Ex­po­nen­tial Idle Guides

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 ori­gin­ally made by XLII, now main­tained by the sim team 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 T1Ra­tio.

The­ory 2: Dif­fer­en­tial Cal­cu­lus #

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

The­ory 3: Lin­ear Al­gebra #

T3 has the most vari­ables of any the­ory and has be­ne­fit­ted the most from player-cre­ated strategies. T3Play2 is cur­rently the best act­ive strategy above e350\(\rho\). T3ρ2C23d is an easier ver­sion be­ing 10-15% slower.

T3SNAX is cur­rently the best idle strategy above e200\(\rho\). T3SNAX2 is sim­ilar to T3SNAX but is de­signed to be QoL com­pat­ible.

The­ory 4: Poly­no­mi­als #

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

The­ory 5: Lo­gistic Func­tion #

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

The­ory 6: In­teg­ral Cal­cu­lus #

T6 is the most power­ful the­ory in en­dgame, with sev­eral play­ers reach­ing well over e1300𝜏 on their main save, some even over e1400𝜏.

The­ory 7: Nu­mer­ical Meth­ods #

T7 is the third-weak­est the­ory in en­dgame, ahead of T2 and T8.

The­ory 8: Chaos The­ory #

T8 is the weak­est the­ory in en­dgame and has the most act­ive strategy of any main the­ory (T8PlaySo­l­arSwap).

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.

Cus­tom The­ory 2: Se­quen­tial Lim­its #

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

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.

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.

Cus­tom The­ory 5: Frac­tional In­teg­ra­tion #

The fifth/​sixth of­fi­cial cus­tom the­ory, ab­bre­vi­ated FI, was first planned by Snaeky, coded by Gen, and bal­anced with help by XLII. It was re­leased on April 1, 2024, in the same up­date as Fractal Pat­terns and the CT tau con­tri­bu­tion rate in­crease by x4.

Cus­tom The­ory 6: Fractal Pat­terns #

The fifth/​sixth of­fi­cial cus­tom the­ory, ab­bre­vi­ated FP, was cre­ated by XLII. It was re­leased on April 1, 2024, in the same up­date as Frac­tional In­teg­ra­tion and the CT tau con­tri­bu­tion rate in­crease by x4.

Cus­tom The­ory 7: Riemann Zeta Func­tion #

The sev­enth of­fi­cial cus­tom the­ory, ab­bre­vi­ated RZ, was cre­ated by prop and re­leased on Decem­ber 15, 2024.

The­ory 1: Re­cur­rence Re­la­tions #

T1Sol­arXLII xexxx #

T1Sol­arXLII xexxx
Be­low xexxx Above xexxx
q1 When q1cost × (6 + lvl % 10) < q2 cost
And q1 cost × (15 + lvl % 10) < c4 cost
And ρ > 5 × cost
q2 When ρ > 1.11 × cost
c1 See T1Ra­tio
c2 See T1Ra­tio
c3 See T1Ra­tio
c4 ✔️

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:

T1Ra­tio #

T1Ra­tio
q1 When ρ > 10 × cost
q2 When ρ > 1.11 × cost
c1 When ρ > 10 × c2ra­tio × cost
c2 When ρ > c2ra­tio × cost
c3 When ρ > c3ra­tio × cost
c4 ✔️



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

For the \(c_2\) Ra­tio:

\(c_2\) Ratio

For the \(c_3\) Ra­tio:

c3 Ra­tio
ρ < e300 1
e300 < ρ < e450 1.1
e450 < ρ < e550 2
e550 < ρ < e655 5
ρ > e655 10


Strategy Cred­its:

T1C34 #

T1C34
q1 ✔️
q2 ✔️
c1
c2
c3 ✔️
c4 ✔️

T1C4 #

T1C4
q1 ✔️
q2 ✔️
c1
c2
c3
c4 ✔️

The­ory 2: Dif­fer­en­tial Cal­cu­lus #

T2MC #

T2MC
Pub­lic­a­tion Mul­ti­plier be­low 1150 Pub­lic­a­tion Mul­ti­plier 1150-2250 Pub­lic­a­tion Mul­ti­plier 2250-2900 Pub­lic­a­tion Mul­ti­plier 2900-4650 Pub­lic­a­tion Mul­ti­plier above 4650
q1 ✔️ ✔️ ✔️ ✔️
q2 ✔️ ✔️ ✔️
q3 ✔️ ✔️
q4 ✔️
r1 ✔️ ✔️ ✔️ ✔️
r2 ✔️ ✔️ ✔️
r3 ✔️ ✔️
r4 ✔️

Strategy Cred­its:

The “MC” in T2MC stands for Mul­tivari­ableCoast.

T2MC­Alt2 #

T2M­C­Alt2
Pub­lic­a­tion Mul­ti­plier be­low 550 Pub­lic­a­tion Mul­ti­plier 550-2050 Pub­lic­a­tion Mul­ti­plier 2050-2700 Pub­lic­a­tion Mul­ti­plier 2700-3500 Pub­lic­a­tion Mul­ti­plier above 3500
q1 ✔️ ✔️ ✔️ ✔️
q2 ✔️ ✔️ ✔️
q3 ✔️ ✔️
q4 ✔️
r1 ✔️ ✔️ ✔️ ✔️
r2 ✔️ ✔️ ✔️
r3 ✔️ ✔️
r4 ✔️

Strategy Cred­its:

The “MC” in T2MC stands for Mul­tivari­ableCoast.

