Magic Math: Ixalan

Ixalan contains several cards that I found interesting from a mathematical perspective. Let’s run some numbers!

How Often Will Rowdy Crew Be a 5/5?

Drawing three cards and discarding two at random is usually better than drawing a card. After all, when you have a low-value card in hand (an excess land, a 1-drop creature in the late game, an extra copy of a legend, or a dead removal spell against control), then you have a chance of upgrading it. Another benefit is filling up your graveyard. If you have Earthshaker Khenra or Gate to the Afterlife in your deck, then there is some value in the discard clause.

A deck building restriction is that Rowdy Crew should be at the top of your curve. After all, if the last card in your hand is Glorybringer, then you risk making your hand worse, not better. But in the right deck, Rowdy Crew could be a bigger Rogue Refiner and a reasonable way to top off your curve.

To figure out how often Rowdy Crew will grow into a 5/5, we need to make some assumptions on the composition of our deck. Let’s start with a Ramunap Red deck.

Ramunap Red

24 lands
10 instants
26 creatures

Starting with this deck, I imagined the following: I set one Rowdy Crew aside, draw any number of cards to represent my opening hand and first couple of draw steps, and finally put the ability of Rowdy Crew on the stack. The numbers for this thought experiment will be close to the ones you will observe in an actual game when you cast Rowdy Crew. I could condition further and take into account mulligans, land drops, and so on, but I chose not to do that because the resulting interpretation would become muddled, the analysis might no longer be tractable, and the increase in accuracy would be negligible.

To determine the probability of a +2/+2 boost, I started with a list of the various three-card combinations (such as “one instant, two creatures”) that you could draw and applied the multivariate hypergeometric distribution to determine the likelihood of each. Then, for every three-card combination I determined the probability that two random cards would share a card type. For example, if you started with an empty hand and drew instant, creature, creature, then you’d have a 1/3 chance of discarding both creatures. Finally, I multiplied and added everything together. It was relatively easy to run the calculations in an Excel sheet.

Click to enlarge.

The results are as follows.

Cast Rowdy Crew on an empty hand: 36.3% probability of getting the +2/+2 boost
Cast Rowdy Crew with 1 land in hand: 37.9% probability of getting the +2/+2 boost
Cast Rowdy Crew with 1 instant in hand: 26.3% probability of getting the +2/+2 boost
Cast Rowdy Crew with 1 creature in hand: 38.7% probability of getting the +2/+2 boost

Based on these numbers, you may think of Rowdy Crew as a 3.7/3.7 creature on average. That’s not a particularly impressive size, so I am somewhat skeptical whether it will replace Hazoret, the FerventChandra, Torch of Defiance, or even Sand Strangler in the 4-drop slot of Ramunap Red. But it’s also not out of the question. In particular, I could see a Rowdy Crew build with Scrapheap Scrounger and Dragonskull Summit for extra graveyard synergies.

Comparing Rowdy Crew to the alternative 4-drops of Ramunap Red depends on the context of the format. If exile spells like Cast Out or Vraska’s Contempt become popular, then Rowdy Crew may prove to be best because it helps you grind against these 1-for-1 removal cards. But if the metagame doesn’t feature as many clean answers to Hazoret, which seems likely, then Hazoret should remain the superior choice. We’ll have to wait and see.

Let’s analyze another potential home for Rowdy Crew.

Jeskai God-Pharaoh’s Gift

22 lands
22 creatures
6 artifact creatures (Walking Ballista and Hollow One)
4 sorceries (Cathartic Reunion)
6 artifacts (Gate to the Afterlife and God-Pharaoh’s Gift)

Using the same approach as before—just way more tedious this time—I was able to find the following numbers.

Cast Rowdy Crew on an empty hand: 36.8% probability of getting the +2/+2 boost
Cast Rowdy Crew with 1 land in hand: 36.7% probability of getting the +2/+2 boost
Cast Rowdy Crew with 1 (nonartifact) creature in hand: 40.9% probability of getting the +2/+2 boost
Cast Rowdy Crew with 1 artifact creature in hand: 44.2% probability of getting the +2/+2 boost
Cast Rowdy Crew with 1 sorcery in hand: 20.4% probability of getting the +2/+2 boost
Cast Rowdy Crew with 1 (noncreature) artifact in hand: 27.4% probability of getting the +2/+2 boost

So the larger variety of card types is harmful to some extent, but the dual nature of the artifact creatures makes up for it. Ultimately, the numbers are pretty similar to the ones for Ramunap Red, so it would still be fair to think of Rowdy Crew as a 3.7/3.7 creature on average.

