# The Definitive Guide to Sequencing in Pokemon TCG — Applying Mathematics & More

Sequencing is one of the most important aspects of any trading card game. It’s especially relevant in the Pokemon TCG and its current formats because the opponent has no ways to interfere during your turn (cue reference to Power Spray from 2010). Knowing how to sequence properly will make you a better player.

Since this is my first piece here, I’ll give a brief introduction. My name’s Xander Pero and I’ve been playing since 2009. My best achievement is third place at the 2017 World Championships, and I’m also a three-time Regional champion. Outside of Pokemon, I’m a university student studying industrial engineering.

This is part one of two on sequencing. In these two articles, I’ll go over some of the classic sequencing problems in Pokemon. While this list is certainly not exhaustive, it should provide enough context for you to learn the reasoning behind each decision.

Learning the guidelines of sequencing is more useful than memorizing each situation.

Unlike other lines of decision-making during a turn, sequencing is easily optimizable. An example of something that takes longer calculation time would be mapping your six Prizes; do I Boss’s Orders the Dedenne-GX now or use Professor’s Research to set up further? Experienced players will have a feel for what’s correct in that situation, but to actually map out the probabilities would take too much time.

My friends know that I’m a huge math nerd. Optimization, data analysis – actually anything math-related – are my favorite subjects within my major. Here I’ll sometimes use specific probabilities to establish a quantifiable difference between proper and improper sequencing; other times I’ll heuristically describe differences. Think of specific probabilities as a citation and an introduction to probability theory.

## General Principles

The first step in any problem is to identify what you want to achieve. The order for any scenario will be determined based on what you want.

As a quick example, let’s say you win the game if you hit Boss’s Orders off of Stellar Wish. You have one copy remaining in a 15-card deck. You should first play Tag Call – and any other cards that decrease cards remaining – before using Stellar Wish. With three cards removed, the probability of winning increases from 0.33 to 0.42! Though that increase is less than 0.1, many decisions throughout the game add up.

The above example is elementary because most decisions don’t result in an immediate win or loss. Each outcome will tip a scale one way or the other. Also, you’ll usually have cards that interact with each other and different types of effects. Acro Bike versus Trainers’ Mail, the first example, has overlapping card effects. Acro Bike can get any card and Trainers’ Mail is limited to Trainers, obviously. They overlap in the Trainers section. An example of non-overlapping cards is Guzma & Hala and Viridian Forest.

Another piece of information to consider is card priority. Let’s say you’d like an Energy and a Boss’s Orders. You absolutely need the Energy to use any attack, so the Energy is more important than the Boss’s Orders. You may have a higher probability of hitting both with one ordering, but a higher probability of hitting an Energy card with a different ordering. This falls under risk aversion and digging for the Energy specifically may be the better play.

A final thing to remember is that certain decks will have different sequencing guidelines, like Mewtwo & Mew-GX and Welder decks. You might maximize the probability of drawing the Naganadel-GX by using Viridian Forest before Professor’s Research, but you reduce the probability of ending the turn with Naganadel-GX in your discard pile. The same logic applies for Welder and Giant Hearth. You can increase the probability of drawing Welder by using Giant Hearth before Dedenne-GX, but you’re unlikely to draw two Fire Energy off of Dedechange as well. Playing a few games with the deck will reveal these exceptions.

## Acro Bike versus Trainers’ Mail

Though both of these cards are relegated to Expanded, their interaction is timelessly instructive. Below is a list with different scenarios. Assume you only have one Acro Bike and one Trainers’ Mail in hand with 10 cards left in deck. You have no other means to dig for cards in your hand, either.

1. Looking for a non-Trainer: Trainers’ Mail first
2. Looking for a Trainer: Acro Bike first
3. Looking for a non-Trainer and Trainer with one of each remaining: either order
4. Looking for a non-Trainer and a one-of Trainer with more non-Trainers remaining: either order
5. Looking for a one-of non-Trainer and a Trainer with more Trainers remaining: Trainers’ Mail first
6. Looking for a non-Trainer and Trainer with two or more remaining of each: Trainers’ Mail first is always equal or better

Note: When I say, “with more X remaining,” I mean outs to that card. Example: One Fire Energy, Two Boss’s Orders

Now I imagine, being a CFB Pro subscriber, you’ve played Pokemon for a long enough time to realize the first two of six teachings. When you want a non-Trainer, you can potentially thin an unwanted Trainer from your deck with the Trainers’ Mail. The second one takes a bit more thinking but makes intuitive sense. By playing Acro Bike and then Trainers’ Mail, you see six unique cards. When you play Trainers’ Mail first, you’re potentially shuffling the “whiff” cards back to the top of the deck.

