Modern Amulet Mulligans: Can it Titan?

Welcome back to another Donkin’s Digest!
Today we’re going to be talking about something that has been at the top of mind lately: Amulet! For the sake of brevity and a bit of a disclaimer, I won’t explain how this deck wins. It just does. I am not the best Amulet player, but I do think that I have something of value I can share today. This article is going to take a very deep + technical dive on what a keepable hand looks like for Amulet, and we’ll also discuss how new card The Mycosynth Gardens improves our odds of drawing a game winning hand.

What Is Our Goal When Keeping a Hand? 🤔

Can it Titan? That is the question we should all be asking ourselves when we look at an opening hand with Amulet. But what does that mean? Your opening hand should have the possibility of casting a Primeval Titan on turn 4 if on the play, and turn 3 if on the draw. If you look at your opening hand, and it’s completely made, then there’s nothing more to talk about. You just get to keep and be happy. But as we all know, not every hand is perfect. So what do we do? Should we always mulligan? If not, how do we know when to keep when our hand isn’t perfect? This is where things start to get interesting (and pretty complicated)! If our goal is to make a turn-three titan on the draw or a turn-four titan if on the play, we will always have three additional draws towards resources that help us accomplish our goal!

We’re going to start doing a lot of math very soon. So to make sure we are all on the same page, I’m going to share with you my current main deck for Amulet (as of 1/31/2023) and give you a few terms that will help you follow along.

Defining Terms

• Green Bounce-Land (GBL)
  • This is pretty simple. In order to cast Primeval Titan on turn 4 if we are on the play or turn 3 if we are on the draw (just going to refer to that as ‘our goal’ from now on) We are almost always going to need a green bounce-land. There will be some hands where you can get by without one, but for the sake of simplicity, let’s assume we need a green bounce-land to hit our goal.
• Single-Ramp (S)
  • These cards ramp you by one land. One card, One land. In order to reach our goal, we need to do a bit of ramping. The amount of ramp needed to reach our goal correlates to how many amulets we have. A good rule of thumb: No Amulets? 3 Ramp. 1 Amulet? 2 Ramp. 2 Amulets? No Ramp.  This heuristic is pretty good and easy to remember. I hope it helps!
• Double-Ramp (D)
  • These cards let us play more than one land. One card, two (or more) lands. While you may be thinking that Dryad of the Ilysian Grove only lets us play one additional land, it’s important to keep our goal in mind. We want to play Titan on turn three or four. This means that if we play a Dryad on turn two, it will give us two additional land drops before our window for casting Titan when on the draw.
  • When applicable, I’ll abbreviate “Single-Ramp or Double-Ramp” to just Ramp. The situation calls for any ramp spell!
• Amulet (A)
  • The namesake card of the deck. Its important to note that Urza’s Saga is effectively an Amulet in keep analysis as it will turn itself into an Amulet when it’s time to execute. Saga, however is not a live top-deck for achieving our goal, that will come into play later in our analysis.
• Titan (T)
  • The other namesake card of the deck, and the most important one. Not much else to say about this category. Remember to pay for your Pact triggers, and when nature calls, run.

Time To Go Big-Brain Mode

Today I’ll be using a hypergeometric calculator to help asses the odds of drawing a particular card or cards. If you want a quick refresher on how hypergeometric calculations work, you can check out this Youtube video I made some years ago. We are also going to be using multivariate hypergeometric distribution, which you can read about in a great article by Frank Karsten, to analyze specific hands. Let us get to work! Just keep in mind though, that these calculations don’t take many variables into account, such as companion revealed, post-board games, matchup strategy etc. This information is mostly a tool to develop heuristics and good habits quickly — you will learn when and how to deviate from them on your own through practice.

What Are We Missing?

I was going to start this section off talking about the odds of drawing a perfect hand, but it was way too much of a headache, and wasn’t worth the effort. Its instead much more practical to try and calculate the odds of drawing the pieces we are missing. When evaluating an Opening hand, always determine what pieces are missing in order to execute. Remember, we have 3 chances to draw the missing piece!

Here’s a data table I made for us to use. Within this table, “n” represents the number of cards we’ve drawn. for example if n=2, then we’ve drawn two cards (turn 3 on the play, turn 2 on the draw) and the percentage represents the odds we get a particular card within the first two draw-steps. Also, I’ve adjusted the odds for the Amulet column because Saga doesn’t work beyond playing it on turn one.

