The­or­ies 5-8

Guide writ­ten by LE★Baldy, Snaeky, & Xelaroc. Con­tri­bu­tions from the Amaz­ing Com­munity.

This guide is cur­rently un­der­go­ing change. Keep in mind, strategies may change.

Feel free to use the gloss­ary as needed.

Gradu­ation rout­ing #

Re­mem­ber to fol­low our rout­ing ad­vice from the in­tro­duc­tion to gradu­ation.

Stu­dent rout­ing with R9 #

All rout­ing fol­lows the stu­dent cal­cu­lator (by Niedzielan, AfuroZamurai, and Milla) and star cal­cu­lator (by Eaux Ta­cous#1021). When you are not push­ing \(f(t)\) you should al­ways have the 9th re­search op­tion maxed (after The­ory 8). When push­ing \(f(t)\), you should be R9 swap­ping (be­low).

There is also the the­ory sim­u­lator by XLII, which works both be­low max mile­stones and after max mile­stones for all the­or­ies.

How to push F(t) with R9 swap­ping #

Mem­or­ize your stu­dent dis­tri­bu­tions with and without 10/​20/​30 R9 stu­dents. Use the stu­dent cal­cu­lator if needed. You will com­monly see people refer to this as R9 seap­ing as a long held name of the strategy.

1. Wait till \(f(t)\) stops grow­ing with stu­dents in R9 push­ing \(\tau\).
2. Start ac­cel (prefer­ably keep it between prestiges).
3. Po­ten­tially sit here to stack t for big­ger \(\varphi_2\) when you have stu­dents in \(\varphi_2\). Only do this when you are near a gradu­ation mark. This is not use­ful if you will not swap into \(\varphi_2\).
4. Re­spec all 10/​20/​30 stu­dents from R9.
5. Wait for the auto­prestige to prestige and swap back stu­dents to R9.
6. Re­peat.

This method al­lows you to push \(f(t)\) with al­most no loss of R9 up­time or push­ing power. This is harder with fewer levels of R9 but still helps if you get used to it.

R9 auto­prestige ex­pres­sion #

You can find the auto­prestige used for R9 Swap­ping here: Equa­tion. If you don’t have this ex­pres­sion, then you will have to manu­ally prestige each swap.

Ref­er­ence R9 Swap­ping Auto­prestige Ex­plan­a­tion

The­ory 1 #

You will not touch this the­ory un­til after ee14k. Once you be­gin push­ing T1 after ee14k, be­gin us­ing the The­ory Sim and the The­ory Sim Guide to give the best strategy and mul­ti­plier for the next pub­lic­a­tion.

The­ory 2 #

This the­ory will be used as overnight un­til 1e350 Tau where it will not be touched un­til after ee14k. See our earlier guide for an over­view for the­ory 2.

The­ory 3 #

See our earlier guide for an over­view for the­ory 3.

The­ory 4 #

See our earlier guide for an over­view for the­ory 4.

The­ory 5 (40σ / 9k) #

The­ory 5 Over­view #

The­ory 5 is based on lo­gistic func­tion. This the­ory is slow early, but be­comes very power­ful later on in the game. It is re­com­men­ded to keep push­ing this the­ory as high as pos­sible be­fore reach­ing ee14k ft. Make sure to care­fully read the be­ha­viour of c1 and c2 vari­ables in this the­ory, as the be­ha­viours are quite unique.

The­ory 5 Equa­tion De­scrip­tion #

\(\dot{\rho} = q_1^{1.15}q_2q\)

\(\dot{q} = (c_1/​c_2)q(c_3^{1.1} - q/​c_2)\)

The first line states that the rate of change of rho is the product of some \(q_1, q_2, q\). Note that \(q_1\) and \(q_2\) are vari­ables that you can buy dir­ectly, while \(q\) is a sep­ar­ate vari­able that is based off something else.

