Dominaria contains several cards I found interesting from a mathematical perspective. I already analyzed Mox Amber here; Wizard’s Lightning, Wizard’s Retort, and Ghitu Lavarunner here; and Weatherlight, Legendary Sorceries, Zahid, and Belzenlok here, but there are so many more. Let’s run some numbers!

How Often Will Powerstone Shard Tap for Multiple Mana?

Cloudpost is banned in Modern, but the deck building cost to fit Cloudpost, Glimmerpost, and Vesuva into your mana base was almost negligible. Powerstone Shard takes a spell slot, but it’s still an interesting effect.

Let’s run some numbers to gain insight. In Standard, the likelihood of having multiple Powerstone Shards depends on how many tutor effects like Trophy Mage or Whir of Invention you run. (You may count four Glint-Nest Crane as one additional copy of Powerstone Shard and four The Antiquities War as two additional copies of Powerstone Shard as well.) Counting each tutor effect as an extra copy of Powerstone Shard, the distribution of the number of Powerstone Shards drawn in your top 11 cards is as follows. (The top 11 cards represent turn 4 on the draw and turn 5 on the play.)

4 copies in deck 8 copies in deck 12 copies in deck
Drawing 0 Powerstone Shard by turn 4-5 43.4% 17.6% 6.6%
Drawing 1 Powerstone Shard by turn 4-5 41.6% 36.9% 22.9%
Drawing 2 Powerstone Shard by turn 4-5 13.3% 30.1% 32.3%
Drawing 3 or 4 Powerstone Shard by turn 4-5 1.7% 15.4% 38.2%
Expected mana produced by a drawn Powerstone Shard 1.30 mana 1.78 mana 2.36 mana

The different columns represent different decks. For each deck, the first couple rows indicate how often you’ll draw a certain number of Powerstone Shards in your top 11 cards. For example, in a deck with four Powerstone Shard and four Trophy Mage, you’ll draw exactly one Shard or Mage in your top 11 cards with probability 36.9% and are even luckier in 45.5% of the cases.

The last row is the most interesting one to me: It indicates how much mana a Powerstone Shard produces in expectation. Here, we’re conditioning on the games where we have drawn at least one Shard and/or tutor effect, under the assumption that we’ve been able to cast every Shard or tutor effect drawn in our top 11 cards. As it turns out, with eight effective copies in your deck, each Powerstone Shard taps for 1.78 mana in expectation. That’s the point where I get interested. (It’s 1.92 mana with nine copies and 2.06 mana with ten copies.)

But, you do need to be ramping into really big stuff. If it’s just Approach of the Second Sun you’re after, then two Orazca Relic and five lands would also get you there—no big need for Powerstone Shard. But if it’s enormous Walking Ballistas, expensive kicker costs, or gigantic Pull from Tomorrows you’re after, then you may be able to achieve something, as long as you have at least eight effective copies of Powerstone Shard in your deck.

How might such a deck look? I started by considering Metalwork Colossus, but I think that without Sanctum of Ugin to chain them, this strategy won’t cut it. And without the typical Colossus setup cards like Prophetic Prism, Whir of Invention didn’t look like a great fit either. But four Trophy Mage is still fine, and there are several new tools from Dominaria that inspired me. When I saw Gilded Lotus, my mind sprang to Twiddle Desire and thus to Dramatic Reversal.

Standard Storm Combo

The idea of this deck is to set up a big turn where you untap your mana rocks with Dramatic Reversal, generate loads of mana, and churn through your deck with Pull from Tomorrow. The cost of both spells is reduced by Baral. The set of mana rocks includes Mox Amber, which is iffy with only eight legends, but at least it replaces itself via Jhoira.

Sadly, this list is almost surely not competitive, especially when Abrade and Negate are running around. Builds with Paradox Engine will likely run into similar problems. I can imagine that taking out the Dramatic Reversal package for cards like The Antiquities War will lead to a better deck, but then you also have to cut Baral, which means you have to cut Mox Amber, and eventually you have a completely different deck. I presented my deck to illustrate where Powerstone Shard could really shine, but I’m curious to hear your thoughts on the most competitive direction.

As a final remark, my combo brew aims to win with Aetherflux Reservoir, which can be found via Inventors’ Fair. But there are plenty of other options. You could create infinite mana with Cogwork Assembler (assuming that you start with multiple Powerstone Shards) and sink it into Walking Ballista for the win. Or you could create infinite mana by controlling Scrap Trawler and two Gilded Lotus and then repeatedly copying Paradox Engine with Mirage Mirror: Since the game sees a 5-mana copy hit the graveyard due to the legend rule, Scrap Trawler allows Mirage Mirror to return itself. The possibilities for brewing are nearly endless.

