Solving the mathematics of Factorio Quality: Asteroid Chunk Recycling

Asteroid chunk recycling is one of the the most effective methods to get legendary resources in Factorio. It is one of the more unique quality grinding setups you’ll ever find, but the calculations needed to assess the efficiency of this method are quite straightforward, as it will be shown in this blog post. Please note that I will assume that the reader has read my analysis of pure recyler loops.

Simply a better version of the pure recycler loop Link to heading

The core mechanic that enables asteroid chunk recycling is the ability to reprocess asteroids chunks with quality modules in the crushers:

At first glance, calculating the efficiency of this setup might seem incredibly complicated, but there is a key simplification that will make our lives a whole lot easier: we don’t care about the type of asteroid we get. After all, we can use the very same reprocessing recipes to get whichever type of asteroid chunk is needed.

With this knowledge in hand, we can simply treat the crusher as a recycler with only 2 module slots and a production ratio of 0.8. We can even use the python functions written in the previous blog post to save a bunch of work:

import numpy as np
from pure_recycler_loop import recycler_loop, recycler_matrix

# Let's lean on the functions from pure_recycler_loop.py

def asteroid_crusher_matrix(quality_chance : float) -> np.ndarray:
    return recycler_matrix(quality_chance, production_ratio=0.8)

def asteroid_crusher_loop(
        input_vector : float,
        quality_chance : float,
        quality_to_keep : int = 5) -> np.ndarray:
    return recycler_loop(input_vector, quality_chance, quality_to_keep, production_ratio=0.8)

We have everything needed to do our statistical analysis, but I still want to see how a crusher production matrix looks like (assume that the quality chance $q$ is 5%):

print(asteroid_crusher_matrix(5))

$$ C_{5} = \begin{bmatrix} 0.76 & 0.036 & 0.0036 & 0.00036 & 0.00004 \\ 0 & 0.76 & 0.036 & 0.0036 & 0.0004 \\ 0 & 0 & 0.76 & 0.036 & 0.004 \\ 0 & 0 & 0 & 0.76 & 0.04 \\ 0 & 0 & 0 & 0 & 0 \\ \end{bmatrix}$$

Statistical analysis Link to heading

A good portion of the statistical analysis I’m about to make was already done in the exact same way in the previous blog post, so it’s understandable if you have a feeling of déjà vu.

Number of normal asteroids needed to craft a legendary asteroid Link to heading

indices = list(range(1, 13)) + [12.4]
ratios = [float(1/asteroid_crusher_loop(1, i)[4]) for i in indices]

print(f"{indices=}")
print(f"{ratios=}")
# indices=[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 12.4]
# ratios=[10658.332544237915, 2916.0000561843026, 1231.4670876459584, 643.8607287181613, 384.0000047970801, 250.63328840475552, 174.71534904370753, 128.05539424941833, 97.6291795243653, 76.83200027295233, 62.06060624879639, 51.22968018283643, 47.698631555272826]

---
config:
  theme: dark
---
xychart-beta
    title "Normal items needed to craft a legendary item"
    x-axis "Quality chance (%)" [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 12.4]
    y-axis "Number of normal items" 0 --> 11000
    line [10658.332544237915, 2916.0000561843026, 1231.4670876459584, 643.8607287181613, 384.0000047970801, 250.63328840475552, 174.71534904370753, 128.05539424941833, 97.6291795243653, 76.83200027295233, 62.06060624879639, 51.22968018283643, 47.698631555272826]

---
config:
  theme: dark
---
xychart-beta
    title "Normal items needed to craft a legendary item (starting from 5%)"
    x-axis "Quality chance (%)" [5, 6, 7, 8, 9, 10, 11, 12, 12.4]
    y-axis "Number of normal items" 0 --> 400
    line [384.0000047970801, 250.63328840475552, 174.71534904370753, 128.05539424941833, 97.6291795243653, 76.83200027295233, 62.06060624879639, 51.22968018283643, 47.698631555272826]

Efficiency of asteroid chunk recycling for every quality level Link to heading

In the charts below, the lines are color-coded to match their quality level’s color (green for common, blue for rare, etc).

indices = list(range(1, 13)) + [12.4]

uncommon  = [float(asteroid_crusher_loop(100, i, 2)[1]) for i in indices]
rare      = [float(asteroid_crusher_loop(100, i, 3)[2]) for i in indices]
epic      = [float(asteroid_crusher_loop(100, i, 4)[3]) for i in indices]
legendary = [float(asteroid_crusher_loop(100, i, 5)[4]) for i in indices]

