The­ory Strategies Ex­plained

Guide writ­ten by spqcey. Con­tri­bu­tions from every­one who has in­nov­ated new strategies and the Amaz­ing Com­munity.

Feel free to use the gloss­ary as needed.

The the­ory sim (sim­u­la­tion) we cur­rently re­com­mend was made by XLII and can be ac­cessed here.

Mod­u­lus Ex­plan­a­tion #

A “%” can be seen in the table of some strategies. The “%” in­dic­ates mod­u­lus, which to sim­plify, is the re­mainder of a di­vi­sion.

So,

13 % 10 = 3

21 % 10 = 1

20 % 10 = 0

Be­cause the mod­u­lus used is mod10, and we use a base 10 num­ber­ing sys­tem, we can look at the last di­git of the level to eas­ily find the res­ult of the mod­u­lus.

So, 2145 % 10 = 5

List of Strategies #

Of­fi­cial The­or­ies

The­ory 1: Re­cur­rence Re­la­tions

T1 is one of the the­or­ies that be­ne­fit the most from act­ive strategies. T1Sol­arXLII xexxx im­proves slightly on T1AI.

• T1Sol­arXLII xexxx  —  Very Act­ive

• T1Ra­tio  —  Act­ive

• T1AI  —  Act­ive

• T1C34  —  Idle

The­ory 2: Dif­fer­en­tial Cal­cu­lus

T2 is the only the­ory without an act­ive strategy after all mile­stones are bought, des­pite at­tempts.

• T2MC  —  Semi-Idle

• T2MS  —  Act­ive

• T2Coast-X%  —  Semi-Idle

• T2  —  Idle

The­ory 3: Lin­ear Al­gebra

T3 has the most vari­ables of any the­ory and has be­ne­fit­ted the most from player-cre­ated strategies. T3SNAX is cur­rently the best idle strategy above e200\(\rho\). T3SNAX2 is sim­ilar to T3SNAX but is de­signed to be QoL com­pat­ible.

• T3Play2  —  Very Act­ive

• T3Play  —  Very Act­ive

• T3SNAX  —  Semi-Idle

• T3SNAX2  —  Act­ive

• T3Noρ1C13rcvNoC12  —  Semi-Idle

• T3Noρ1C13rcv  —  Semi-Idle

• T3NoC11C13C21C33d  —  Act­ive

• T3NoC11C13C21C33  —  Idle

• T3NoC13C33d  —  Act­ive

• T3NoC13C33  —  Idle

• T3NoC11C13C33d  —  Act­ive

• T3NoC11C13C33  —  Idle

• T3NoC13C32C33d  —  Act­ive

• T3NoC13C32C33  —  Idle

• T3C11C12C21d  —  Act­ive

• T3C11C12C21  —  Idle

The­ory 4: Poly­no­mi­als

T4 is the third-strongest the­ory in en­dgame, only be­hind T5 and T6.

• T4C3d66  —  Act­ive

• T4C3d  —  Act­ive

• T4SNAX  —  Semi-Idle

• T4C3dC12rcv  —  Act­ive

• T4C3  —  Idle

• T4C123d  —  Act­ive

• T4C123  —  Idle

• T4C12d  —  Act­ive

• T4C12  —  Idle

• T4C56d  —  Act­ive

• T4C56  —  Idle

• T4C4d  —  Act­ive

• T4C4  —  Idle

• T4C5d  —  Act­ive

• T4C5  —  Idle

• T4  —  Idle

The­ory 5: Lo­gistic Func­tion

T5 is the second-strongest the­ory in en­dgame, only be­hind T6.

• T5AI  —  Act­ive

• T5Idle xexxx  —  Semi-Idle

• T5  —  Idle

The­ory 6: In­teg­ral Cal­cu­lus

T6 is the most power­ful the­ory in en­dgame, with sev­eral play­ers reach­ing well over e1100𝜏 on their main save, some even over e1200𝜏.