How Reliable is Adventurous Impulse?

This card is basically Oath of Nissa without the planeswalker rider, but that should still be good enough to see Standard play. It’s the green Opt. You can find my original analysis on Oath of Nissa here. There, I showed that in a deck with 24 lands and 26 creatures, you will be able to pick up at least one card in 99.7% of the cases and a creature in 83.2% of the cases. In other words, for creature-heavy green decks, the chance to miss is almost zero, and you’ll usually be able to pick a creature in the late game.

As a minimum number of combined creatures and lands for Adventurous Impulse, I recommend 42. With the corresponding 24 lands and 18 creatures, you would be able to pick up at least one card with probability 97.9% and a creature with probability 67.2%.

What’s the Optimal Mix of Rat Colonies and Swamps?

I set up a simulation to figure it out. Of course I did.

As it turns out, if you merely want to maximize your goldfish kill, then the optimal deck is 35 Rats and 25 Swamps, both on the play (with an expected kill-turn of 5.029) and on the draw (with an expected kill-turn of 5.006).

Goldfishing is a little silly, since every hand with two Swamp and three Rat Colony automatically secures a turn-5 kill. But even if you give your opponent two free removal spells, 35 Rats and 25 Swamps remain optimal on the draw (with an expected kill-turn of 6.196).

If you’re crazy enough to unleash the Rats in Standard, then you could add some fun spells.

Standard Mono-Black Rats

If Tetsuko Umezawa, Fugitive were black, I would have included that card as well, but it doesn’t mesh well with Cabal Stronghold. Either way, this deck can never beat Ixalan’s Binding, and it’s surely not going to dominate Standard anytime soon. But it does look like a ton of fun to bring to an FNM.

What’s the Best Mix of Swamps and Cabal Strongholds?

Cabal Stronghold is worse than Cabal Coffers because it won’t allow you to cast your 6-drops ahead of schedule. But free late-game mana is still appealing for the right deck.

Either way, a deck building question arises: In 24-land mono-black, should you play three or four Cabal Stronghold? After running the numbers, I concluded that four is probably best, unless your curve tops out at 7 and/or you want to improve your chances of casting a turn-3 Dread Shade. More detailed analysis below.

With six lands: You’ll have 7 mana if and only if you have exactly one Cabal Stronghold. The probability of drawing exactly one Cabal Stronghold in your first six lands is 45.4% in a 21 Swamp + three Cabal Stronghold mana base and 46.1% in a 20 Swamp + four Cabal Stronghold mana base. This difference is negligible, whereas an extra Swamp will be appreciated by Dread Shade decks.

With seven lands: You’ll have nine mana only with one or two Cabal Stronghold. The probability of drawing this many Cabal Strongholds among your first seven lands is 64.7% with 3 Cabal Strongholds and 71.7% with four Cabal Strongholds. This difference could be relevant if you plan to cast Torment of Hailfire or want to do nonsense like casting Secret Salvage and two Rat Colony on the same turn.

With eight lands: You’ll have 11 mana with one or three Cabal Stronghold and 12 mana with two Cabal Stronghold. In either case, you can kick Josu Vess, Lich Knight. The probability of having at least 10 mana with your first eight lands is 72.3% with three Cabal Strongholds and 82.2% with four Cabal Strongholds. Unsurprisingly, if you want to reliably kick Josu Vess, Lich Knight, then you should play four Cabal Stronghold.

The Mirari Conjecture is Beautiful

The beauty of math is that given unambiguous axioms and definitions, things can be proven with absolute certainty. But there are ideas that, even though they appear likely to be true, no one has been able to formally prove. Such ideas are called conjectures.

In the real world, there are several famous conjectures for which a proof would earn you a million dollars. The most interesting to me is the conjecture that P is not equal to NP, which very loosely states that riddles with easily-verifiable solutions (such as in cryptography) cannot be solved quickly. Meanwhile on Dominaria, the most famous conjecture appears to revolve around the nature of the Mirari’s magic. To me, it seems pretty simple: It copies spells. But perhaps the formal statement of the conjecture is more intricate.

Either way, the card features one of my favorite pieces of artwork in the set, and the wallpaper makes for a perfect phone background. Thanks, James Arnold!