T2MC­Alt3 #

T2M­C­Alt3
Pub­lic­a­tion Mul­ti­plier be­low 750 Pub­lic­a­tion Mul­ti­plier 750-1700 Pub­lic­a­tion Mul­ti­plier 1700-2650 Pub­lic­a­tion Mul­ti­plier 2650-3700 Pub­lic­a­tion Mul­ti­plier above 3700
q1 ✔️ ✔️ ✔️ ✔️
q2 ✔️ ✔️ ✔️
q3 ✔️ ✔️
q4 ✔️
r1 ✔️ ✔️ ✔️ ✔️
r2 ✔️ ✔️ ✔️
r3 ✔️ ✔️
r4 ✔️

Strategy Cred­its:

The “MC” in T2MC stands for Mul­tivari­ableCoast.

T2MS #


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

T2QS #

T2QS
q1 ✔️
q2 ✔️
q3 ✔️
q4 ✔️
r1 ✔️
r2 ✔️
r3 ✔️
r4 ✔️

Start the pub­lic­a­tion with mile­stones in the 1→2→3→4 or­der.

When your pub­lic­a­tion mul­ti­plier reaches a cer­tain point, switch your mile­stones to the 3→4→1→2 or­der.

The pub­lic­a­tion mul­ti­plier when you should switch is de­term­ined as fol­lows, where \(\rho\) is \(\rho\) at the end of your last pub­lic­a­tion:

Pub­lic­a­tion Mul­ti­plier
ρ < e75 10
e75 < ρ < e150 200
e150 < ρ < e200 600
e200 < ρ < e225 100
ρ > e225 25

T2 #

T2
q1 ✔️
q2 ✔️
q3 ✔️
q4 ✔️
r1 ✔️
r2 ✔️
r3 ✔️
r4 ✔️

The­ory 3: Lin­ear Al­gebra #

T3Play2 #

T3­Play2
Re­cov­ery up un­til e1 away from re­covered * From e1 away from re­covered un­til 1.2 Pub Mult Pub­lic­a­tion Mult 1.2-2.4 Coast­ing (pub mult > 2.4)
b1 When ⅛ of c31 cost
b2 When ⅕ of c32 cost When ⅕ of c32 cost When ⅛ of c12 cost ✔️
b3 When ⅛ of c23 cost When ⅛ of c23 cost When ⅛ of c23 cost ✔️
c11
c12 When 1/​100 of c32 cost When 1/​100 of c32 cost ✔️ ✔️
c13
c21
c22 When ⅖ of c32 cost When ⅖ of c32 cost When ⅛ of c12 cost
c23 ✔️ ✔️ ✔️ ✔️
c31 ✔️
c32 ✔️ ✔️ When ⅛ of c12 cost
c33 When 1/​10 of c23 cost When 1/​10 of c23 cost



Strategy Cred­its:



* 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 #

T3­Play
Re­cov­ery (pub mult < 1) Pub­lic­a­tion Mult 1-2 Coast­ing (pub mult > 2)
b1 When cost is ⅛ of c31 cost When cost is ⅛ of c31 cost
b2 When cost is 2/​9 of c12 cost, 4/​9 of c22 cost When cost is ¼ of c32, c12 cost, ½ of c22 cost ✔️
b3 When cost is ⅛ of c23 cost, 4/​9 of c33 cost When cost is ⅛ of c23 cost, ½ of c33 cost When cost is ⅛ of c23 cost, ½ of c33 cost
c11
c12 ✔️ ✔️ ✔️
c13
c21
c22 When cost is ½ of c32, c12 cost When cost is ½ of c32, c12 cost ✔️
c23 ✔️ ✔️ ✔️
c31 ✔️ ✔️
c32 ✔️ ✔️ ✔️
c33 When cost is 9/​32 of c23 cost When cost is ¼ of c23 cost When cost is ¼ of c23 cost

Strategy Cred­its:

T3SNAX #

T3SNAX
Re­cov­ery (pub mult < 1) Tau Gain (pub mult > 1)
b1 ✔️
b2 ✔️ ✔️
b3 ✔️ ✔️
c11
c12 ✔️ ✔️
c13
c21
c22 ✔️ ✔️
c23 ✔️ ✔️
c31 ✔️
c32 ✔️
c33 ✔️

Strategy Cred­its:

T3SNAXdC12 #

T3SNAXd­C12
Re­cov­ery (pub mult < 1) Tau Gain (pub mult > 1)
b1 ✔️
b2 ✔️ ✔️
b3 ✔️ ✔️
c11
c12 When cost × 100 < c32 cost ✔️
c13
c21
c22 ✔️ ✔️
c23 ✔️ ✔️
c31 ✔️
c32 ✔️
c33 ✔️

Strategy Cred­its:

T3SNAX2 #

T3SNAX2
Re­cov­ery (pub mult < 1) Tau Gain (pub mult > 1)
b1 When cost is 1/​10 of ρ1
b2 When cost is ⅓ of ρ2 When cost is ⅓ of ρ2
b3 When cost is ⅕ of ρ3 When cost is ⅕ of ρ3
c11
c12 When cost is 1/​100 of ρ2 ✔️
c13
c21
c22 ✔️ When cost is ⅛ of ρ2
c23 ✔️ ✔️
c31 ✔️
c32 ✔️
c33 When cost is 1/​10 of ρ3

Strategy Cred­its:

T3ρ2C23d #

T3ρ2C23d
b1
b2 When cost is ⅓ of min(c12 cost, c22 cost, c32 cost)
b3 When cost is 1/​9 of c23 cost
c11
c12 ✔️
c13
c21
c22 ✔️
c23 ✔️
c31
c32 ✔️
c33