• T6AI  —  Act­ive

• T6SNAX x.xxexxx  —  Semi-Idle

• T6NoC34d  —  Act­ive

• T6NoC34  —  Idle

• T6NoC1234d  —  Act­ive

• T6NoC1234  —  Idle

• T6NoC345d  —  Act­ive

• T6NoC345  —  Idle

• T6C4d  —  Act­ive

• T6C4  —  Idle

• T6C3d  —  Act­ive

• T6C3  —  Idle

The­ory 7: Nu­mer­ical Meth­ods

T7 is the third-weak­est the­ory, ahead of T2 and T8. T7PlaySpqceyX is a slightly faster ver­sion of T7Play-25.

• T7PlaySpqceyX  —  Act­ive

• T7Play-25  —  Act­ive

• T7C3d  —  Act­ive

• T7C12d  —  Act­ive

• T7C12  —  Idle

• T7C123d  —  Act­ive

• T7NoC12  —  Idle

• T7NoC123  —  Idle

• T7NoC124  —  Idle

• T7NoC1234  —  Idle

• T7  —  Idle

The­ory 8: Chaos The­ory

T8 is the weak­est the­ory in en­dgame and has the most act­ive strategy of any the­ory (T8PlaySo­l­arSwap) T8 also has the largest num­ber of strategies (20+ total), but only those I have seen re­com­men­ded by the sim are shown here.

Older the­ory sim­u­lat­ors may re­turn the at­tractor as part of the strategy name (Ex­ample: T8R in­stead of T8). As al­ways, we re­com­mend us­ing XLII’s sim. We have since de­cided to re­move the at­tractor let­ter from strategy names.

• T8PlaySo­l­arSwap  —  VERY Act­ive

• T8Play  —  Act­ive

• T8SNAX  —  Semi-Idle

• T8NoC35d  —  Act­ive

• T8NoC35  —  Idle

• T8NoC5d  —  Act­ive

• T8NoC5/T834  —  Idle

• T8NoC3d  —  Act­ive

• T8NoC3  —  Idle

• T8d  —  Act­ive

• T8  —  Idle

Of­fi­cial Cus­tom The­or­ies

In chro­no­lo­gical or­der of re­lease.

Cus­tom The­ory 1: Wei­er­straß Sine Product

The first of­fi­cial cus­tom the­ory, ab­bre­vi­ated WSP, was cre­ated by xelaroc and re­leased on Janu­ary 15, 2022.

• WSPAI  —  Act­ive

• WSPdStopC1  —  Act­ive

• WSPd  —  Act­ive

• WSPStopC1  —  Semi-Idle

Cus­tom The­ory 2: Se­quen­tial Lim­its

The second of­fi­cial cus­tom the­ory, ab­bre­vi­ated SL, was cre­ated by el­lip­sis and re­leased on Janu­ary 22, 2022.

• SLMS  —  Act­ive

• SLMSd  —  Very Act­ive

• SLStopAd  —  Act­ive

• SLStopA  —  Idle

Cus­tom The­ory 3: Euler’s For­mula

The third/​fourth of­fi­cial cus­tom the­ory, ab­bre­vi­ated EF, was first planned by Snaeky, coded by pea­nut, and bal­anced with help by XLII. It was re­leased on May 4, 2022, in the same up­date as CSR2 and the tau cap in­crease.

• EFAI  —  Act­ive

• EFSNAX  —  Semi-Idle

• EFd  —  Act­ive

• EF  —  Idle

Cus­tom The­ory 4: Con­ver­gents to √2

The third/​fourth of­fi­cial cus­tom the­ory, ab­bre­vi­ated CSR2 or CS2, was cre­ated by Sol­arion. It was re­leased on May 4, 2022, in the same up­date as Euler’s For­mula and the CT tau cap in­crease from e100 to e150.

• CSR2XL x.xx  —  VERY Act­ive pre-e500\(\rho\), Act­ive post-e500\(\rho\)

• CSR2d  —  Act­ive

• CSR2  —  Idle

Of­fi­cial The­or­ies

The­ory 1: Re­cur­rence Re­la­tions #

T1Sol­arXLII xexxx #

The “xexxx” is re­turned by the re­com­men­ded the­ory sim­u­lator. For ex­ample, it could say “3e647”.