Is that good enough, especially when double-red can be tough on the mana? Maybe. The deck has to look for a replacement for Insolent Neonate somehow, and given that you need to retain a high creature count for Minister of Inquiries, I can see Rowdy Crew earning a few slots.

How Reliable Is Sunbird’s Invocation?

Well, that depends on the converted mana cost of your spells and on the number of hits in your deck. Here is a handy chart.

Click to enlarge.

The numbers in this chart should be considered as a priori expectations. To explain what I mean by that, consider the Approach of the Second Sun dream scenario: If you hit another copy in the top 7, then the one you cast from your hand wins the game on the spot. Frequently, you’ll trigger Sunbird’s Invocation for 7 when you have 45 cards or so remaining in your library. If those 45 contain 3 Approaches, then you’d have a 40.0% probability of hitting one. If those 45 contain only 2 Approaches, then you’d only have 29.0%. So clearly it depends on the situation at hand. But without seeing your opening hand or the contents of your draw steps, i.e., at the very start of the game, you’d have approximately a 32.5% probability of hitting another Approach when you trigger Sunbird’s Invocation for 7. That’s the number shown in the chart. The reason for providing these numbers is that they can yield insight for the deck building process.

The insight that I gained is that Sunbird’s Invocation’s reliability is poor. The Approach of the Second Sun dream in particular seems questionable, as you’ll “miss” in more than 2/3 of the games. Then again, “missing” in this context will often mean a Fumigate or Glimmer of Genius, which is a fine consolation prize—free spells are never to be underestimated. But is that worth the addition of a 6-mana card to a deck with an already-high curve? To me, this is dubious.

There might be potential for abuse in a wacky ramp/combo brew, perhaps with Deeproot Champion. Still, 6 mana is a lot for a card that doesn’t affect the board immediately and that is not guaranteed to provide you with any value on the next turn. You need a very specific deck to take advantage of Sunbird’s Invocation.

How Many Dinosaurs Can You Expect to Hit with Gishath, Sun’s Avatar?

Gishath is tailor-made to be the end-boss of a Jurassic Park ramp deck. Its ability is quite powerful.

Click to enlarge.

So in a deck with Gishath and 9 other Dinosaurs, if you connect for the full 7 damage, you’d hit 1.07 Dinosaurs on average. That seems like a good amount of value for a deck that isn’t all-in on Dinosaurs. Once you go up to 18 other Dinosaurs, you can expect to net more than 2 free creatures if you hit for the full 7 damage and still nearly 1 free creature if you connect for only 3 damage. Sounds good!

A downside of Gishath is that it matches up poorly against instant-speed removal spells. If your opponent kills it before it can deal damage, then you have nothing to show for your 8-mana investment. So Gishath looks more like a sideboard card to me. Imagine a ramp deck that aims to blank opposing removal spells by having only 4 Carnage Tyrants as main-deck win conditions. Against such a deck, opponents may board out all of their spot removal spells, in which case you may wish to transition into a more aggressively-slanted Dinosaur midrange deck for games 2 and 3. That’s where Gishath (perhaps flanked by Ripjaw Raptor) could make an impact.

How Many Islands or Merfolk Do You Need for Kumena’s Speaker?

Since this is an early-game card, let’s focus on the games where one of the cards in your top 9 (which corresponds to turn 3 on the play and turn 2 on the draw in a game without mulligans) is a Kumena’s Speaker. For those games, I am interested in the probability of having at least one Island or Merfolk in those top 9 cards, taking into account that at least one of your cards (but not necessarily the first one you draw) is already taken up by a Kumena’s Speaker. In other words, I’ll determine a conditional probability in the same way as I did for Orator of Ojutai.

Click to enlarge.

