Cantrip Calculations

When constructing blue decks in Legacy, playing four copies of Brainstorm is a given for almost all strategies. The only somewhat common exception to this is Merfolk, which values mana efficiency in relation to damage dealt more highly than the overall consistency Brainstorm offers.

If additional filtering effects are wanted, the choice is mostly between Sensei’s Divining Top and Ponder. Top is usually chosen in decks that are looking to play longer games, whereas Ponder gets the nod over Top in more aggressive strategies. This makes perfect sense, considering Ponder is much more efficient short-term, whereas Top is so powerful long-term that it can work as an engine on its own. (see: Miracles)

Let’s focus on the decks that choose to run Ponder over Top. If those are looking for additional draw spells, Gitaxian Probe is usually the first choice. Again, this makes sense, seeing as the decks that benefit the most from the situationally free spell are either combo decks or those that rely on velocity, like Delver strategies.

Sometimes, there are decks that want even more of this kind of effect. These decks then turn to Preordain to fill out their slots. The number of people who run Preordain over Ponder is extremely low. What I want to do today is take a look at the digging power of these cards (Brainstorm, Ponder, Preordain & Gitaxian Probe) individually and how to optimise the power of our cantrips if we have multiples.

My assumptions are fairly simple. There are x cards left in our deck and we’re digging for one of y cards, where that may either be the amount of copies of a single card we have left in our deck or the amount of cards from a certain group of cards we have left in our deck. Obviously both x and y are positive integers with (y > 0) & (x-y > 3). (If (y = 0) we are guaranteed to miss and if (x-y ≤ 3) we are guaranteed to hit with any two cantrips.)

First, let’s briefly tackle Gitaxian Probe. In terms of pure digging power, it should be clear that Probe is the worst of the bunch. It only goes one card deep, whereas all the other cantrips go three cards deep (with the exception of Ponder, which does a little more). But Gitaxian Probe can be cast without costing any mana while also providing information, which in turn helps you to make better decisions with all your other cantrips.

Apart from corner case scenarios where we want to hide one or more cards on the top of our library to draw into later, Probe should always be the first cantrip we cast. If we have Gitaxian Probe and Ponder, casting Probe first will also increase our chances of hitting because we go four cards deep and then two, rather than three and two.

With that out of the way, let’s look at a couple scenarios.

1. We have one cantrip.

With Brainstorm, here’s the probability of not finding what we’re looking for:

Brainstorm

(A short explanation on why I calculate like this. If you are looking for the probability of finding at least one copy, that’s simply (1-p) and this is also the easiest way to calculate that. The reason I opt to display the probabilities for missing on our cantrips is that I will later compare probabilities and this makes for much cleaner and simpler comparisons.)

The probability for Preordain is exactly the same, but for Ponder, we also have to factor in the ability to shuffle and draw a random card:

Ponder

Conclusion: Clearly, Ponder is the best card in this scenario. Duh.

2.1 We have two cantrips – 2 Brainstorm

If we have just two copies of Brainstorm, there aren’t any choices, assuming we need to cast both in one turn and don’t get to see any additional cards. We simply go four cards deep:

Preordain+Draw

Just for completeness’ sake, let’s also take a look at what happens if we have two Brainstorms and a fetchland. Please note that I am working under the assumption that the land we fetch is not one of the cards we are looking for. Quite simply, there are two possibilities: Leading with Brainstorm or leading with the fetchland. Here are the odds of missing when leading with the fetchland:

Fetch+Brainstorm+Brainstorm

By comparison, the probability of missing when we use our fetchland after the first Brainstorm:

Brainstorm+Fetch+Brainstorm

The solution:

Fetch+Brainstorm+BrainstormVsBrainstorm+Fetch+Brainstorm

Conclusion: If we have two Brainstorms and a fetchland, we should use our fetchland after the first Brainstorm.

2.2 We have two cantrips – 1 Brainstorm, 1 Ponder

For a combination of Brainstorm and Ponder, here’s the probability of missing when casting Ponder first:

Ponder+Brainstorm

And this is what we get when casting Brainstorm first:

Brainstorm+Ponder

The solution is actually incredibly simple:

Ponder+BrainstormVsBrainstorm+Ponder

(In this scenario, it is better to cast Ponder first.)

Interestingly, because the next two cards we see when leading with Ponder won’t increase our odds, casting Brainstorm first actually becomes better when we see two or more additional cards. Here are the odds for casting Brainstorm, casting Ponder and then drawing two more cards:

Brainstorm+Ponder+Draw+Draw

Again, the comparison is very simple:

Ponder+Brainstorm+2DrawVsBrainstorm+Ponder+2Draw

Again, we should also consider if having a fetchland changes the way we should play.