The latter four are a bit more confusing. The probabilities become rougher with more than one card because you have to consider hitting at least one out, and also conditioning on the Acro Bike when it’s first. The probability of hitting a Trainer decreases when Acro Bike is played first and discards a Trainer with its second card.

A quick exercise proves the third point. Once again there are 10 cards in deck, with one non-Trainer and one Trainer needed to win.

### Evidence of #3

```Acro Bike first: p(AB)*p(TM) -> (2/10)*(4/9) = 0.0889

Trainers’ Mail first: p(TM)*p(AB) -> (4/10)*(2/9) = 0.0889```

Both of these come out to the same thing. Even though Acro Bike discards two cards from the deck, leaving eight cards in deck, the second term is still 4/9 because the Trainer could’ve been in the second slot of the Acro Bike.

The fourth is absolutely baffling. My first guess was that playing Acro Bike first was correct, but it actually doesn’t matter. The easiest way to see this is pattern is by looking at the other extreme: nine non-Trainers and one Trainer. With Trainers’ Mail first, you have a 4/10 chance and then a guaranteed Acro Bike. With Acro Bike first, the initial setup has a 4/10 chance that the Trainer is in slots three through six.

For my sections on the remaining situations, I’ll assume that the needed non-Trainer is an Energy so that it’s easier to understand. “Energy” and “Trainer” are more distinguishable!

### Evidence of #4

#### Case One: 10 cards, 2 Energy, 1 Trainer

```Acro Bike first: (2/10*1/9)*(4/8) + (2*2/10*8/9)*(7/8*4/8) = 1/6 = 0.1667

p(2E)*p(TM|2E) + p(1E)*p(TM|1E)

Trainers’ Mail first: (4/10)*(1-(7/9)*(6/8)) = 0.1667

p(TM)*p(E>=1)```

#### Case Two: 14 cards, 4 Energy, 1 Trainer

```Acro Bike first: (4/14*3/13)*(4/12) + (2*4/14*10/13)*(9/10*4/12) = 2/13 = 0.1538

Trainers’ Mail first: (4/14)*(1-(9/13)*(8/12)) = 2/13 = 0.1538```

I calculated two cases to show it wasn’t a single occurrence. The difficult math arises when you play Acro Bike first because you draw cards without replacement. You have to factor in the possibility of hitting two Energy or one Energy with the Acro Bike by conditioning: splitting up the outcomes into two cases based on their respective probability. Because the second Energy lies outside of the Acro Bike in case 1E, the Trainer has a probability 7/8 of being outside of the Acro Bike, then a probability of 4/8 of being in the Trainers’ Mail.

The fifth one is intuitive once you realize what changes by adding an additional Trainer card. You increase the probability of a successful Trainers’ Mail, meaning that the scarcer resource is the Energy. As the number of Trainer outs increases towards its upper limit, the probability of hitting the Trainers’ Mail approaches one. If the probability is one, it’s as if you’re searching it out of the deck. If you can understand why you’d play Trainers’ Mail for seven outs of 10, then you can understand why it’s first for five or three.

Six is the trickiest to think about. A simple case is two Energy and two Trainers. Using the knowledge of previous examples, you can guess that Trainers’ Mail first makes sense. Now imagine there are four Energy and two Trainers – it’s not so intuitive then. Or even better, how about 40 cards, 10 Energy, and two Trainers?

### Evidence of #6

#### Case 1: 10 cards, 2 Energy, 2 Trainers

```Acro Bike first: (2/10*1/9)*(1-6/8*5/7*4/6*3/5) + (2*2/10*8/9)*((6/8)*(1-6/8*5/7*4/6*3/5) + (2/8)*(4/8)) = 0.2714

p(2E)*p(T>=1|2E) + p(1E)*(p(1E,0T|1E)*p(T>=1|1E,0T) + p(1E,1T|1E)*p(T>=1|1E,1T))

Trainers’ Mail first: (1-8/10*7/9*6/8*5/7)*(1-7/9*6/8) = 0.2778

p(TM>=1)*p(E>=1)```

#### Case 2: 10 cards, 4 Energy, 2 Trainers

```Acro Bike first: (4/10*3/9)*(1-6/8*5/7*4/6*3/5) + (2*4/10*6/9)*((4/6)*(1-6/8*5/7*4/6*3/5) + (2/6)*(4/8)) = 0.4730

Trainers’ Mail first: (1-8/10*7/9*6/8*5/7)*(1-5/9*4/8) = 0.4815```

#### Case 3: 40 cards, 10 Energy, 2 Trainers

```Acro Bike first: (3/52)*(1-36/38*35/37*34/36*33/35) + (5/13)*((28/30)*(1-36/38*35/37*34/36*33/35) + (2/30)*(4/38)) = 0.08682

Trainers’ Mail first: (1-38/40*37/39*36/38*35/37)*(1-29/39*28/38) = 0.08694```

The calculations aren’t much different than #4. In layman’s terms, it looks more complicated because there’s the additional conditioning of a Trainer card being lost during the initial Acro Bike. It also takes longer to write out with the at least one probability appearing multiple times, indicated by the (1-x) components.

If there’s one thing to take away from this, it’s to always play Trainers’ Mail first when you’re looking for a non-Trainer card in a vacuum.

Trainers’ Mail first is always equal or better – even when you also want a Trainer – in maximizing the probability of hitting what you need.

As a disclaimer, certain situations may call for playing Acro Bike first because of the discard knowledge or uncertainty, which I’ll cover later. In those situations, play Acro Bike first. These are merely guidelines that provide the tools to make better informed decisions. It’s up to you to apply the guidelines correctly.

## Great Ball versus Quick Ball

Let’s get a break from that nonsense and look at something easier to digest. You’ve played Eternatus VMAX, so you know about Great Ball, Quick Ball, and Pokemon Communication. We’ll ignore Pokemon Communication for now because it’s a universal Pokemon search card. Instead let’s break down the different scenarios between Great Ball and Quick Ball.

1. Looking for an Evolution: Quick Ball first
2. Looking for an Evolution and a Basic: Quick Ball first
3. Looking for two Basics: Great Ball first

These are pretty easy to break down. When you need an Eternatus VMAX, you should thin your deck of a Basic before going with Great Ball. #2 follows the same principle. You can guarantee the Basic you desire with Quick Ball, then look for the Eternatus VMAX with Great Ball afterward. Lastly, #3 will happen when you’re looking to set up your board initially or in reaching for a Knock Out. Because playing Quick Ball immediately would actually remove a desired target from the deck, use Great Ball first.

Here’s a tougher scenario in which there’s more uncertainty. Imagine you need a Galarian Zigzagoon and Eternatus VMAX to win the game. However, you have Crobat V left in your deck that could draw you more cards. In which order do you play your “Ball” cards?

This is an example where the redo potential of Crobat V trumps the small gain in percentage from thinning. You lose out on equity by playing Great Ball first but gain useful information. If you miss the Eternatus VMAX, you can get Crobat V and hope to draw into it or another search card. Not to mention the fact that you can potentially hit Crobat V or Galarian Zigzagoon off of the first Great Ball, meaning you’re already halfway there before Dark Asset.

## Sequencing Based on Uncertainty

Uncertainty plays a large part in sequencing, especially in the early and middle stages of a game. Sequencing based on uncertainty is different than sequencing based on probabilities.

The main reason to focus on uncertainty over probability is to lessen the probability of your bad outcomes.

Maximin and minimax strategies from game theory make a brief connection here. If your optimal policy is to reduce your worst possible outcome, you’re employing a maximin/minimax strategy.

Uncertainty has a rough hierarchy. Plain draw cards are the most uncertain; you have no idea what you’ll get. Cards like Acro Bike or Underground Expedition – those that can grab a subset of seen cards – are next. These have some degree of certainty with choice, but don’t search the entire deck. Trainers’ Mail and Jirachi are next, but in my opinion could fall before the previous category. It’s certain that you’re locked into a Trainer, but uncertain in that you could whiff. My logic is that Acro Bike sees fewer cards, meaning the outcome is more uncertain. If Trainers’ Mail said reveal the top 20 cards of your deck, you can all but guarantee the Trainer card you want. Direct search cards are next: Quick Ball, Viridian Forest, Tag Call, and many others. Lastly, there’s universal search: Teammates, Rosa, and Computer Search.

The biggest deck in recent history that sequences on uncertainty is Archie’s Blastoise. Its only goal was to land the coveted turn-one Archie’s Ace in the Hole. This was achieved by reducing uncertainty from the opening hand such that you could map out a way to dump your entire hand. Battle Compressor first, then Trainers’ Mail, then Ultra Ball. Battle Compressor found their way in there, depending on what you needed to hit. If you needed to hit more dig cards off of a Trainers’ Mail, you’d play Battle Compressor first to dump the Blastoise, Archie’s Ace in the Hole, and an Energy. If you were uncertain, then you’d hold it.

## Effects in Play

The last piece of information to consider is knowledge of cards already on the board. These include Viridian Forest, Persian-GX, Jirachi, Oranguru – anything with an effect once in play. The rule of thumb is to leave these until as late in the turn as makes sense. This gives you the most information for these effects. Would you rather be forced into using Stellar Wish without knowledge of your cards from Professor’s Research? Before seeing those seven cards, you don’t know if you would draw a Switch or Scoop Up Net to move the Jirachi. Additional information always helps in decision-making.

That being said, Abilities shouldn’t always be left for last. If you’re lacking direction, such as how to play out a turn, an immediate Stellar Wish is grounding. With one uncertain effect out of the way, you can play out the rest of your turn optimally. You might not have hit the card you wanted, but you’re moving forward with the turn in a safer fashion. You could’ve dug for the Reset Stamp with Professor’s Research and Stellar Wish, but after missing the Stellar Wish you decided to play Marnie instead.

### Oranguru

Perhaps the most versatile on the board Ability is Primate Wisdom. At worst, you get the card you would’ve drawn a turn early. In other scenarios, you can combo putting the card back with other effects, namely Mr. Mime and Zacian V. There are three scenarios worth considering.

1. Intentionally leaving a card on top of the deck
2. Combining Primate Wisdom with another Ability/card
3. Returning a bad card to the deck

Intentionally putting a card on top of the deck has great uses in countering an opposing Marnie. If you’re worried about your Boss’s Orders being Marnie’d away, simply put it on top of the deck at the end of your turn. Obviously, this would require leaving Primate Wisdom until after you’ve finished messing with the deck.

Next comes combining Primate Wisdom with other effects. The moment you do this during your turn depends on the uncertainty of the other effect. If you’re pairing it with Smooth Over, you should do other things first if possible. If it’s the Mr. Mime plus Jirachi Prism combo, then any time works.

Lastly, you should shuffle the deck afterward when putting back a bad card. Without doing this, you essentially nullify your draw for the following turn. If the current situation calls for different sequencing you must weigh it against the value from an additional card on the following turn. That being said, it’s usually correct to always leave a shuffle effect for after Primate Wisdom.

Case: 15 cards in deck, one Boss’s Orders to win on this turn or next turn

```Cherish Ball then Primate Wisdom: 1/14 = 0.0714

Primate Wisdom then Cherish Ball: 2/15 = 0.1333```

Regardless of your capabilities to dig on the following turn, saving the Cherish Ball until after Primate Wisdom increases your probability of drawing the card you want. In any given turn, you increase your options on the following turn by saving the shuffle effect.

## Conclusion

One of Pokemon’s many intricacies is in sequencing. Cases with a specific goal will call for direct maximization with probability, whereas others without a clear plan usually follow the path of reduced uncertainty. It’s important to know both cases.

Part two of sequencing will release later this week and will go over more scenarios. Similar to Acro Bike versus Trainers’ Mail, there will be an in-depth section on Dedenne-GX and Professor’s Research. Expect additional cards and their interactions too!

Scroll to Top