If you are con­fused about the “%”, read about it here.

Strategy Cred­its:

• Sol­arion and XLII for cre­at­ing and re­fin­ing this strategy off of the ex­ist­ing T1AI.
• rus9384#1864 for im­ple­ment­ing coast­ing and a con­stant pub­lic­a­tion cycle for this strategy.

T1Ra­tio #

Do not buy \(c_1\) or \(c_2\) after e300\(\rho\).

For the \(c_2\) Ra­tio:

For the \(c_3\) Ra­tio:

Strategy Cred­its:

• XLII

T1AI #

T1C34 #

The­ory 2: Dif­fer­en­tial Cal­cu­lus #

T2MC #

Strategy Cred­its:

• rus9384#1864
• XLII, whose the­ory sim­u­lator was used to de­velop and test this strategy.
  

The “MC” in T2MC stands for Mul­tivari­ableCoast.

T2MS #

T2MS is the mile­stone swap strategy for T2, which can be found over here.

T2Coast-X% #

If the sim re­com­mends this, it will re­turn a per­cent­age in the box: T2Coast-☐%

We re­com­mend us­ing T2MC in­stead of this strategy.

T2 #

The­ory 3: Lin­ear Al­gebra #

T3Play2 #

Strategy Cred­its:

• Playspout for cre­at­ing this strategy.
• Sol­arion for \(c_{12}\) over­push idea.
• xelaroc, whose sim was used to verify Playspout’s res­ults
• rus9384#1864 for chan­ging phase 3 start from pub mult 1 to pub mult 1.2, and chan­ging phase 4 start from 2 to 2.4
• XLII, whose sim was used by rus to test T3Play2 modi­fic­a­tions, in­clud­ing the one stated above.

* An earlier ver­sion of the strategy did not buy any \(ρ_1\) vari­ables. This is only around 1% slower than the strategy shown above, which does buy \(c_{31}\) and \(b_1\) in early parts of the strategy. It’s up to you whether you think that a 1% im­prove­ment is worth it or not.

T3Play #

Strategy Cred­its:

• Playspout for cre­at­ing this strategy.
• xelaroc, whose sim was used to verify Playspout’s res­ults

T3SNAX #

Strategy Cred­its:

• Snaeky for the idea.
• XLII for sim­u­lat­ing the strategy and find­ing op­timal ra­tios.

T3SNAX2 #

Strategy Cred­its:

• Snaeky for the idea.
• XLII for sim­u­lat­ing the strategy and find­ing op­timal ra­tios.
  
  It is de­signed to be an easier ver­sion of cur­rent t3 strategies.
  
  This is the only act­ive SNAX strategy.

T3Noρ1C13rcvNoC12 #

Strategy Cred­its:

• xelaroc
• Playspout

T3Noρ1C13rcv #

Strategy Cred­its:

• xelaroc
• Playspout

T3NoC11C13C21C33d #

T3NoC11C13C21C33 #

T3NoC13C33d #

T3NoC13C33 #

T3NoC11C13C33d #

T3NoC11C13C33 #

T3NoC13C32C33d #

T3NoC13C32C33 #

T3C11C12C21d #

T3C11C12C21 #

The­ory 4: Poly­no­mi­als #

T4C3d66 #

When the the­ory sim­u­lator re­com­mends this strategy, it will show something like: T4C3d66 \(q_1\): 377 \(q_2\): 252”.

The levels it shows after \(q_1\) and \(q_2\) are the last levels you should buy them to in this pub­lic­a­tion.

If you are con­fused about the “%”, read about it here.

The “66” in the name T4C3d66 refers to the ⅔ ra­tio \(q_2\) is bought at re­l­at­ive to \(c_3\).

Strategy Cred­its:

• XLII
• rus9384#1864

T4C3d #

T4SNAX #

Strategy Cred­its:

• Snaeky
• XLII

T4C3dC12rcv #

T4C3 #

T4C123d #

This strategy is typ­ic­ally seen at lower \(\rho\) and is even­tu­ally out­paced by T4C3d.

