely dependent on your tiebreakers.

Precise calculations are nearly impossible to make because it is difficult to factor in drops, draws and pair downs. However, I have posted a link that should be pretty good at telling you the threshold record for a top cut in the reference section. The algorithm that it uses it is just the Pascal Triangle using proportions.

 

Deckbuilding Math

I want to share one of the best examples for application of the Hypergeometric distribution in Magic, the card Collected Company. Below is a table that shows the number of creatures in a 60-card deck and the associated probabilities of missing with Collected Company (hitting 0 creatures), getting only 1 creature, and getting a good Company with 2 creatures.

Creatures P(0) P(1) P(2)
27 2.2% 12.8% 85%
28 1.8% 11.3% 86.9%
29 1.5% 9.8% 88.7%
30 1.2% 8.5% 90.3%
31 0.9% 7.4% 91.3%
32 0.7% 6.3% 93%
33 0.6% 5.3% 94.1%

 

What do those numbers mean? “Elementary my dear Watson,” say you are playing in the best MTG circuit of the Midwest, the NRG Series, and you decide to use Collected Company in all formats – playing 9 rounds of Modern and 9 more rounds of Pioneer. That means 18 rounds and assuming you go to game three every round, 54 games of Magic. If you are playing 27 creatures in your deck, chances are, you are going to brick a company in one of your games. But that’s not all this table tells us, it also tells us that even if we go as high as 33 creatures, in about 5% of the games, which means about 3 games in a weekend, we are going to hit only one creature with our Company.

Let’s look at another example, say we are the worst kind of people, the very kind of people that play Azorius control, and to make life choices that will cause other people to be miserable, we also chose to play 80 cards for Yorion. We are in the middle of a game with a Dig Through Time and a Counterspell in hand. Our opponent has a Tarmogoyf in play and is casting another Tarmogoyf that will represent lethal next turn. We have three mana, which means we can only cast one spell this turn. Should we counter the Goyf or try to hit a Supreme Verdict to get a two-for-one?

In this case, we have an 80-card deck, that is our “population size” parameter, our outs are the 4 Supreme Verdict in our deck, that is the “number of successes in the population” parameter, and we are going to look at 7 cards with DTT, that is our “population sample” parameter and we need at least 1 verdict, that is the “number of success in sample” parameter. Putting those values into a Hypergeometric Distribution calculator will show that our chances of finding the out are 31%. Essentially, we could Counterspell the Goyf and buy ourselves a turn to find an answer for the Goyf already on the table or we could roll the dice and get a 31% of solving our problems and getting some card advantage in the process.

Quick tip, if you are in the middle of a game of Magic, you can estimate an approximate probability by doing (4/80)*7 = 35%. Remember, this is an estimation, and they are good ways to get an approximate and meaningful result in a few seconds, but they may not correspond to actual probability values.

I have used the magnificent work of Frank Karsten, PhD. to write this part about how many lands you should play in your deck. If you want to know more about it, please, check out the references.

For a low mana curve deck, with all spells costing either 1 or 2 mana (a good example of such a deck is Burn in Modern), you can easily play 18-20 lands. Your chances of flooding will be pretty low and you will hardly have issues with “not enough lands” (just be careful keeping those risky 1-land hands with lots of 2-mana cards in it).

For Aggro decks, somewhere between 20-22 lands should be enough. Those decks are very similar to Burn in Modern but will have one or more other spells with mana value of 3 (or a Snapcaster Mage which essentially requires 3 mana to be played effectively). A good example of such a deck would be Death’s Shadow in Modern.

For Midrange decks, try playing 23-24 lands. A good old example of such a deck is Jund (also, you must be very brave to register Jund nowadays, good luck), you do need to hit land drops 3 and 4 very consistently to be able to cast Liliana into BBE.

For Control decks, you will need somewhere between 25-28 lands. Your chances of flooding are going to be very high indeed, however you are drawing a bunch of cards and they all have a pretty high impact on the board (control cards tend to trade 2-for-1 and 3-for-1 on a regular basis). Even if you are casting less spells, because they are more impactful, you should be fine.

In regards to Mana Dorks and other non-land cards that produce mana, you can count those as being 0.75 of a land for the purposes of deckbuilding. For example, Mono Green Aggro can play 8 Mana Elves in addition to 20 lands and be fine in the matters of mana. In regards to Mana Sinks, don’t be afraid to play more lands than what’s recommended above if you can make good use of your mana. A great example of this is the best deck in Modern, Hardened Scales (no arguing, tier S). It has a lot of X mana spells, activated abilities that cost mana, and Urza’s Saga. Even though the deck has a top curve of 2, I am super happy with my list that plays 24 lands + Springleaf Drum.

Last and perhaps the most important part of this article: don’t let your performance or experience in a tournament make you think you should be playing more or less lands than what you are actually playing. Your bad experience of getting mana screwed or mana flooded is something called variance, and it happens. In fact, you should expect to get mana screw or mana flood at least one game in an 8-round tournament. Don’t let a small sample of data guide your decisions, trust an expert instead 😉

 

Conclusion

I hope this article was fun for you to read, if it wasn’t, I am sure it will still be useful as a reference for future moments in your Magic life when trying to navigate through this world full of numbers.

 

References

If you want an easy-to-use tool to figure out if you can ID in a tournament, or if you have chances of getting prizes, I use this one a lot, it is great.

https://sixprizes.com/top-cut-calculator/

 

The good-old Hypergeometric distribution calculator. It is useful for a whole bunch of things, such as figuring out the probability of hitting land drop number X by turn Y, how many cards you should play in a Collected Company deck, your chances of drawing a particular card in a game, and many other applications!

https://stattrek.com/online-calculator/hypergeometric

 

Here’s to you, Frank Karsten, Ph.D. Great guy, great writer and very knowledgeable in Magic and Mathematics. If you are interested in digging deep, I recommend this read:

https://strategy.channelfireball.com/all-strategy/mtg/channelmagic-articles/how-many-lands-do-you-need-to-consistently-hit-your-land-drops/

 

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Antonio Zanutto