T3ρ2C23 #

T3ρ2C23
b1
b2 ✔️
b3 ✔️
c11
c12 ✔️
c13
c21
c22 ✔️
c23 ✔️
c31
c32 ✔️
c33

T3ρ2C23C33d #

T3ρ2C23C33d
b1
b2 When cost is ⅓ of min(c12 cost, c22 cost, c32 cost)
b3 When cost is 1/​9 of c23 cost
c11
c12 ✔️
c13
c21
c22 ✔️
c23 ✔️
c31
c32 ✔️
c33 ✔️

T3ρ2C23C33 #

T3ρ2C23C33
b1
b2 ✔️
b3 ✔️
c11
c12 ✔️
c13
c21
c22 ✔️
c23 ✔️
c31
c32 ✔️
c33 ✔️

T3NoC11C13C21C33d #

T3NoC11C13C21C33d
b1 When cost is 1/​8 of c31 cost
b2 When cost is 1/​5 of other ρ2 vari­ables’ cost (c12, c22, c32)
b3 When cost is 1/​8 of c23 cost
c11
c12 ✔️
c13
c21
c22 ✔️
c23 ✔️
c31 ✔️
c32 ✔️
c33

T3NoC11C13C21C33 #

T3NoC11C13C21C33
b1 ✔️
b2 ✔️
b3 ✔️
c11
c12 ✔️
c13
c21
c22 ✔️
c23 ✔️
c31 ✔️
c32 ✔️
c33

T3NoC13C33d #

T3NoC13C33d
b1 When cost is 1/​10 of min(c21 cost, c31 cost)
b2 When cost is ¼ of min(c12, c22, c32)
b3 When cost is 1/​10 of c23 cost
c11 ✔️
c12 ✔️
c13
c21 ✔️
c22 ✔️
c23 ✔️
c31 ✔️
c32 ✔️
c33

T3NoC13C33 #

T3NoC13C33
b1 ✔️
b2 ✔️
b3 ✔️
c11 ✔️
c12 ✔️
c13
c21 ✔️
c22 ✔️
c23 ✔️
c31 ✔️
c32 ✔️
c33

T3NoC11C13C33d #

T3NoC11C13C33d
b1 When cost is 1/​10 of min(c21 cost, c31 cost)
b2 When cost is ¼ of min(c12, c22, c32)
b3 When cost is 1/​10 of c23 cost
c11
c12 ✔️
c13
c21 ✔️
c22 ✔️
c23 ✔️
c31 ✔️
c32 ✔️
c33

T3NoC11C13C33 #

T3NoC11C13C33d
b1 ✔️
b2 ✔️
b3 ✔️
c11
c12 ✔️
c13
c21 ✔️
c22 ✔️
c23 ✔️
c31 ✔️
c32 ✔️
c33

T3NoC13C32C33d #

T3C13C32C33d
b1 When cost is ⅛ of min(c11 cost, c21 cost, c31 cost)
b2 When cost is ⅕ of min(c12 cost, c22 cost)
b3 When cost is ⅛ of c23 cost
c11 ✔️
c12 ✔️
c13
c21 ✔️
c22 ✔️
c23 ✔️
c31 ✔️
c32
c33

T3NoC13C32C33 #

T3C13C32C33
b1 ✔️
b2 ✔️
b3 ✔️
c11 ✔️
c12 ✔️
c13
c21 ✔️
c22 ✔️
c23 ✔️
c31 ✔️
c32
c33

T3C11C12C21d #

T3C11C12C21d
b1 When cost is ⅐ of min(c11 cost, c21 cost)
b2 When cost is ⅐ of c12 cost
b3
c11 ✔️
c12 ✔️
c13
c21 ✔️
c22
c23
c31
c32
c33

T3C11C12C21 #

T3C11C12C21
b1 ✔️
b2 ✔️
b3
c11 ✔️
c12 ✔️
c13
c21 ✔️
c22
c23
c31
c32
c33

The­ory 4: Poly­no­mi­als #

T4C3d66 #

T4C3d66
Re­cov­ery (pub mult < 1) Tau Gain (pub mult > 1)
c1
c2
c3 ✔️ ✔️
c4
c5
c6
q1 When 10 + q1lvl % 10 times cheaper than min(q2 cost, c3 cost) When 10 + q1lvl % 10 times cheaper than min(q2 cost, c3 cost)
q2 ✔️ When cost is ⅔ of c3 cost

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 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:

T4C3coast #

T4C3­coast
c1
c2
c3 ✔️
c4
c5
c6
q1 ✔️
q2 ✔️

When the the­ory sim­u­lator re­com­mends this strategy, it will show something like: T4C3coast \(q_1\): 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.

T4C3 #

T4C3
c1
c2
c3 ✔️
c4
c5
c6
q1 ✔️
q2 ✔️

T4C3dC12rcv #

T4C3d­C12rcv
Re­cov­ery (pub mult < 1) Tau Gain (pub mult > 1)
c1 When cost is 1/​10 of c2 cost
c2 ✔️
c3 ✔️ ✔️
c4
c5
c6
q1 When cost is 1/​10 of q2 cost When cost is 1/​10 of q2 cost
q2 ✔️ ✔️

T4C356dC12rcv #

T4C356d­C12rcv
Re­cov­ery (pub mult < 1) Tau Gain (pub mult > 1)
c1 When cost is 1/​10 of c2 cost
c2 ✔️
c3 ✔️ ✔️
c4
c5 ✔️ ✔️
c6 ✔️ ✔️
q1 When cost is 1/​10 of q2 cost When cost is 1/​10 of q2 cost
q2 ✔️ ✔️

T4C456dC12rcvMS #

T4C456d­C12r­cvMS
Re­cov­ery (pub mult < 1) Tau Gain (pub mult > 1)
c1 When cost is 1/​10 of c2 cost
c2 ✔️
c3
c4 ✔️ ✔️
c5 ✔️ ✔️
c6 ✔️ ✔️
q1 When cost is 1/​10 of q2 cost When cost is 1/​10 of q2 cost
q2 ✔️ ✔️

Mile­stone swap­ping strategy

T4C123d #

T4C123d
c1 When cost is 1/​10 of c2 cost
c2 ✔️
c3 ✔️
c4
c5
c6
q1 When cost is 1/​10 of q2 cost
q2 ✔️

This strategy is typ­ic­ally seen at lower \(\rho\) and is even­tu­ally out­paced by T4C3d66.