T4C123 #

This strategy is typ­ic­ally seen at lower \(\rho\) and is even­tu­ally out­paced by T4C3.

T4C12d #

T4C12 #

T4C56d #

T4C56 #

T4C4d #

T4C4 #

T4C5d #

T4C5 #

T4 #

The­ory 5: Lo­gistic Func­tion #

T5AI #

This act­ive strategy does not work well with this method of show­ing the­ory strategies. Please visit this guide page to learn how to per­form T5AI.

T5Idle xexxx #

IM­PORT­ANT: Do act­ive \(c_2\) buy­ing (buy­ing \(c_2\) when \(q\) is­n’t in­creas­ing, and x10 for the first few seconds of the pub­lic­a­tion) for the first few minutes of the pub­lic­a­tion.

The “xexxx” is re­turned by the re­com­men­ded the­ory sim­u­lator. For ex­ample, it could say “3e647”.

Strategy Credit:

• XLII

T5 #

The­ory 6: In­teg­ral Cal­cu­lus #

T6AI #

If you are con­fused about the “%”, read about it here.

T6SNAX x.xxexxx #

The “x.xxexxx” is re­turned by the re­com­men­ded the­ory sim­u­lator. For ex­ample, it could say “3e647”.

Strategy Cred­its:

• Snaeky
• XLII

T6NoC34d #

T6NoC34 #

T6NoC1234d #

Note: If you need to, you can idle re­cov­ery by auto­buy­ing \(q_1\) and \(r_1\) un­til the the­ory has passed its pre­vi­ous pub­lic­a­tion point. This will re­duce your rates by only ~1% re­l­at­ive to buy­ing \(q_1\) and \(r_1\) at e1 ra­tio the en­tire pub­lic­a­tion.

T6NoC1234 #

T6NoC345d #

This strategy is some­times seen at lower taus but not at higher taus.

T6NoC345 #

This strategy is some­times seen at lower taus but not at higher taus.

T6C4d #

T6C4 #

T6C3d #

Both T6C3d and T6C3 don’t have \(r_1\) or \(r_2\) lis­ted be­cause the two strategies only ap­pear on the sim be­low the mile­stones that give \(r_1\) and \(r_2\).

T6C3 #

Both T6C3d and T6C3 don’t have \(r_1\) or \(r_2\) lis­ted be­cause the two strategies only ap­pear on the sim be­low the mile­stones that give \(r_1\) and \(r_2\).

The­ory 7: Nu­mer­ical Meth­ods #

T7PlaySpqceyX #

If there is no num­ber after T7­PlaySpqcey, then use this table:

If there is a num­ber after T7PlaySpqcey (Ex­ample: T7PlaySpqcey10 or T7PlaySpqcey100), then use this table:

While the “X” is only dir­ectly in the \(c_2\) buy­ing con­di­tion, be­cause the \(c_1\) buy­ing con­di­tion is based on the cost of \(c_2\), the “X” also in­dir­ectly af­fects how much \(c_1\) is bought.
Re­gard­less of the “X” value, the \(c_1\) buy­ing con­di­tion re­mains “When cost is 1/​10 of \(c_2\) cost”

The num­ber after T7PlaySpqcey oc­curs more at lower \(\tau\) val­ues.

If you are con­fused about the “%”, read about it here.

Strategy Cred­its:

• Playspout for \(q_1\), \(c_4\), \(c_5\), \(c_6\) buy­ing strategy
• spqcey for the ad­di­tion of \(c_3\)
• Snaeky for num­ber after T7PlaySpqcey for if \(c_2\) and \(c_1\) are bought, and if so, what ra­tio
• XLII for sim­u­lat­ing Snaeky’s idea

T7Play-25 #

T7Play-25 has a “25” due to the two 25% (¼) ra­tios

Strategy Cred­its:

• Playspout

T7C3d #

T7C12d #

T7C12 #

T7C123d #

T7NoC12 #

T7NoC123 #

T7NoC124 #

T7NoC1234 #

T7 #

The­ory 8: Chaos The­ory #

T8PlaySo­l­arSwap #

Mile­stone Swap:
Every 34 seconds, re­move one level from the first mile­stone then im­me­di­ately add the level back.