If we start by activating our fetchland, we already know that leading with Ponder will be better, as that will only change the size of our library and the calculations we have done before have already shown us what is true for all x.

Similarly, casting Ponder first and then using a fetchland before casting Brainstorm is strictly worse than using the fetchland before Ponder. So the real question is whether it is better to lead with Brainstorm, use a fetchland and then Ponder or to fetch first and then to proceed as before.

The probability of missing when leading on Brainstorm:

Brainstorm+Fetch+Ponder

The probability of missing when cracking the fetchland first and then casting Ponder followed by Brainstorm:

Fetch+Ponder+Brainstorm

And this is the result:

Brainstorm+Fetch+PonderVsFetch+Ponder+Brainstorm

Conclusion: Ponder should be cast before Brainstorm if we draw up to one additional card. If we draw two or more additional cards, Brainstorm should be cast first. If we have a fetchland, a Ponder and a Brainstorm with no additional draws, we should lead with cracking the fetchland.

2.3 We have two cantrips – 1 Brainstorm, 1 Preordain

For this one, we don’t have to do any math to know which is the better option. If we cast Brainstorm first, we get to see four cards, but if we lead with Preordain, we get to see six cards. However, we want to be able to compare them with the other combinations to find out which combination of cantrips we should cast if we’re given a choice.

Preordain+Brainstorm

The answer to our next question, what to do when we also have a fetchland, is pretty simple, but let’s still include both options. First, our probability of missing when cracking the fetch first and then casting Preordain:

Fetch+Preordain+Brainstorm

Now, our probability of missing for using Brainstorm first and then fetching before casting Preordain:

Brainstorm+Fetch+Preordain

The solution:

Fetch+Preordain+BrainstormVsBrainstorm+Fetch+Preordain

Conclusion: If we have a combination of Preordain and Brainstorm, we should always cast Preordain first.

2.4 We have two cantrips – 2 Ponder

Even simpler than our last combination, we don’t have any options here at all. But again, we still want to be able to do our comparison later on. Here’s the probability of missing without a fetchland:

Ponder+Ponder

And here with a fetchland:

Fetch+Ponder+Ponder

2.5 We have two cantrips – 1 Ponder, 1 Preordain

Let’s start with Ponder into Preordain, because that’s simply the same as Ponder into Brainstorm:

Ponder+Brainstorm

And now Preordain into Ponder:

Preordain+Ponder

And the solution:

Ponder+PreordainVsPreordain+Ponder

Again, because there’s no Brainstorm involved, we don’t need to figure out when to crack our fetchland, but we still want to be able to do comparisons:

Conclusion: If we have Ponder and Preordain, we should cast Preordain first.

2.6 We have two cantrips – 2 Preordain

Finally, we have another combination without options and it’s also the simplest of all combinations, because we just go six cards deep. This is the probability of missing on two Preordains if we don’t have a fetchland:

Preordain+Brainstorm

…and the probability of missing if we do have one:

Fetch+Preordain+Brainstorm

Summing Up

I realise I have omitted scenarios where we can cast three or more cantrips, but those are much more complex than the ones we looked at today. I might work on them at some point in the future. Further, we have not actually looked at the comparisons between the combinations of cantrips we could cast, but I have tiered them.

The first tier is 2 Ponder and 1 Preordain 1 Ponder. The second tier of combinations is Ponder + Brainstorm and Preordain + Brainstorm/Preordain. The lowest tier is just Brainstorm + Brainstorm, which is always the worst combination of cantrips to use for digging. I will get into how to further split these tiers shortly, but first a couple notes. Firstly, these tiers stay the same with our without fetchlands, as those don’t have any effects on cantrip power outside of double Brainstorm scenarios, considering they just remove one card from our library and we have calculated with variable libraries. Secondly, Preordain into Preordain can be worse than Preordain into Brainstorm with hands where we benefit from positioning a card on top of our library.

And once more, please keep in mind these tiers are only looking at digging power and there might be other reasons to use or hold cantrips, such as positioning cards on the top of our library, not filling our graveyard at the earliest opportunity or saving life on Gitaxian Probe and fetchlands.