T4C123 #

T4C123
c1 ✔️
c2 ✔️
c3 ✔️
c4
c5
c6
q1 ✔️
q2 ✔️

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

T4C12d #

T4C12d
c1 When cost is 1/​10 of c2 cost
c2 ✔️
c3
c4
c5
c6
q1
q2

T4C12 #

T4C12
c1 ✔️
c2 ✔️
c3
c4
c5
c6
q1
q2

T4C56 #

T4C56
c1
c2
c3
c4
c5 ✔️
c6 ✔️
q1 ✔️
q2 ✔️

T4C4 #

T4C4
c1
c2
c3
c4 ✔️
c5
c6
q1 ✔️
q2 ✔️

T4C5 #

T4C5
c1
c2
c3
c4
c5 ✔️
c6
q1 ✔️
q2 ✔️

T4 #

T4
c1 ✔️
c2 ✔️
c3 ✔️
c4 ✔️
c5 ✔️
c6 ✔️
q1 ✔️
q2 ✔️

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 #

T5Idle xexxx
Be­fore xexxx ρ After xexxx ρ
q1 ✔️ ✔️
q2 ✔️ ✔️
c1 ✔️
c2 ✔️ ✔️
c3 ✔️ ✔️

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:

T5 #

T5
q1 ✔️
q2 ✔️
c1 ✔️
c2 ✔️
c3 ✔️

The­ory 6: In­teg­ral Cal­cu­lus #

T6AI #

T6AI
q1 When 7 + q1lvl % 10 times cheaper than min(q2, r2, c5)
q2 ✔️
r1 When 5 + r1lvl % 10 times cheaper than min(q2, r2, c5)
r2 ✔️
c1 When 8 + c1lvl % 10 times cheaper than c2
c2 As the pub­lic­a­tion pro­gresses, buy less and
less c2 (and there­fore less c1 too)
c3
c4
c5 ✔️

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

How does \(c_2\) pur­chas­ing works ex­actly?

In the sim, the ra­tio at which \(c_2\) is bought to min(\(q_2\), \(r_2\), \(c_5\)) is defined as the term ra­tio between the \(c_5\) term and the \({c_1}^{1.15}{c_2}\) term, which equates to \({c_5}r/{​c_1}^{1.15}{c_2}\). If this ra­tio is 2, then the sim will buy \(c_2\) when 2 times cheaper than min(\(q_2\), \(r_2\), \(c_5\)).

T6SNAX x.xxexxx #

T6SNAX x.xxexxx
Be­fore x.xxexxx After x.xxexxx
q1 ✔️ ✔️
q2 ✔️ ✔️
r1 ✔️ ✔️
r2 ✔️ ✔️
c1 ✔️
c2 ✔️
c3
c4
c5 ✔️ ✔️

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

Strategy Cred­its:

T6NoC34d #

T6NoC34d
q1 When cost is 1/​10 of q2 cost
q2 ✔️
r1 When cost is 1/​10 of r2 cost
r2 ✔️
c1 When cost is 1/​10 of c2 cost
c2 ✔️
c3
c4
c5 ✔️

T6NoC34 #

T6NoC34
q1 ✔️
q2 ✔️
r1 ✔️
r2 ✔️
c1 ✔️
c2 ✔️
c3
c4
c5 ✔️

T6NoC1234d #

T6NoC1234d
q1 When 7 + q1lvl % 10 times times cheaper than min(q2, r2, c5,)
q2 ✔️
r1 When 5 times cheaper than min(q2, r2, c5,)
r2 ✔️
c1
c2
c3
c4
c5 ✔️



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 #

T6NoC1234
q1 ✔️
q2 ✔️
r1 ✔️
r2 ✔️
c1
c2
c3
c4
c5 ✔️

T6NoC345d #

T6NoC345d
q1 When cost is 1/​10 of q2 cost
q2 ✔️
r1 When cost is 1/​10 of r2 cost
r2 ✔️
c1 When cost is 1/​10 of c2 cost
c2 ✔️
c3
c4
c5

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

T6NoC345 #

T6NoC345
q1 ✔️
q2 ✔️
r1 ✔️
r2 ✔️
c1 ✔️
c2 ✔️
c3
c4
c5

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

T6C4d #

T6C4d
q1 When cost is ⅕ of min(c4 cost, q2 cost, r2 cost)
q2 ✔️
r1 When cost is ⅕ of min(c4 cost, q2 cost, r2 cost)
r2 ✔️
c1
c2
c3
c4 ✔️
c5

T6C4 #

T6C4
q1 ✔️
q2 ✔️
r1 ✔️
r2 ✔️
c1
c2
c3
c4 ✔️
c5

T6C3d #

T6C3d
q1 When cost is ⅓ of min(q2 cost, c3 cost)
q2 ✔️
c1
c2
c3 ✔️
c4
c5

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 #

T6C3
q1 ✔️
q2 ✔️
c1
c2
c3 ✔️
c4
c5

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:

T7­PlaySpqcey
q1 When cost is ¼ of c6 cost
c1
c2
c3 When cost is 1/​10 of c6 cost
c4 When cost is 1/​10 of c6 cost
c5 When cost is ¼ of c6 cost
c6 ✔️

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

T7­PlaySpqceyX
q1 When cost is ¼ of c6 cost
c1 When cost is 1/​10 of c2 cost
c2 When X times cheaper than c6 cost
c3 When cost is 1/​10 of c6 cost
c4 When cost is 1/​10 of c6 cost
c5 When cost is ¼ of c6 cost
c6 ✔️

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:

T7C3d #

T7C3d
q1 When cost is 1/​10 of c3 cost
c1
c2
c3 ✔️
c4
c5
c6

T7C12d #

T7C12d
q1 When cost is 1/​10 of c2 cost
c1 When cost is ⅛ of c2 cost
c2 ✔️
c3
c4
c5
c6

T7C12 #

T7C12
q1 ✔️
c1 ✔️
c2 ✔️
c3
c4
c5
c6

T7C123d #

T7C123d
q1 When cost is 1/​10 of min(c2 cost, c3 cost)
c1 When cost is ⅛ of min(c2 cost, c3 cost)
c2 ✔️
c3 ✔️
c4
c5
c6

T7NoC12 #

T7NoC12
q1 ✔️
c1
c2
c3 ✔️
c4 ✔️
c5 ✔️
c6 ✔️

T7NoC123 #

T7NoC123
q1 ✔️
c1
c2
c3
c4 ✔️
c5 ✔️
c6 ✔️

T7NoC124 #

T7NoC124
q1 ✔️
c1
c2
c3 ✔️
c4
c5 ✔️
c6 ✔️

T7NoC1234 #

T7NoC1234
q1 ✔️
c1
c2
c3
c4
c5 ✔️
c6 ✔️

T7 #

T7
q1 ✔️
c1 ✔️
c2 ✔️
c3 ✔️
c4 ✔️
c5 ✔️
c6 ✔️

The­ory 8: Chaos The­ory #

T8PlaySo­l­arSwap #

T8­PlaySo­l­arSwap
c1 When 5 + c1lvl % 10 times cheaper than
min(c2 cost, c4 cost)
c2 ✔️
c3 When cost is ⅖ of min(c2 cost, c4 cost)
c4 ✔️
c5 When cost is ⅖ of min(c2 cost, c4 cost)

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:

T8Play #

T8­Play
c1 When cost is ⅛ of min(c2 cost, c4 cost)
c2 ✔️
c3 When cost is ⅖ of min(c2 cost, c4 cost)
c4 ✔️
c5 When cost is ¼ of min(c2 cost, c4 cost)



Strategy Cred­its:

T8SNAX #

T8SNAX
Pub­lic­a­tion Mul­ti­plier < 1.6 Pub­lic­a­tion Mul­ti­plier 1.6-2.3 Pub­lic­a­tion Mul­ti­plier > 2.3
c1 ✔️
c2 ✔️ ✔️ ✔️
c3 ✔️ ✔️
c4 ✔️ ✔️ ✔️
c5 ✔️ ✔️



Strategy Cred­its:

T8NoC35d #

T8NoC35d
c1 When 10 times cheaper than
min(c2 cost, c4 cost)
c2 ✔️
c3
c4 ✔️
c5

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

T8NoC35 #

T8NoC35
c1 ✔️
c2 ✔️
c3
c4 ✔️
c5

T8NoC5d #

T8NoC5d
c1 When cost is 1/​10 of min(c2 cost, c4 cost)
c2 ✔️
c3 ✔️
c4 ✔️
c5

T8NoC5 #

T8NoC5
c1 ✔️
c2 ✔️
c3 ✔️
c4 ✔️
c5

T8NoC3d #

T8NoC3d
c1 When cost is 1/​10 of min(c2 cost, c4 cost)
c2 ✔️
c3
c4 ✔️
c5 ✔️

T8NoC3 #

T8NoC3
c1 ✔️
c2 ✔️
c3
c4 ✔️
c5 ✔️

T8d #

T8d
c1 When cost is 1/​10 of c2 cost
c2 ✔️
c3 ✔️
c4 ✔️
c5 ✔️

T8 #

T8
c1 ✔️
c2 ✔️
c3 ✔️
c4 ✔️
c5 ✔️

Wei­er­straß Sine Product #

WSPdStopC1 #

WSP­dStopC1
First 15 Seconds of Pub­lic­a­tion Rest of Pub­lic­a­tion
q1 When 8 + q1lvl % 10 times cheaper than min(q2 cost, n cost, c2 cost) When 8 + q1lvl % 10 times cheaper than min(q2 cost, n cost, c2 cost)
q2 ✔️ ✔️
n ✔️ ✔️
c1 ✔️ When cost 1/​Ra­tio of min(q2 cost, n cost, c2 cost)
c2 ✔️ ✔️


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:


c1 Ra­tio
ρ < e25 1
e25 < ρ < e40 3
e40 < ρ < e200 10
e200 < ρ < e400 50
e400 < ρ < e700 1000
ρ > e700 Do not buy c1 after 15 seconds into pub­lic­a­tion


Strategy Cred­its:

WSPStopC1 #

WSP­StopC1
First 15 Seconds of Pub­lic­a­tion Rest of Pub­lic­a­tion
q1 ✔️ ✔️
q2 ✔️ ✔️
n ✔️ ✔️
c1 ✔️ If ρ < e450, ✔️
If ρ > e450, ❌
c2 ✔️ ✔️


Strategy Cred­its:

WSP #

WSP
q1 ✔️
q2 ✔️
n ✔️
c1 ✔️
c2 ✔️

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:

Ra­tios for Swap­ping
State 1 State 2 State 3
e25ρ-e50ρ Un­til 5x cost/​ρ Ra­tio Un­til 4x cost/​ρ Ra­tio Un­til up­grade is bought
e50ρ-e75ρ Un­til 7x cost/​ρ Ra­tio Un­til 6x cost/​ρ Ra­tio Un­til up­grade is bought
e75ρ-e100ρ Un­til 12x cost/​ρ Ra­tio Un­til 10x cost/​ρ Ra­tio Un­til up­grade is bought
e100ρ-e150ρ Un­til 20x cost/​ρ Ra­tio Un­til 15x cost/​ρ Ra­tio Un­til up­grade is bought
e150ρ-e175ρ Un­til 8x cost/​ρ Ra­tio Un­til 6x cost/​ρ Ra­tio Un­til up­grade is bought
e175ρ-e200ρ Un­til 1.5x cost/​ρ Ra­tio Un­til up­grade is bought Skip
e200ρ-e275ρ Un­til 3x cost/​ρ Ra­tio Skip Un­til up­grade is bought
e275ρ-e300ρ Un­til 2x cost/​ρ Ra­tio Skip Un­til up­grade is bought

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

SLMS
Pub­lic­a­tion Mult < 4 Pub­lic­a­tion Mult 4-7.5 Pub­lic­a­tion Mult > 7.5
a1 ✔️
a2 ✔️
b1 ✔️ ✔️
b2 ✔️ ✔️



Strategy Credit:

SLMSd #


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

SLMSd
Pub­lic­a­tion Mult < 4 Pub­lic­a­tion Mult 4-7.5 Pub­lic­a­tion Mult > 7.5
a1 If a1lvl % 3 = 0, then ✔️
If not, buy when 2 × (a1lvl % 3) times cheaper than a2 cost
a2 ✔️
b1 If b1lvl % 4 = 0 or 1, then ✔️
If not, buy when b1lvl % 4 times cheaper than b2 cost
If b1lvl % 4 = 0 or 1, then ✔️
If not, buy when b1lvl % 4 times cheaper than b2 cost
b2 ✔️ ✔️


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

Strategy Credit:

SLStopAd #

SLSto­pAd
Pub­lic­a­tion Mult < 4.5 Pub­lic­a­tion Mult 4.5-6 Pub­lic­a­tion Mult > 6
a1 If a1lvl % 3 = 0, then ✔️
If not, buy when 2 × (a1lvl % 3) times cheaper than a2 cost
a2 ✔️
b1 If b1lvl % 4 = 0 or 1, then ✔️
If not, buy when b1lvl % 4 times cheaper than b2 cost
If b1lvl % 4 = 0 or 1, then ✔️
If not, buy when b1lvl % 4 times cheaper than b2 cost
b2 ✔️ ✔️


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:

SLStopA #

SLStopA
Pub­lic­a­tion Mult < 4.5 Pub­lic­a­tion Mult 4.5-6 Pub­lic­a­tion Mult > 6
a1 ✔️
a2 ✔️
b1 ✔️ ✔️
b2 ✔️ ✔️


Strategy Cred­its:

Euler’s For­mula #

EFAI #

EFAI
Re­cov­ery (pub mult < 1) Tau Gain (pub mult > 1)
✔️ ✔️
q1 When cost × (10 + lvl % 10) < q2 cost When cost × (10 + lvl % 10) < q2 cost
q2 ✔️ ✔️
b1 ✔️ When ⅕ of a2 cost
b2 ✔️ When ⅕ of a2 cost
c1 ✔️ When ⅕ of a3 cost
c2 ✔️ When ⅕ of a3 cost
a1 When cost × (4 + (lvl % 10)/​2) < q2 cost When cost × (4 + (lvl % 10)/​2) < q2 cost
a2 ✔️ ✔️
a3 ✔️ ✔️

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:

EFSNAX #

EF­SNAX
Re­cov­ery (pub mult < 1) Tau Gain (pub mult > 1)
✔️ ✔️
q1 ✔️
q2 ✔️ ✔️
b1 ✔️
b2 ✔️
c1 ✔️
c2 ✔️
a1 ✔️ ✔️
a2 ✔️ ✔️
a3 ✔️ ✔️


Strategy Cred­its:


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

EFd #

EFd
✔️
q1 When cost is 1/​10 of q2 cost
q2 ✔️
b1 ✔️
b2 ✔️
c1 ✔️
c2 ✔️
a1 When cost is ¼ of q2 cost
a2 ✔️
a3 ✔️

EF #

EF
✔️
q1 ✔️
q2 ✔️
b1 ✔️
b2 ✔️
c1 ✔️
c2 ✔️
a1 ✔️
a2 ✔️
a3 ✔️

Con­ver­gents to √2 #

CSR2XL x.xx #

CSR2XL x.xx
Be­fore x.xx Pub­lic­a­tion Mul­ti­plier After x.xx Pub­lic­a­tion Mul­ti­plier
q1 When cost × 10 < min(q2c, nc, c2c), where c = cost
q2 ✔️(if q2 has a sim­ilar cost to either n or c2,
pri­or­it­ize the other vari­able over q2)
c1 When cost × 10 < min (q2c, nc, c2c)
n ✔️(if n and c2 have sim­ilar costs, pri­or­it­ize c2)
c2 ✔️

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:

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:

Ra­tio
ρ < e45 Could­n't find any good ra­tio here, just swap
when you are very close to the up­grade. - XLII
e45 < ρ < e80 4
e80 < ρ < e115 8
e115 < ρ < e220 20
ρ > e220 40

Strategy Cred­its:

CSR2d #

CSR2d
q1 When cost × 10 < min(q2, n, c2), where c = cost
q2 ✔️
c1 When cost × 10 < min(q2, n, c2), where c = cost
n ✔️
c2 ✔️

CSR2 #

CSR2
q1 ✔️
q2 ✔️
c1 ✔️
n ✔️
c2 ✔️

Frac­tional In­teg­ra­tion #

FId #

FId
q1 When cost × (1 + lvl % 23) < min(q2 cost, K cost)
q2 ✔️
K ✔️
m ✔️
n When cost × (1 + lvl % 11) < min(q2 cost, K cost, m cost)

Mile­stone rout­ing in­form­a­tion

This strategy does not use the 3rd level of the g(x) mile­stone un­til you un­lock the last mile­stone point at e1150ρ.