If you are con­fused about the “%”, read about it here.

Strategy Cred­its:

• Playspout for the vari­able buy­ing strategy
• Sol­arion for the mile­stone swap idea

T8Play #

Strategy Cred­its:

• Playspout

T8SNAX #

Strategy Cred­its:

• Snaeky
• XLII

T8NoC35d #

If you are con­fused about the “%”, read about it here.

T8NoC35 #

T8NoC5d #

T8NoC5/T834 #

T8NoC3d #

T8NoC3 #

T8d #

T8 #

Of­fi­cial Cus­tom The­or­ies

Wei­er­straß Sine Product #

WSPAI #

Strategy Cred­its:

• xelaroc

WSPdStopC1 #

If you are con­fused about the “%”, read about it here.

The “Ra­tio” stated in \(c_1\) is de­term­ined as fol­lows, where \(\rho\) is \(\rho\) at the end of your last pub­lic­a­tion:

Strategy Cred­its:

• xelaroc for test­ing/​cre­at­ing the strategy
• Snaeky for the stop \(c_1\) idea
• XLII for some modi­fic­a­tions in­clud­ing adding mod­u­lus and \(c_1\) changes

WSPd #

Strategy Cred­its:

• xelaroc

WSPStopC1 #

Strategy Cred­its:

• xelaroc for sim­u­lat­ing it/​adding it to the sim
• Snaeky for the stop \(c_1\) idea
• XLII for \(c_1\) modi­fic­a­tion

Se­quen­tial Lim­its #

SLMS #

Mile­stone Swap:

The mile­stone state will be de­scribed with four num­bers. These de­scribe the pri­or­ity of the mile­stone. For ex­ample, 4→3→1→2 means you should pri­or­it­ize buy­ing fourth mile­stone levels first, then after the fourth mile­stone is maxed, then you buy third mile­stone levels, and then first mile­stone, then second mile­stone.

There are 3 mile­stone states:

State 1: 4→3→1→2 (pri­or­it­izes boost for the \(e-\gamma\) term)

State 2: 2→1→4→3 (pri­or­it­izes boost for \(\dot\rho_2\))

State 3: 1→2→4→3 (pri­or­it­izes boost for in­stant­an­eous \(\rho\) gain, aka \(\rho_2\) ex­po­nent)

This strategy swaps mile­stones de­pend­ing on how far away the next \(b_1\) or \(b_2\) up­grade is.

The goal of this strategy is:

1. Buy \(b_1\)/\(​b_2\)
2. Swap mile­stones into State 1 to fully util­ize the boost from the new \(b_1\)/\(​b_2\) up­grade.
3. After some time (the ex­act amount of time will be de­scribed later in the strategy), we want to swap to State 2. This way we boost \(\dot\rho_2\) so we get more \(\rho_2\) quicker.
4. When we are closer to the next \(b_1\)/\(​b_2\) up­grade, swap to State 3 so we util­ize the boost that we just got for \(\rho_2\). This will get us to the next \(b_1\)/\(​b_2\) up­grade a lot quicker.
5. Re­peat.
   
   

This may seem dif­fi­cult to ex­ecute, but pub­lic­a­tions in SL are 1 to 1.5 hours long after e50\(\rho\)-e300\(\rho\). This means there can be up to 5-10 minute gaps between \(b_1\)/\(​b_2\) up­grades later in the pub­lic­a­tion. You will also get used to the swap­ping ra­tios and dur­a­tion quickly be­cause of the con­sist­ent pub­lic­a­tion lengths. At the very end of pub­lic­a­tions, you will also not have to mile­stone swap, as we will use State 3 only, to get the last \(\rho\) boost be­fore pub­lish­ing.

As long as \(\rho\) < e175, we will swap between the three states, after that its just the first two states.

Note: NEVER swap into State 2 after 4.5 pub­lic­a­tion mul­ti­plier. Do State 3 in­stead for that time.

