This is a mathematical-heavy, min-maxing guide for advanced players. I will explain how the manipulation works, why it works, how it affects your RNG and how to execute it! Some things in this specific guide can be slightly off.

Contents

## Guide to Scroll-RNG Manipulation

### Introduction

Still, I am confident in the correctness of the general information displayed here (the formula was provided by a developer and some of the effects of the manipulation were guessed after discussing how it works). Nonetheless, the principle and effectiveness of Scroll Manipulation stay the same!

**Abbreviations**

- RNG = Random Number Generation (aka. “chance”).
- Dual Scroll(s) = Scrolls that can be assigned to only two colours.
- Main-stat = The stat/colour that has the highest amount of scrolls.
- Off-stats = The stats/colours that have fewer scrolls than your main-stat.
- GoodScroll = A Dual Scroll that has your main-stat(s) (aka. “lucky scroll”).
- GoodChance = The chance for finding a goodScroll.
- BadScroll = A Dual Scroll that doesn’t have your main-stat(s).
- BadChance = The chance for finding a badScroll.

### What is Scroll-Manipulation?

Scroll Manipulation is the act of changing the RNG of the Dual Scrolls so you receive more lucky Dual Scrolls that increase your main-stat instead of your off-stats. It is a min-maxing strategy but isn’t required to win the game!

Dual Scrolls are usually skewed towards your off-stats (so Tactics & Survival if you go for Brutality), but you can change this behaviour by leaving the first biome while having two stats with an equal amount of stats (e.g. 3-3-1).

This is only viable for the transition from Prisoners’ Quarters to any second biome as having two equal stats past the first biome is a bad idea and makes this strategy not worth it. If you were successful in the manipulation then this trick can increase your main-stat by up to 2 stats for the price of having less hp (because your off-stats are lower).

### How do you perform Scroll-Manipulation?

Scroll Manipulation is done by investing the two Power Scrolls in Prisoners’ Quarters into one colour of your choice (as you would do in any normal run) and having a 2-off-stat amulet (0-2-0) so your stats look like 3-3-1!

Starting from 3BC, every amulet will have 2 stats in PQ because their minimum level will be 6 (which adds the second stat). This amulet can then be switched out in the 2nd biome.

That’s the ideal set-up, but what if you can’t find a 2-off-stat amulet? Is investing a Power Scroll into an off-stat worth it? It’s still a gamble and you can’t control the chances to the extent that you can guarantee something will or will not happen, but what are the numbers?

If you find an amulet with one main-stat and one off-statt (1-1-0) and split your Power Scrolls (so you’re 3-3-1), then you will need to find **two** lucky Dual Scrolls to make a net profit. The probability of ending up with that profit of one extra scroll is ~1.71x higher if you manipulate the RNG than if you don’t (29.75% for 2 lucky Dual Scrolls with manipulation vs. 17.36% for *at least *1 lucky Dual Scroll without manipulation)! Keep in mind you get 2 lucky scrolls in only ~30% of your attempts and you might even end up with one less main-stat in ~20% of your attempts where you won’t find any lucky Dual Scrolls!

What about investing a Power Scroll into your off-stat so it’s *higher* than your main-stat? You have the option to do this when you find a Rift or Cursed Chest in PQ (your stats might look like 3-4-1), or do it with an amulet that increases both your off-stats by one each (0-1-1, and split the Power Scrolls between Brut & Tact → 2-3-2).

You will come out neutral (by getting one lucky Dual Scroll) 8.68% of the time but you will gain one extra scroll (aka. get two lucky Dual Scrolls) in 91.12% of your attempts!

Investing one Power Scroll into your off-stat (here Tactics → 3-4-1) gives you a chance of 91.12% to gain one extra stat, which is ~5.25x higher than if you invest that scroll into your main-stat (here Brutality → 4-3-1; 17.36% to get at least 1 lucky Dual Scroll, of which 16.53% is for 1 lucky Dual Scroll and 0.83% is for 2 lucky Dual Scrolls).

### Base Equation

The base numbers for good/bad scrolls are:

- x: goodChance = 10
- y: badChance = 100

The numbers for good- and badScrolls change depending on how many bad- and goodScrolls you find:

- goodChance (x) = x + (40 • badScroll)
- badChance (y) = y • (3 • goodScroll)

This equation is used to calculate the probabilities for both scroll types:

- goodScroll: x ÷ (x + y)
- badScroll: y ÷ (x + y)

Note: This has been provided by a dev.

### Calculations

With the equation above we can calculate the probabilities of 2, 1 and 0 scrolls that increase your main stat!