Looking at this figure, 14 other Islands or Merfolk seem to be a good target. For example, a blue-green deck with 5 Island, 4 Kumena’s Speaker, 4 Deeproot Champion, and 2 Merfolk Branchwalker can rely on having an early game 2/2 with sufficient (90.9%) consistency. Note that I substracted 1 Kumena’s Speaker to get to 14, the number of other Islands or Merfolk in your deck.

How Many Pirates Do You Need for Lookout’s Dispersal?

If you’re interested in casting this in the early game, then it comes down to pretty much the same question as for Kumena’s Speaker. So 14 Pirates may be a nice target. But don’t be blind to this number—evaluate everything in the context of your deck.

If your deck aims to cast Lookout’s Dispersal on turn 2 to stifle the opponent’s development, then you may be better off with 8 Pirates if they all cost 1 mana than with 14 Pirates if they all cost 2 mana. But if your goal is to cast a 2-drop and Lookout’s Dispersal on turn 4 to double-spell, then it would be the other way around. As always, it just really depends on what your deck is trying to accomplish. Either way, the figure above can tell you what to expect.

How Reliable Is Commune with Dinosaurs?

Removing Commune with Dinosaurs from a 60-card deck and putting it on the stack, I can find the following probabilities.

Click to enlarge.

As long as you’re content with any land of any color, Commune with Dinosaurs will be somewhere in between Traverse the Ulvenwald and Oath of Nissa in terms of how consistently it will act as a mana source. But the additional ability to grab a Dinosaur when you’ve mana-flooded (with no delirium strings attached) is very powerful.

Commune with Dinosaurs reminds me of Ancient Stirrings, a centerpiece of R/G Tron and Bant Eldrazi decks in Modern. Also, Dinosaurs are ancient, so it’s a fitting comparison.

In a deck with 10 or more Dinosaurs, Commune with Dinosaurs will be a pivotal driver of consistency, and I expect it to see a lot of play if R/G Dinosaurs becomes a contender in Standard.

How Many Creatures Do You Need for Growing Rites of Itlimoc?

If you want to be 89.8% consistent to hit a creature, which seems like a reasonable goal, then for a 60-card deck you should aim for 25 creatures at minimum. You’d have 88.5% with 24 creatures and 91.0% with 26 creatures.

For Standard, my first impression of this enchantment is that it feels like a win-more card. Opponents usually try to destroy your creatures, so if you already have four, do you really need the extra help? Then again, Hanweir Militia Captain frequently transformed in a game that wasn’t won yet, so there may be some hope. Still, a potential home for Growing Rites of Itlimoc needs a good mana sink, and I’m not sure if the tools are available to build around this in Standard.

For Commander, it may be easier to meet the transformation condition. With a 99-card deck, you can get the same level of 89.8% consistency with 42 creatures. That number, by the way, is always a good answer to any question.

Is 20 Damage Appropriate for Star of Extinction?

I love the design and flavor of Star of Extinction. 7 mana may prove to be too steep a cost for competitive play, but I’m still hoping to see some Modern-legal Stuffy Doll/Boros Reckoner brews in the coming months.

Star of Extinction also prompted some questions. First, is 20 the largest number ever printed on a Magic card? A quick check revealed that this was not the case. The number 20 had appeared on various cards before:

Even larger numbers have also appeared on Magic cards. Examples include Providence, Serra Ascendant, and Helix Pinnacle. The largest number that I could find is 100,000, from the Portal Three Kingdoms card Borrowing 100,000 Arrows.

So in the grand scheme of things, 20 doesn’t feel like all that much. Which is actually kind of weird because this is not just a comet or asteroid crashing into Ixalan—it’s a friggin’ star!

The smallest stars in our universe, red dwarves, still have 7-8% the mass of our Sun. That percentage may be small, but if you would put such a low mass star next to the Earth or even Jupiter, it will still be way more massive. As this picture from NASA shows, it isn’t even close!

Now, maybe the gravitational constant is different in Magic’s multiverse such that achieving nuclear fusion is easier and smaller stars are possible. Who knows. I mean, instantaneous travel between planes is also possible. But I cannot escape the idea that a red dwarf star smashing into Ixalan would be very bad news for the plane as a whole, and 20 damage seems like a lowball offer. Would it really have killed R&D to have this deal 1,000,000 damage instead?

A missed opportunity, I say.

2 thoughts on “Magic Math: Ixalan”

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