With that out of the way, here’s a spreadsheet I compiled to compare leading Preordain and leading Ponder when casting another cantrips afterwards. The x-values are displayed horizontally, increasing from left to right, whereas y-values increase from top to bottom. For readability, I have split the spreadsheet into 5 x 10 grids. For each cell, I subtract the probability of missing with Ponder into another cantrip from the probability of missing with Preordain into another cantrip. Therefore, a negative number indicates leading Ponder is more likely to miss (these cells have green background), whereas a positive number (red background) means leading Preordain is more likely to miss. Black background means x – y ≥ 4 is violated so the combination of values for x and y in that cell doesn’t make sense. Finally, there is one blue cell, for x=12 and y=1. In that case, it does not make a difference which one you cast, as the probabilities are equal.

Now, before you learn the entire spreadsheet in order to maximise your cantrip power, let me make it a little easier for you:

-for a library of 54 cards or more, leading Preordain is better so long as x – y ≤ 5

-for a library of 28-54 cards, leading Preordain is better so long as x – y ≤ 6

-for a library of 20-27 cards, leading Preordain is better so long as x – y ≤ 7

-for a library of 14-19 cards, leading Preordain is better so long as x – y ≤ 8

-for a library of 13 cards, leading Preordain is better so long as x – y ≤ 9

-for a library of 12 cards or less, leading Preordain is always better, except for x = 12 and y = 1 (as described above)

Thanks for reading.

J

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10 thoughts on “Cantrip Calculations

  1. excellent article. i had spent so much time focused on brainstorm, fetch, ponder because it provides the best overall card selection that i had considered that im losing value in pure digging situations. any chance we’ll get an article like this with top math soon? i know that’s a large endeavor.

    1. Thank you very much, glad you liked it.
      The Brainstorm, fetch, Ponder scenario is the main selling point of this article, in my opinion. We’re also so used to that order that just stick to it in all situations.

      Care to explain a little what you are looking for in regards to Top? I might be missing something here, but it seems to me like you just activate it whenever you can and don’t really need to cast any additional cantrips anymore, but I might be missing something here.

      1. i guess it can be simplified rather easily. i guess what i what im really not positive about is when it is most efficient to top (assuming we dont have access to infinite mana) while digging for a card in between cantrips but when i actually think about it those situations are less complex then i thought (top after brainstorm or topped ponder sees one extra, top after shuffled ponder or fetch see 3 extra, top after preordain see 3 unless both were scried to top, plus you need to rethink life decisions about putting top and preordain in the same deck

        off of that, it also would be interesting to see the mana efficiency of keeping drawing cantrips to continue the chain (ie if you ponder into a ponder, you theoretically see one extra card for the cost of one mana if you choose not to shuffle on the original ponder)

        1. I think the decisions involving Top come down to logic rather than probabilities. Top basically functions like a Brainstorm, only it’s reusable. But you also want to keep it on the board as long as possible, so I think the actual contents of your hand are also a very important factor, i.e. if the cantrips have any use while they’re in your hand.

          So the question isn’t really how can see the most cards (which is more or less answered here by treating Top activations as Brainstorm) but rather if seeing additional cards is worth replacing your cantrips with random cards.

  2. So you write that

    And this is a big mistake. Here is an example for you: x=30, y=2. 25/27>(28/30)*(27/29)

    1. Yes, you are right, good thing you noticed that. The mistake probably came about when I rewrote and restructured the entire article the day I published it. Also, I did not do any of the math myself, so I might have typed in something incorrectly at some point. The good news is that this is only relevant when you have more cantrips than you can cast and doesn’t influence the order in which you would cast your cantrips. I am going to edit all of that tomorrow.
      Either way, thank you very much for pointing this out.

    2. Again, thank you for pointing out the mistake. I have fixed the article and included a spreadsheet to shed light on the situations where Preordain into Ponder is better vs. the situations where Ponder into Ponder is better.

  3. Welcome)
    “-for a library of 54 cards or more, leading Preordain is better so long as x – y ≤ 5”
    You mean x-y≥5

    1. No, that would mean leading Preordain never leads to worse results for x ≥ 54. The spreadsheet calculates like this: P(missing Preordain lead) – P(missing Ponder lead). Therefore, as stated in the article, a negative number means missing with Ponder is more likely. For 54 cards, this means y ≥ 49. This means x – y ≤ 5. (Specifically: 54-49=5 & 54-50=4, which violates x – y ≥ 5.)

  4. now I understand what you mean, but I think many people won’t. In our case X is 53 or less (because we draw 7 cards at the begining), and usually x-y>9. So most of the times PONDER IS BETTER THAN PREORDAIN

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