print(f"{uncommon=}")
print(f"{rare=}")
print(f"{epic=}")
print(f"{legendary=}")
# uncommon=[3.4615384615260094, 6.6666666666465115, 9.642857142830861, 12.413793103415511, 14.999999999958789, 17.41935483866937, 19.687499999959517, 21.818181818139728, 23.82352941171918, 25.714285714234272, 27.49999999995658, 29.189189189135714, 29.83957219245846]
# rare=[0.465976331353591, 1.1111111110940393, 1.8941326530371276, 2.782401902468548, 3.749999999959269, 4.7762747137958605, 5.844726562452839, 6.942148760279247, 8.057958477466594, 9.18367346933998, 10.312499999943766, 11.439006574092652, 11.887957905523466]
# epic=[0.06272758306376097, 0.1851851851747421, 0.3720617711167263, 0.623641805702194, 0.9374999999701643, 1.3096237118236027, 1.7351531982007549, 2.2088655146028944, 2.7254859555867896, 3.2798833818761275, 3.8671874999424296, 4.48285392765304, 4.736111571954229]
# legendary=[0.009382330798851889, 0.034293552803847605, 0.08120395796876448, 0.15531309335629098, 0.26041666663694074, 0.3989893028560425, 0.572359561880939, 0.7809120505857308, 1.0242839375283508, 1.3015410245193235, 1.6113281249513514, 1.9519934519426527, 2.096496209768022]

---
config:
  theme: dark
---
xychart-beta
    title "Efficiency of asteroid chunk recycling"
    x-axis "Quality chance (%)" [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 12.4]
    y-axis "Efficiency (%)" 0 --> 30
    line [3.4615384615260094, 6.6666666666465115, 9.642857142830861, 12.413793103415511, 14.999999999958789, 17.41935483866937, 19.687499999959517, 21.818181818139728, 23.82352941171918, 25.714285714234272, 27.49999999995658, 29.189189189135714, 29.83957219245846]
    line [0.465976331353591, 1.1111111110940393, 1.8941326530371276, 2.782401902468548, 3.749999999959269, 4.7762747137958605, 5.844726562452839, 6.942148760279247, 8.057958477466594, 9.18367346933998, 10.312499999943766, 11.439006574092652, 11.887957905523466]
    line [0.06272758306376097, 0.1851851851747421, 0.3720617711167263, 0.623641805702194, 0.9374999999701643, 1.3096237118236027, 1.7351531982007549, 2.2088655146028944, 2.7254859555867896, 3.2798833818761275, 3.8671874999424296, 4.48285392765304, 4.736111571954229]
    line [0.009382330798851889, 0.034293552803847605, 0.08120395796876448, 0.15531309335629098, 0.26041666663694074, 0.3989893028560425, 0.572359561880939, 0.7809120505857308, 1.0242839375283508, 1.3015410245193235, 1.6113281249513514, 1.9519934519426527, 2.096496209768022]

The fact that we don’t even need a separate chart for epic and legendary efficiency speaks volumes about how good asteroid recycling is. If you have normal T3 quality modules the efficiency will be 0.26%, otherwise if you happen to have legendary T3 quality modules, the efficiency improves by an order of magnitude to 2.0964%¹.

Number of crushers required per quality level Link to heading

When you build these asteroid reprocessing ships you can probably wing it with the amount of crushers you use for every quality level and be completely fine. However, we can do better with the tools we have at our disposal. Let’s assume that we’re feeding a full belt of asteroid chunks to the system:

print(asteroid_crusher_loop(1, 12.4)[:4])
# [3.34224599 0.9973119  0.3973248  0.1582925 ]

As you might recall from the previous blog post, the first four values of this function represent the internal flow of the system. Knowing this, we can infer that in order to fully process a full green belt of asteroid chunks we need:

• Enough common asteroid crushers to process 3.34 belts (~371 crushers).
• Enough uncommon asteroid crushers to process 1 belt (~111 crushers).
• Enough rare asteroid crushers to process 0.4 belts (~45 crushers).
• Enough epic asteroid crushers to process 0.16 belts (~18 crushers).

Building a ship with this many crushers would be a massive undertaking (and it would require a fusion plant big enough to power Liechtenstein²). Instead of constraining the asteroid recycling setup by the input belt, let’s limit ourselves by the number of epic asteroid crushers in the setup:

print(asteroid_crusher_loop(1, 12.4)[:4] / asteroid_crusher_loop(1, 12.4)[3])
# [21.1143675   6.30043693  2.51006712  1.        ]

For every epic asteroid crusher in the setup we need:

• 2.51 rare asteroid crushers.
  • x2.51 increase relative to the number of epic asteroid crushers.
• 6.3 uncommon asteroid crushers.
  • x2.51 increase relative to the number of rare asteroid crushers.
• 21.11 common asteroid crushers.
  • x3.35 increase relative to the number of uncommon asteroid crushers.

My personal asteroid crusher count recommendation Link to heading

Having only one epic asteroid crusher is quite annoying because you have to switch between 3 different recipes on demand. I personally prefer having an asteroid crushing setup with 3 epic asteroid crushers, one for each recipe:

print(asteroid_crusher_loop(1, 12.4)[:4] * 3 / asteroid_crusher_loop(1, 12.4)[3])
# [63.3431025  18.9013108   7.53020136  3.        ]

Here is my personal recommendation:

• 60 common asteroid crushers (I’d organize them in 4 rows of 15).
• 18 uncommon asteroid crushers
• 9 rare asteroid crushers (slight overkill, but I want to keep every number divisible by 3)
• 3 epic asteroid crushers (one for each asteroid type)

Next step: recycler-assembler loops Link to heading

The only quality grinding setup that remains to be analysed is the recycler-assembler loop. That will be the goal of the next (and final) blog post of this series.