FI #

FI
q1 ✔️
q2 ✔️
K ✔️
m ✔️
n ✔️

Mile­stone rout­ing in­form­a­tion

This strategy does not use the 3rd level of the g(x) mile­stone un­til you un­lock the last mile­stone point at e1150ρ.

FIMSd #

FIMSd
q1 When cost × (1 + lvl % 23) < min(q2 cost, K cost)
q2 ✔️
K ✔️
m ✔️
n When cost × (1 + lvl % 11) < min(q2 cost, K cost, m cost)

Mile­stone swap­ping strategy

When buy­ing a new level of q2, swap n and m mile­stones to q1 ex­po­nent to build up q.

Swap back to m and n when your q got mul­ti­plied by a ra­tio (since you star­ted the swap­ping phase) de­pend­ing on your q1 lvl % 23:

q1 lvl % 23 q ra­tio
1-4 4
5-9 3
10-19 2.5
20+ 2

Strategy Cred­its:

FIMS #

FIMS
q1 ✔️
q2 ✔️
K ✔️
m ✔️
n ✔️

Mile­stone swap­ping strategy

When buy­ing a new level of q2, swap n and m mile­stones to q1 ex­po­nent to build up q.

Swap back to m and n when your q got mul­ti­plied by a ra­tio (since you star­ted the swap­ping phase) de­pend­ing on your q1 lvl % 23:

q1 lvl % 23 q ra­tio
1-4 4
5-9 3
10-19 2.5
20+ 2

Strategy Cred­its:

FIdPer­maSwap #

FId­Per­maSwap
q1 When cost × (1 + lvl % 23) < min(q2 cost, K cost)
q2 ✔️
K ✔️
m ✔️
n When cost × (1 + lvl % 11) < min(q2 cost, K cost, m cost)

Per­maSwap strategy

Be­fore reach­ing e1076ρ in your pub­lic­a­tion, do not use the 3rd level of the g(x) mile­stone.

Upon reach­ing e1076ρ, switch a mile­stone point into the 3rd level of the g(x) mile­stone, at the cost of re­set­ting q.

FIPer­maSwap #

FIPer­maSwap
q1 ✔️
q2 ✔️
K ✔️
m ✔️
n ✔️

Per­maSwap strategy

Be­fore reach­ing e1076ρ in your pub­lic­a­tion, do not use the 3rd level of the g(x) mile­stone.

Upon reach­ing e1076ρ, switch a mile­stone point into the 3rd level of the g(x) mile­stone, at the cost of re­set­ting q.

FIMS­dPer­maSwap #

FIMS­dPer­maSwap
q1 When cost × (1 + lvl % 23) < min(q2 cost, K cost)
q2 ✔️
K ✔️
m ✔️
n When cost × (1 + lvl % 11) < min(q2 cost, K cost, m cost)

Mile­stone swap­ping strategy

When buy­ing a new level of q2, swap n and m mile­stones to q1 ex­po­nent to build up q.

Swap back to m and n when your q got mul­ti­plied by a ra­tio (since you star­ted the swap­ping phase) de­pend­ing on your q1 lvl % 23:

q1 lvl % 23 q ra­tio
1-4 4
5-9 3
10-19 2.5
20+ 2

Per­maSwap strategy

Be­fore reach­ing e1076ρ in your pub­lic­a­tion, do not use the 3rd level of the g(x) mile­stone.

Upon reach­ing e1076ρ, switch a mile­stone point into the 3rd level of the g(x) mile­stone, at the cost of re­set­ting q.


Strategy Cred­its:

FIMSPer­maSwap #

FIM­SPer­maSwap
q1 ✔️
q2 ✔️
K ✔️
m ✔️
n ✔️

Mile­stone swap­ping strategy

When buy­ing a new level of q2, swap n and m mile­stones to q1 ex­po­nent to build up q.

Swap back to m and n when your q got mul­ti­plied by a ra­tio (since you star­ted the swap­ping phase) de­pend­ing on your q1 lvl % 23:

q1 lvl % 23 q ra­tio
1-4 4
5-9 3
10-19 2.5
20+ 2

Per­maSwap strategy

Be­fore reach­ing e1076ρ in your pub­lic­a­tion, do not use the 3rd level of the g(x) mile­stone.

Upon reach­ing e1076ρ, switch a mile­stone point into the 3rd level of the g(x) mile­stone, at the cost of re­set­ting q.


Strategy Cred­its:

Fractal Pat­terns #

FPmod­BurstC1MS #

FP­mod­Burst­C1MS
✔️
c When cost × (1 + lvl % 100) < min(c2 cost, s cost)
(Auto­buy if you can af­ford the next lvl%100=1 be­fore the next c2 or s)
c2 ✔️(if c2 and s have sim­ilar costs, pri­or­it­ize s)
q1 When cost × (1 + lvl % 10) < min(q2 cost, s cost)
q2 ✔️(if q2 and s have sim­ilar costs, pri­or­it­ize s)
r1 ✔️
n ✔️
s ✔️

Mile­stone swap­ping strategy (e700ρ+)

After you un­lock the s mile­stone, you can per­form mile­stone swap­ping un­til s > 2. Al­tern­ate between hav­ing the s mile­stone on and off. When s > 2, keep the mile­stone on everytime. Mile­stone swap­ping is no longer re­quired once you can reach s > 2 quickly.