The ra­tios for swap­ping are as fol­lows, where you enter the next state when the ra­tio: \(\frac{min(b_1 cost, b_2 cost)}{\rho}\) is lower than the ra­tio provided un­der the header:

cost in the table refers to min­imum(\(b_1\) cost, \(b_2\) cost)

This may seem con­fus­ing, but let’s take the first row as an ex­ample.

In the first phase (e25\(\rho\) - e50\(\rho\)) you should have your mile­stones in State 1 un­til \(\rho\) is 1/​5 of min(\(b_1\) cost, \(b_2\) cost).
Then, swap to State 2 un­til \(\rho\) is 1/​4 of min(\(b_1\) cost, \(b_2\) cost).
After that, swap to State 3 un­til you get the up­grade.

Note: You do not have to fol­low those ra­tios ex­actly. It does not make much dif­fer­ence if you do it slightly dif­fer­ently.

The num­bers are just im­port­ant to give the idea for around where you want to swap.
e.g. in e200\(\rho\)+ range, you only want to swap when close to next \(b_1\)/\(​b_2\). and e75-e150\(\rho\) you swap very early.
These num­bers also tell you that you never want to be in State 2 for a long time.

Vari­able Buy­ing

Strategy Credit:

• XLII

SLMSd #

For the mile­stone swap­ping de­tails, read the above strategy, SLMS. This strategy only mod­i­fies the vari­able buy­ing strategies.

If you are con­fused about the “%”, read about it here.

Strategy Credit:

• XLII

SLStopAd #

If you are con­fused about the “%”, read about it here.

The boost from doub­lings (\(a_1\) to \(a_2\), \(b_1\) to \(b_2\)) is rarely enough to jus­tify choos­ing this strategy over the idle strategy SLStopA.

Strategy Cred­its:

• xelaroc
• Playspout
• rus9384#1864 for \(b_1\), \(b_2\) coast­ing at 6 pub­lic­a­tion mul­ti­plier
• XLII for buy­ing ra­tios change and in­teg­ra­tion of mod­u­lus

SLStopA #

Strategy Cred­its:

• xelaroc
• Playspout
• rus9384#1864 for \(b_1\), \(b_2\) coast­ing at 6 pub­lic­a­tion mul­ti­plier

Euler’s For­mula #

EFAI #

When the the­ory sim­u­lator re­com­mends this strategy, it will show something like: “EFAI \(q_1\): 1223 \(q_2\): 144 \(a_1\): 382”. The levels it shows after each vari­able are the last levels you should buy them to in this pub­lic­a­tion.

If you are con­fused about the “%”, read about it here.

Strategy Cred­its:

• The Amaz­ing Com­munity (EFAI is a product of a bunch of ideas from vari­ous play­ers)

EFSNAX #

Strategy Cred­its:

• Snaeky
• XLII
• Gaunter for buy­ing \(a_1\) after re­cov­ery when past e150\(\rho\) strategy

* If be­low e150\(\rho\), don’t auto­buy \(a_1\) after re­cov­ery.

EFd #

EF #

Con­ver­gents to √2 #

CSR2XL x.xx #

The “x.xx” is re­turned by the re­com­men­ded the­ory sim­u­lator. For ex­ample, it could say “CSR2XL 2.85”, which would mean turn off auto­buy on all vari­ables at 2.85 pub­lic­a­tion mul­ti­plier.

Mile­stone Swap (pre e500\(\rho\))

Start pub­lic­a­tion with mile­stones in \(c_2\)/\(​c_2\) ex­po­nent.

Swap to \(q_1\) ex­po­nent when one of these is true:

• \(\rho\) × Ra­tio > \(c_2\) cost
• \(\rho\) × \(\frac{Ra­tio}{2}\) > \(n\) cost
• \(\rho\) × 2 > \(q_2\) cost and pub­lic­a­tion mul­ti­plier > 1

Fol­low the sim’s ad­vice for when to start coast­ing.

The “Ra­tio” stated in the \(q_1\) swap con­di­tions is de­term­ined as fol­lows, where \(\rho\) is \(\rho\) at the end of your last pub­lic­a­tion:

Strategy Cred­its:

• XLII

CSR2d #

CSR2 #

FI #

FId #