One thing that is worth noting is that there is a 21% chance of having two Amulet/Sagas in your opening hand. These types of hands generate our most explosive starts. With that in mind, I would personally suggest taking mulligans on hands that have no amulets. The amount of equity you gain by keeping a hand that has zero amulets and needs one vs. taking a mulligan to try and find two amulets is just over 8%. That is to say, a hand with no Amulet makes a Titan + Amulet on time only 8% more often than drawing a starting hand with two amulets. This means you should mulligan hands of seven cards without amulet 92% of the time.

So what hands should we not mulligan to try to find double Amulet? Besides single-Amulet hands that have it all, any hand that is only missing Titan or missing only a bounce land should be a keep. If on the play, you can add a hand that needs any ramp spell to that list. For example, if you’re missing only a Titan, there is a 39% chance to hit the target. Compare that to finding a hand with double-Amulet and you’ll see a 18% difference. 18% is pretty significant, and it’s where I draw the line. Missing a bounce-land has a difference of 26%, meaning you are more than twice as likely to make a turn three or four Titan than you are to mulligan into a hand with double-Amulet.

Why Is Double-Amulet So Good?

Double-Amulet starts are good because they enable us to accomplish our goal without needing any ramp spells. if we untap with two mana and two Amulets, we can play a single bounce-land to make 6 mana. Not only that, but a Titan with two Amulets in play can do a lot of powerful things that a Titan with just one Amulet cannot. For example, it can attack for 20 damage while a Titan with one Amulet can only attack for 8. A hand with two Amulets is also much easier to keep as there are fewer pieces required to percolate. Since a ramp card is not needed to achieve our goal when we have Double-Amulet, the penalty for taking a mulligan does not matter all too much. For example, an opening hand with one Amulet needs a Titan, a bounce-land, and one or two ramp pieces. A hand with two Amulets requires only the bounce land and the Titan. Take a look at the table bellow to see what I’m talking about.

What I’m trying to get at with this table is that you are more likely to open a hand that is going to get there provided it has two Amulets. Not only does it put Titan into play, but it also packs way more of a punch. Greater odds of success with fewer pieces means we mulligan well.

How Does Gardens Change the Math?

The Mycosynth Gardens is an incredible card for Amulet because it gives the deck access to more copies of Amulet of Vigor. At the cost of our Dryad + Valakut back-up plan, we can turn Amulet into a deck that has ‘the nuts’ way more often then before. I’ll let the numbers in this table speak for themselves.

In the words of Minibata (known Amulet Discord frequent) “WOW!” That’s an incredible amount of equity. The odds of drawing two Amulets based on this table (subtracting the odds of double-counted scenarios — statistics mumbo jumbo) results in a roughly 37% chance to open a hand with the ability to at the very least create a scenario that will put a Titan on the battlefield on turn-three or turn-four with two copies of Amulet in play. That’s an increase of about sixteen percent.

This is really significant! If you remember earlier when I mentioned that you should mulligan a hand without amulet about 92% of the time? well now you are more likely to mulligan into a hand with double-Amulet than you are to draw a copy of Amulet in the first three turns! With Gardens, every hand of seven cards without Amulet is a mulligan.

That’s A Lot Of Amulets.dek (2/9/2023)

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Explaining Some of The Card Choices

Before I sign off, I want to give a brief explanation for the changes of the new list.

• We cut Dryad because it doesn’t do busted enough things with double-Amulet when compared to Azusa.
• We added Journey over Azusa 2-4 because Azusa provides interaction windows for cards like Lightning Bolt that can disrupt the combo. It can also find some basic forests which is nifty. This is my sauce and you’re not allowed to hate it if you read this far into the article.
• More T-West and a Pact of Negation due to our ability to generate a lot of extra mana when we have two or more Amulets in play. This gives us increased threat density, protection for when we go all in, and more ways to find Cavern now that Map is out of the picture.
• Sideboard Veil of Summer is a personal taste. I think the card is very powerful and likely useful in a Scam + Creativity field.

Lastly, I want to say thanks to my friend Tucker for helping me with some of the mathematics, and thank you to my good friends in the Amulet discord for their curiosity and thirst for innovation! Lastly, thank you for reading.

Until next time,
Ryan “GR_DONKIN”