The base chances for a singular goodScroll & badScroll without manipulation are:

- goodScroll: 10 ÷ (10 + 100) = 1/11 ≈ 9.1%
- badScroll: 100 ÷ (10 + 100) = 10/11 ≈ 90.9%

Probabilities without manipulation:

- 2 lucky scrolls: (1/11 • 1/11) = 1/121 ≈ 0.83%
- 1 lucky scroll: (1/11 • 10/11) + (10/11 • 1/11) = 20/121 ≈ 16.53%
- 0 lucky scrolls: (10/11 • 10/11) = 100/121 ≈ 82.64%

For the manipulation probabilities (for 3-3-1) you need to split the 10/11 (chance for a badScroll) into half because you have two main-stats in that scenario (with 3-3-1 stats you can find Brut-Tact (goodScroll), Brut-Surv (lucky badScroll because it contains Brut) and Tact-Surv (unlucky badScroll because it *doesn’t* contain Brut)).

Probabilities for a singular scroll with manipulation (3-3-1):

- goodScroll: 10 ÷ (10 + 100) = 1/11 ≈ 9.1%
- Lucky badScroll: 50 ÷ (10 + 100) = 5/11 ≈ 45.45%
- Unlucky badScroll: 50 ÷ (10 + 100) = 5/11 ≈ 45.45%

Probabilities with manipulation (3-3-1):

- 2 lucky scrolls: (1/11 • 1/11) + (1/11 • 5/11
**¹**) + (5/11**¹**• 1/11) + (5/11**¹**• 5/11**¹**) = 36/121 ≈ 29.75% - 1 lucky scroll: (1/11 • 5/11
**²**) + (5/11**²**• 1/11) + (5/11**¹**• 5/11**²**) + (5/11**²**+ 5/11**¹**) = 60/121 ≈ 49.59% - 0 lucky scrolls: (5/11
**²**• 5/11**²**) = 20/121 ≈ 20.66%

¹ = badScroll **with** your main-stat

² = badScroll **without** your main-stat

For the manipulation probabilities (for 3-4-1) you need to split the 1/11 (chance for a goodScroll) into half because you have two off-stats in that scenario (with 3-4-1 stats you can find Brut-Tact (lucky goodScroll), Brut-Surv (badScroll) and Tact-Surv (unlucky goodScroll)).

Probabilities for a singular scroll with manipulation (3-4-1):

- Lucky goodScroll: 5 ÷ (10 + 100) = 1/22 ≈ 4.55%
- Unlucky goodScroll: 5 ÷ (10 + 100) = 1/22 ≈ 4.55%
- badScroll: 100 ÷ (10 + 100) = 10/11 ≈ 90.91%

Probabilities with manipulation (3-4-1):

- 2 lucky scrolls: (1/22
**¹**• 1/22**¹**) + (1/22**¹**• 20/22) + (20/22 • 1/22**¹**) + (20/22 • 20/22) = 441/484 ≈ 91.12% - 1 lucky scroll: (1/22
**¹**• 1/22**²**) + (1/22**²**• 1/22**¹**) + (1/22**²**• 20/22) + (20/22 • 1/22**²**) = 42/484 ≈ 8.68% - 0 lucky scrolls: (1/22
**²**• 1/22**²**) = 1/484 ≈ 0.21%

¹ = goodScroll **with** your main-stat

² = goodScroll **without** your main-stat

### Summary of All Probabilities

Without Scroll-Manipulation:

- 2 lucky scrolls: 1/121 ≈ 0.83%
- 1 lucky scroll: 20/121 ≈ 16.53%
- 0 lucky scrolls: 100/121 ≈ 82.64%

With Scroll-Manipulation (3-3-1):

- 2 lucky scrolls: 36/121 ≈ 29.75%
- 1 lucky scroll: 60/121 ≈ 49.59%
- 0 lucky scrolls: 20/121 ≈ 20.66%

With Scroll-Manipulation (3-4-1):

- 2 lucky scrolls: 441/484 ≈ 91.12%
- 1 lucky scroll: 42/484 ≈ 8.68%
- 0 lucky scrolls: 1/484 ≈ 0.21%

### Notes

- We assume that all Dual Scrolls for a biome spawn with the same probabilities and that the chance doesn’t instantly change after one of the two Dual Scrolls has been assigned.
- The Dual Scrolls will be assigned in the loading screen to the actual biome. You can’t switch out your amulet in the passage/transition area to the biome as there is another loading screen coming up!

- If you happen to have two main-stats (3-1-3), then only Dual Scrolls that can be assigned to
**both**main-stats will be considered a goodScroll. This means your chance to get additional lucky Dual Scrolls will be more likely!- Remember, the base chance to get a goodScroll is 9.1% per scroll, but with manipulation you can get a Dual Scroll that increases your main-stat without negatively affecting your RNG! We referred to those scrolls as “lucky badScrolls”!
*I found confirmation from a dev, this is indeed how it works.*

- Even if you do not pick up a Dual Scroll or invest a goodScroll into an off-stat, the probabilities will still change for the worse as the chances are calculated upon entering a biome!
*This is an educated**guess**, but it does make sense when considering how much simpler it is to code this. Technically you could test not picking up the goodScrolls to see if you get more of them in the next biomes.*

I hope you found this useful!

## Be the first to comment