For more in­form­a­tion about the FP mile­stone swap­ping strategy, check here.


Strategy Cred­its:

FPdMS #

FP­dMS
✔️
c When cost × (1 + lvl % 100) < c2 cost
c2 ✔️
q1 ✔️
q2 ✔️
r1 ✔️
n ✔️
s ✔️

Mile­stone swap­ping strategy (e700ρ+)

After you un­lock the s mile­stone, you can per­form mile­stone swap­ping un­til s > 2. Al­tern­ate between hav­ing the s mile­stone on and off. When s > 2, keep the mile­stone on everytime. Mile­stone swap­ping is no longer re­quired once you can reach s > 2 quickly.

For more in­form­a­tion about the FP mile­stone swap­ping strategy, check here.

FP #

FP
✔️
c1 ✔️
c2 ✔️
q1 ✔️
q2 ✔️
r1 ✔️
n ✔️
s ✔️

Riemann Zeta Func­tion #

RZdBH #

RZdBH
c1 When c1lvl < 4 × c2lvl + 2
c2 ✔️
w1 When cost × 5 < min(w2 cost, w3 cost)
w2 ✔️
w3 ✔️
b ✔️(Don't buy un­til t>16)

When the the­ory sim­u­lator re­com­mends this strategy, it will show something like:
RZdBH t=388.85 c1: 3091 c2: 773.

The value after t is the t value at which the black hole should be set. We re­com­mend put­ting the threshold 0.01 above this value to avoid pre­ci­sion is­sues.

The levels it shows after c1 and c2 are the last levels you should buy them to in this pub­lic­a­tion.

Strategy cred­its:

RZBH #

RZBH
c1 ✔️
c2 ✔️
w1 ✔️
w2 ✔️
w3 ✔️
b ✔️

When the the­ory sim­u­lator re­com­mends this strategy, it will show something like:
RZBH t=462.07 c1: 3091 c2: 773.

The value after t is the t value at which the black hole should be set. We re­com­mend put­ting the threshold 0.01 above this value to avoid pre­ci­sion is­sues.

The levels it shows after c1 and c2 are the last levels you should buy them to in this pub­lic­a­tion.

Strategy cred­its:

RZSpir­alSwap #

RZ­Spir­alSwap
c1 When c1lvl < 4 × c2lvl + 2
c2 ✔️
w1 When cost × 5 < w2 cost
w2 ✔️
b ✔️(Don't buy un­til t>16)

Spir­alSwap

To per­form Spir­alSwap, switch mile­stones to pri­or­it­ize c1 ex­po­nent when ζ is close to zero (when the graph is close to the ori­gin) and to pri­or­it­ize w2 when |ζ| is > 1.

This strategy is ex­tremely act­ive. Us­ing RZdMS is much easier and does­n’t lose that much time com­pared to spir­alswap.

When the the­ory sim­u­lator re­com­mends this strategy, it will show something like:
RZSpir­alSwap c1: 953 c2: 238.

The levels it shows after c1 and c2 are the last levels you should buy them to in this pub­lic­a­tion.

Strategy cred­its:

RZdMS #

RZdMS
c1 When c1lvl < 4 × c2lvl + 2
c2 ✔️
w1 When cost × 5 < w2 cost
w2 ✔️
b ✔️(Don't buy un­til t>16)

When the the­ory sim­u­lator re­com­mends this strategy, it will show something like:
RZdMS swap:197 c1: 953 c2: 238.

Start the pub­lic­a­tion with the w2 mile­stone on. The value after swap is the ρ value were you should switch it to c1 ex­po­nent.

The levels it shows after c1 and c2 are the last levels you should buy them to in this pub­lic­a­tion.

Strategy cred­its:

RZMS #

RZMS
c1 ✔️
c2 ✔️
w1 ✔️
w2 ✔️
b ✔️

When the the­ory sim­u­lator re­com­mends this strategy, it will show something like:
RZMS swap:196 c1: 952 c2: 238.

Start the pub­lic­a­tion with the w2 mile­stone on. The value after swap is the ρ value were you should switch it to c1 ex­po­nent.

The levels it shows after c1 and c2 are the last levels you should buy them to in this pub­lic­a­tion.

Strategy cred­its:

RZd #

RZd
c1 When c1lvl < 4 × c2lvl + 2*
c2 ✔️
w1 When cost × 5 < min(w2 cost, w3 cost)
w2 ✔️
w3 ✔️
b ✔️(Don't buy un­til t>16)

* If you don’t have \(c_1\) ex­po­nent mile­stones, use 4×\(c_2\)lvl + 1 in­stead.

When the the­ory sim­u­lator re­com­mends this strategy, it will show something like:
RZd c1: 953 c2: 238.

The levels it shows after c1 and c2 are the last levels you should buy them to in this pub­lic­a­tion.

Strategy cred­its:

RZ #

RZ
c1 ✔️
c2 ✔️
w1 ✔️
w2 ✔️
w3 ✔️
b ✔️

When the the­ory sim­u­lator re­com­mends this strategy, it will show something like:
RZ c1: 953 c2: 238.

The levels it shows after c1 and c2 are the last levels you should buy them to in this pub­lic­a­tion.

Strategy cred­its: