Is Every Game of Slay the Spire Winnable?

There is a lot of randomness in a game of Slay the Spire. From the cards and relics offered, to the potions dropped, to the map layouts and random encounters, very little remains the same between two playthroughs. This is part of the beauty of the game, forcing you to adapt and make the most out of the resources you have been given in each new attempt. However, with such randomness comes a natural question: can every run of Slay the Spire be won? In this article, I will discuss what I believe to be the current state of knowledge about the answer to this question.

The two interpretations of the question

Before we begin answering, first there is one detail to get straight: the question of whether every game can be won can actually be interpreted in two different ways. Both are valid and interesting questions, and have very different answers. Here are some more precise formulations of each question:

1. “Can a perfect player win every game of Slay the Spire?” (alternatively, “Is there a strategy that achieves a 100% win rate in Slay the Spire?”)
2. “For every seed of Slay the Spire, is there a sequence of decisions that results in a win?”

The first question concerns the performance of a player that makes the theoretically optimal decisions at every point in the game to maximize its win rate, given the information available. Generally, there is no computationally feasible way to actually play optimally, so this question is very difficult to answer. However, in many respects, it is the more useful question to a player looking to maximize their own win rate, as it shows how close they are to playing optimally. As an example for how to answer these types of questions, however, we may look at another game with heavy randomness: Minesweeper.

Minesweeper is a puzzle game where the objective is to find the hidden locations of mines, using the number of surrounding mines as clues. Despite having simple rules, perfect play is still very difficult to achieve. However, even without knowing how to play perfectly, we can still prove that a perfect player cannot win every game. For example, in the Minesweeper game below, the two remaining mines can either be in the squares marked “A” or the squares marked “B”, and a guess must be made. Because such positions are guaranteed to arise in some seeds, not every game of Minesweeper can be won, even following an optimal strategy.

Now, I’ve used the word seed twice, but what does it mean? In a computer game with randomness, a seed is a sequence of letters or numbers which determines all of the randomness in a playthrough. For Minesweeper, this means all the mines are in the same location each time you play a seed. Therefore, each seed is winnable by simply learning which squares do not contain mines. To summarize, for Minesweeper, the answer to our two questions would be:

1. “Can a perfect player win every game?” No.
2. “Can every seed be won?” Yes.

We can go even further, and refine our answer to question 1, by asking how often a perfect player is expected to win. Again, this question is usually impossible to answer exactly, but because any strategy’s win rate is a lower bound for the optimal win rate, we can develop intuition about the answer by analyzing the win rate of players. In particular, for Minesweeper, I found that computer programs have been written to obtain about a 40% win rate on Expert difficulty, which means that with perfect play, you can expect to win at least 40% of the time.

For Slay the Spire, however, answering either question is much more difficult, due to the much higher complexity of the game. For this post, I will mostly focus on the second question, of whether a completely impossible seed exists. This is because the existence of an impossible seed is much easier to definitively establish, and if an impossible seed exists, then obviously a perfect player also cannot win on that impossible seed, also answering question 1. However, at the end of the post, I will also provide my thoughts about the win rate of optimal play.

Randomness in Slay the Spire

Randomness and seeds in Slay the Spire are much more complicated than they are in Minesweeper. However, every random event in a run of Slay the Spire is still determined by the game seed, which can be found on the top right of the screen in-game. Using the same seed and making the same decisions always results in the same outcomes. However, the exact mechanics of the game’s randomness are too complicated to cover here. For the purposes of this post, it is sufficient to note that the following things are almost always the same for a seed, no matter what the player does:

• The map layouts
• The order of the normal and elite fights in each act
• The bonuses offered by Neow at the start of the run (for this post I assume there is a Neow bonus)
• The contents of unknown (?) rooms on the map (e.g. the first unknown room is always an event in this seed)
• The enemy AI for a particular encounter (not counting things like splitting slimes)

And these things can often have a fairly small number of different outcomes depending on prior choices in the run (for example, taking a card from Neow may change the card rewards from the first combat node):

• Card rewards
• Potion drops
• The order of events encountered
• Outcomes of events
• Contents of shops

The important takeaway here is that there is only a very limited amount that the player can manipulate their deck, relics, potions, and encounters, especially early in Act 1. This means that there may really be a possibility for a seed having such bad card, relic, and encounter options that it is completely impossible, no matter what the player does.

What might an unwinnable seed look like?

Since aspects like map layouts, potion drops, events, combats, Neow rewards, and card rewards are mostly static when replaying the same seed, a good starting point is to try to design the worst outcomes for each of these categories, and ask if those outcomes are enough to force the player to lose. For this, we will focus on a particular ruleset of Slay the Spire: Ascension 20 Silent. This choice is made because of the extra difficulty that Silent faces in defeating early elite encounters in Act 1, before the player can tailor their resources towards the specific problems posed by a given seed. In particular:

Silent cannot kill Lagavulin with the starter deck.

By considering the maximum damage that can be dealt with perfect shuffle luck, it is possible to prove that killing Lagavulin is impossible with Silent’s starter deck. She simply cannot deal enough damage before her strength is reduced to -6, and all attacks become ineffective. Let us consider an example fight (Strike=S, Defend=D, Neutralize=N, Survivor=U, Ascender’s Bane=B):

Turn Strength Hand Discard Draw Played Total Damage Dealt
1 0 SDDDDUB   SSSSDN   0
2 0 SSSND DDDDB S SSSN 13 (+21-8)
3 0 SSSND   SSDDDDU SSSN 34 (+21)
4 0 SSDDU SSSND DD SSU 46 (+12)
5 0 SSSDD   SSDDDNU SSS 64 (+18)
6 -2 SSDNU SSSDD DD SSNU 73 (+9)
7 -2 SSSDD   SSDDDNU SSS 85 (+12)
8 -2 SSDNU SSSDD DD SSN 94 (+9)
9 -4 SSSDD   SSDDDNU SSS 100 (+6)
10 -4 SSDNU SSSDD DD SSNU 104 (+4)
11 -4 SSSDD   SSDDDNU SSS 110 (+6)
12 -6 All cards are useless 110

Because we can only play 5 strikes and 1 Neutralize per two turns, the maximum damage dealt to Lagavulin turns out to be 110 on Ascension 20, with the starter deck. However, Lagavulin always has at least 112 HP, so victory is impossible.

The game can force an elite fight on floor 6.

With particularly bad luck, the map generation algorithm can force the player to fight an elite on floor 6, which is the lowest possible floor for elites to spawn. With even more bad luck, the map might not have any shops or rest sites before this forced elite fight. This severely restricts the resources the player can have going into this forced elite fight.

The Silent might not find any useful cards or potions before floor 6

We already know the Silent cannot kill Lagavulin without additional card rewards, due to a lack of damage in her starting deck. Of course, the Silent will not fight Lagavulin until floor 6, but the game may not actually give any useful resources in this period. For example, here is a set of options from Neow which does not help the Silent deal damage:

1. Remove one card
2. Gain 100 gold
3. Gain 250 gold, lose HP
4. Boss swap (for Black Star)

Due to how potion drops work, even after fighting the maximum of five fights before the first elite, there is a 0.72% chance of still not receiving a potion from any of them. Even if a potion drops, many, such as the Dexterity Potion, do not help the Silent deal more damage.

Finally, there are many Silent cards which do not increase the amount of damage that can be dealt to Lagavulin with the basic deck. Some examples include:

• Deflect
• Outmaneuver
• Piercing Wail
• Accuracy
• Blur
• Concentrate
• Escape Plan
• Footwork
• Reflex

It is entirely possible, though extraordinarily unlikely, to be offered only cards from this list in the first five floors of the game, plus from Neow. Therefore, we have the following theoretical scenario for an unwinnable run:

1. Forced Lagavulin elite fight on floor 6, with no shops or rest sites on the first five floors
2. No Neow bonuses, potion drops, or card rewards which help the player deal damage to Lagavulin

If any seed contains all of these components, then that seed cannot be won (on A20 Silent), no matter what the player does. Therefore, it seems likely that an unwinnable seed exists.

Finding an unwinnable seed

Now that we have constructed a set of circumstances that produce an unwinnable seed, the natural next step is to find such a seed. Fortunately, a long time ago, I created a tool for exactly this task. I programmed the search to find seeds with the following parameters, assuming ascension 20 Silent play on PC with full unlocks:

• A forced floor-6 elite fight with no shops or rest sites below it
• No useful Neow rewards or boss relic swaps
• Not many useful damage cards in the first few card rewards (it was too restrictive to filter for no damage cards whatsoever)
• Four forced monster fights before the elite fight, with the fourth fight being a nasty hard pool fight (large slime, gremlin gang, exordium thugs, exordium wildlife)
• No potions that deal damage or increase strength before floor 6

Because the restrictions on potions, damage cards, and even the identity of the elite fight were somewhat relaxed, these seeds were not provably impossible to defeat. To help make up for these relaxed restrictions, I added a requirement for an additional difficult mandatory combat before the first battle. In the end, less than one in 50,000,000 seeds were returned from the search. In order to truly evaluate these seeds, they then had to be manually played through floor 6. I would first attempt the seeds myself, and if I failed to find a way to win in a couple attempts, I would post them as challenges on the Slay the Spire Discord channel. Some readers may have heard about these challenges, which were done three times, once in October of 2019, once in December of 2018, and once while writing this post, in June of 2020. One of the most difficult seeds from 2018 was only solved by taking Grand Finale, but ultimately the entirety of Act 1 was defeated:

The most difficult seed ever found was seed 1Z21MCW (for PC version 2.0). It is just barely possible to kill Lagavulin on floor 6 with at least 9 HP remaining, but it requires a truly unnatural sequence of plays. Lagavulin also spawns with only 113 HP, and the route I took barely killed Lagavulin with a 2-damage strike. If Lagavulin instead had 115 HP, I think it’s plausible that the seed would become impossible. If you want to challenge yourself with a very difficult puzzle, try to beat the seed yourself. I believe Peebs from the Jorbs community Discord channel was the first player to defeat the Corrupt Heart on this seed, proving without a doubt that the seed is not impossible.

I expected that some of the seeds I found would never be solved. In reality, I never found an impossible seed, despite considering several billion seeds. This does not mean that all of these billions of seeds are possible to win, but it does suggest that unbeatable seeds are extraordinarily rare, at least for Silent on Ascension 20. And although I was only searching for seeds where death was inevitable by floor 6, I believe due to the extraordinary number of ways to play through Act 1, if the boss is reachable on a given seed, the entire seed is probably possible to win, even with a heart kill.

Does an unwinnable seed probably exist?

There are only $2^{64}$ possible seeds for Slay the Spire, which is about $1.84*10^{19}$, or 18.4 quintillion seeds. If a random event is much more likely than 1 in $2^{64}$, then we may expect a seed to contain that event. If it is much less likely than 1 in $2^{64}$, there is probably no possible seed which contains the event. So, how likely is our scenario for an unwinnable seed? Assuming there’s five fights available before the elite and at least four forced normal combats, I calculate a rough upper bound on the no-damage Lagavulin scenario as follows, using some quick estimates of each of the components, through various simulations and calculations: $$\begin{equation*} \underbrace{6.6\times10^{-8}}_{\text{No useful cards}} \times \underbrace{1.3\times10^{-4}}_{\text{Bad map layout}} \times \underbrace{0.0072}_{\text{No potions}} \times \underbrace{0.33}_{\text{Lagavulin}} \times \underbrace{0.6}_{\text{Useless event}} \times \underbrace{0.027}_{\text{Useless Neow}} \approx 3.3 \times 10^{-16} \end{equation*}$$

Based on this probability estimate, there is a very high chance that a provably unwinnable seed exists, even if it is too rare to practically find. But even using my current search techniques, I may eventually uncover an impossible seed. Seed 1Z21MCW occurred within the first 4 billion seeds, and was so barren of resources that even a slightly unluckier seed would certainly be truly impossible to defeat.

Do unwinnable seeds exist for other rulesets?

So far, I have been focusing on Ascension 20 (A20) Silent because it is the easiest combination to construct a truly unwinnable seed for. However, it is worth asking the same for other characters and difficulty levels. Based solely on intuition and my experiences with the RNG system in Slay the Spire, here are my current predictions:

• A20 Watcher: An unwinnable seed probably does not exist
• A20 Ironclad and Defect: An unwinnable seed might exist, but it would be much rarer than even on Silent
• A16 and below, all characters except Silent: An unwinnable seed probably does not exist
• A8-20: An unwinnable seed may exist for Silent

Again, when I say here that an unwinnable seed probably does not exist, I mean that while it is possible to contrive of scenarios which cannot be won, those scenarios are so unlikely that they would almost certainly not occur in any of the 18.4 quintillion possible seeds. For example, while it is possible for every Watcher card reward before floor 6 to contain, for example, exactly Collect/Tranquility/Nirvana, this is vanishingly unlikely, and still wouldn’t guarantee a loss to a forced floor-6 super elite, as I believe Watcher can defeat all of them with her starter deck and relics, plus a bit of shuffle luck (which can be further aided in practice by RNG manipulation).

Ironclad, Defect, and Watcher have the advantage over Silent that their starting decks and relics contain much more damage, and therefore early elite fights are less dangerous. This makes it much more likely that they can defeat all early challenges using the resources they are given, even if those resources are exclusively defensive, rather than offensive, in nature. Furthermore, due to how much more difficult fights are on ascension 20 compared to ascension 15, and how difficult it is already to find an unwinnable seed on ascension 20, I think it is likely that no impossible seed exists on ascension 15. However, one possible exception is on Silent, where a forced floor 6 burning elite on, say, Gremlin Nob, might be enough to push a seed over the edge.

What is the win rate of an optimal player?

Based on the above discussion, all I can say so far is that the win rate for an optimal player on A20 Silent must be below 100%. However, without access to optimal play, any other conclusion must be speculation. Also, there are now even more rulesets to consider. For determining the existence of optimal seeds, I believe the decision to fight the Corrupt Heart is practically irrelevant, due to the amount of manipulation that can be done before the Heart must be encountered. However, the Heart makes a real difference in the performance of an actual player, as it represents a loss of resources over the course of the run, plus a pair of very dangerous additional encounters. I will attempt to make an estimate for win rates with and without Heart kills, but some of these combinations have very little data, and so I am forced to speculate. The truth is that nobody knows the real answers to these questions.

Ascension 0 or 1 win rate

I believe all characters have a win rate above 99.99% with optimal play, likely even above 99.999%, with or without the heart. A few months after the game’s early access release, ascension levels did not exist. Back then, I was one of the players competing on the win streak leaderboards, mostly with Celerity, the only streamer who played the game at a similarly high level. Both of us had a combined streak of around 50 with the Ironclad and Silent. The game felt hard to lose when played well, which is why we were pressuring the developers for ways to make the game harder. With the release of ascension levels and a leaderboard reset, both of us basically stopped playing ascension 0 for win rate, and I know of few players who have focused on ascension 0 win rate since. However, because the win rate was already believed to be above 99% with perfect play at the time, and because the game has become easier since that time and game knowledge and strategy have seen so many developments, I am confident that players can win almost every game. The addition of the Heart or additional elites should not change the optimal win rate very much at this difficulty, though ascension 1 may be slightly better for heart kills with perfect play, due to the additional resources available.

Ascension 15 win rate

For a long time, the highest difficulty of the game was ascension 15. Right before ascension 20 was released, there were several streamers competing for ascension 15 win rate or win streaks. The highest recorded win rate I know of was due to Pibonacci, with a 94% win rate over 67 games across Ironclad, Silent, and Defect. Ascension 15 was a remarkably fair game, despite its difficulty, and the best players could win fairly consistently, even without playing perfectly. The game has become slightly easier since that time, and perfect play is significantly stronger than even the best human play, so I estimate the optimal win rate to be significantly higher without Heart kills, at least 98-99% for Ironclad, Silent, and Defect, and at least 99.5% for Watcher.

For Ascension 15, I do expect the Heart to make a real difference in the amount of wins. I know of no recorded statistics for ascension 15 heart kill win rates, so I am forced to speculate based on my experience watching and playing ascension 15. I believe that some percentage of runs which could defeat the act 3 boss would come up slightly short when forced to also fight the heart. The question is how many? I estimate, based slightly on data and largely on intuition, that about 20-30% of those runs which die, die in act 4. If we assume an optimal player has similar performance, and add a slight additional penalty for the difficulty incurred by unlocking act 4, then optimal Heart kill rates might be around 97-98% for Ironclad, Silent, and Defect, and still over 99% for Watcher.

Ascension 20 win rate

This is the difficulty that is mostly attempted by high-level players right now. Unfortunately, it is also frustratingly difficult to find statistics for the performance of ascension 20 players. The only two I can find for ascension 20 heart kills (which aren’t strictly worse than these) are:

• TerrenceM has a 52%/40%/46%/72% win rate with Ironclad/Silent/Defect/Watcher over his last 50 attempts with each character
• Lifecoach has a 94% win rate on Watcher over his last 50 games

Based on watching all of these players, and my own experiences with each character, I expect the optimal win rates to be at least 75% for the Ironclad, 60% for the Silent, 65% for the Defect, and 96% for the Watcher. I suspect these estimates are all also a little bit on the low side. Slay the Spire is an incredibly complex game with an enormous number of decisions to make. Nobody plays perfectly, and everybody makes mistakes in each run which they do not even realize they are making. Very few people also focus exclusively on one character or play to their full potential every game to maximize win rate. Before Lifecoach, I probably would have estimated the Watcher win rate to be closer to 90% when played optimally.

For runs that do not include the heart, I’ll again estimate that about 25% of heart runs which die are deaths in Act 4, and then reduce the deaths by a further 10% due to not having to unlock act 4 or save resources for the Heart fight. This gives us around an 83% win rate for the Ironclad, 73% for the Silent, 75% for Defect, and 97% for the Watcher.

Can I help find an impossible seed?

If you would like to try running a search yourself, check out the hard-seeds branch of SeedSearch. You will have to build it yourself, and the code is very messy in that branch, but it is what I was using to find all of my seeds. An advantage of compiling SeedSearch yourself is that you can also tweak the search parameters however you want, to pursue your own theories. To avoid overlapping with my own search, I also suggest looking at seeds greater than 100 billion. Also, with the configuration I use, less than 1 in 100 million seeds are returned, so don’t expect very many hits. Look at the Readme for the had-seeds branch for more information. Good luck!

Summary

• Impossible seeds probably exist for ascension 20 Silent, but might not exist for other characters or Silent at lower levels.
• No impossible seeds have ever been found, but an impossible ascension 20 Silent seed may be found in the future
• It’s not really possible to extrapolate from any given player to reason about perfect play, but for some reason I still tried
• If I had to guess, I would put the win rate with perfect play well above 50% for all classes on ascension 20 heart, and still over 98% on ascension 15. These numbers are much higher for Watcher.
Written on June 20, 2020