The second line defines this the­ory. It de­scribes the be­ha­viour of a typ­ical lo­gistic func­tion. A lo­gistic func­tion typ­ic­ally has slow growth at the be­gin­ning, then fast growth in the middle, then it flat­tens out at the end. Here we have \(c_1\), which speeds up the rate at which we reach the max­imum value of \(q\). Note it DOES NOT in­crease the ac­tual max­imum value of \(q\) it­self. We also have \(c_2\). This in­creases the max­imum value of \(q\). However, it HALVES the speed at which this max­imum value is reached. There­fore we must be care­ful to not buy too many at once.

\(c_3\) is sim­ilar to \(c_2\) in which buy­ing it in­creases the max­imum value of \(q\). However it does not have the draw­back that \(c_2\) has. So we can buy this un­con­di­tion­ally.

The max­imum value of \(q\) is \(c_2c_3^{1.1}\). Once \(q\) reaches this max­imum value, \(q\) dot will be zero and \(q\) will not grow any­more un­til we buy either more \(c_2, c_3\)

The­ory 5 Vari­able De­scrip­tion #

Ap­prox­im­ate vari­able strengths on \(\dot\rho\) with all mile­stones are as fol­lows:

Bi­cycle Ana­logy (al­tern­at­ive ex­plan­a­tion) of c1 and c2 in­ter­ac­tions #

Think of buy­ing \(c_1\) as throt­tling on the bi­cycle faster. Buy­ing \(c_2\) is sim­ilar to shift­ing the bi­cycle gear up by 1 gear.

If all we do is buy \(c_1\) and never \(c_2\), then we’re stuck in gear 1 forever and make no pro­gress. However, if all we do is buy \(c_2\) and never \(c_1\), then this is sim­ilar to try­ing to ride from highest gear from 0 speed, which we know takes a long time and a lot of ef­fort.

There­fore us­ing the bi­cycle ana­logy, we should buy \(c_2\) only when we have the speed to sup­port it; not too early and not too late. Fur­ther­more, later in the pub­lic­a­tion, we should buy only 1 level of \(c_2\) at a time. We should buy \(c_1\) only right after buy­ing \(c_2\) (shift­ing up gear).

When de­cid­ing when to buy \(c_1, c_2\), think of \(c_1\) as throt­tling a bi­cycle, and \(c_2\) as shift­ing up gear by 1 level.

The­ory 5 strategy #

The strengths of each vari­able are as fol­lows:

\(c_3\) ~= \(q_2\) ≈ \(c_2 \) > \(q_1 \) > \(c_1 \)

Note that \(c_1\) and \(c_2\) have spe­cial in­ter­ac­tions.

Manual buy­ing c2 - READ THIS BE­FORE DO­ING THE STRATEGIES

For step 2 of the semi-idle and act­ive strategies be­low, you should be manu­ally buy­ing \(c_2\). If you’re over e150 \(\rho\), start with buy 10x vari­ables at a time. Oth­er­wise buy 1 level of c2 at a time.

You want to buy \(c_2\) un­til the value of \(q\) in­creases slowly. The pat­tern should be:

1. Buy \(c_2\). \(q\) should in­crease.
2. Once \(q\) in­creases slowly (or stops in­creas­ing), buy more \(c_2\).
   
   

If you buy a \(c_2\) and the value of \(q\) is in­creas­ing even slower than be­fore, this means you bought too many levels of \(c_2\). Wait for \(q\) to sta­bil­ise and then con­tinue buy­ing \(c_2\).

Once you’ve reached within e5 of the pre­vi­ous pub­lic­a­tion point, you can safely auto­buy \(c_2\) for the rest of the pub­lic­a­tion. Re­mem­ber to un­tick \(c_2\) auto­buy at the end of the pub­lic­a­tion when you click ‘pub­lish’!

Idle

It is not re­com­men­ded to idle the­ory 5 (see ex­plan­a­tion on \(c_1\) and \(c_2\) above). Do try the semi-idle strategy be­low.

For each pub­lic­a­tion: auto­buy \(c_3\) and \(q_2\).

1. For the first 10 seconds, auto­buy everything ex­cept \(c_2\).
2. Af­ter­wards, simply auto­buy all un­til pub­lish.

Semi-Idle

Semi idle is sim­ilar to idle, but we do manual \(c_2\) buy­ing and dis­abling \(c_1\) once the the­ory is re­covered.

For each pub­lic­a­tion: auto­buy \(c_3\) and \(q_2\).

1. For the first 10 seconds, auto­buy everything ex­cept \(c_2\).
2. Then we want to manu­ally buy \(c_2\). See Manual Buy­ing c2. Do this un­til it slows down and you’re within about e5 \(\rho\) un­der last pub­lic­a­tion mark.
3. Then we auto­buy all un­til \(\rho \) has reached its pre­vi­ous pub­lic­a­tion value (fin­ished re­cov­ery).
4. Af­ter­wards, de­ac­tiv­ate \(c_1\) and auto­buy the rest un­til pub­lish.

Act­ive

Here’s a simple yet ef­fect­ive act­ive strategy that can be used right un­til en­dgame. If you want more op­tim­ised strategies, please see List of the­ory strategies.

For each pub­lic­a­tion: auto­buy \(c_3\) and \(q_2\). Buy \(c_1\) ONLY right after buy­ing \(c_2\) from steps 2 on­wards. Prac­tic­ally, everytime you buy a level of \(c_2\) from steps 2 on­wards, you’d buy about 5-6 levels of \(c_1\).
Note that for faster speed, for the first part of step 2, you may buy 10 levels at a time

1. For the first 10 seconds, auto­buy everything ex­cept \(c_2\).
2. Then we want to manu­ally buy \(c_2\). See Manual Buy­ing c2.
3. Then we auto­buy \(c_3, q_2, c_2\). Out of these 3 vari­ables, find the one with the cheapest cost. Then buy \(q_1\) un­til its cost ex­ceeds 15% of the cheapest vari­able you found above. Buy \(c_1\) ONLY right after buy­ing a level of \(c_2\).
4. Once the the­ory has re­covered to its pre­vi­ous pub­lic­a­tion mark, you can slowly put less em­phasis on \(c_1\). When in doubt, have \(c_1\)’s cost be sim­ilar to \(q_1\)’s cost. Con­tinue do­ing step 2 un­til pub­lish.

The­ory 5 mile­stone route #

All mile­stones into the 2nd mile­stone to un­lock \(c_3\). Then into 1st mile­stone be­cause \(q_1\) vari­able is higher value than \(c_3\) vari­able, fi­nally put the rest into the last mile­stones. This the­ory does not have a known ef­fect­ive mile­stone swap­ping strategy.

Com­ment­ary

(By Snaeky)

No com­ment­ary

(By Snaeky)

(By Playspout)

T5 will al­ways give its best res­ults from act­ive play. However, after step 3, you can still get good res­ults while auto­buy­ing \(q_1\) and manu­ally pur­chas­ing \(c_1\) every 10-15min. This makes the the­ory slightly less act­ive and easier to deal with.

Warn­ing: Do not overnight this the­ory. It has ter­rible de­cay after passing a good pub­lic­a­tion mark and will not give good res­ults. T5i is only vi­able very late/​en­dgame.

Ad­di­tional in­form­a­tion

Pur­chase \(c_2\) when \(1.5q > c_2*c_3^{m_3}\). \(m_3\) is the num­ber of mile­stone 3.

\(q\) be­gins to slow down when you reach \(2q > c_2*c_3^{m_3}\).

Strategy con­struc­ted by: Snaeky, Marks, Baldy, and Nerdy

The­ory 6 (45σ / 10k) #

T6 Over­view #

T6 has the low­est de­cay of all the the­or­ies. It will be second place to T5 un­til about e750 and is the only the­ory that can get to \(>e1100τ\). You should overnight this and T4 after you get your T2 to \(e350+\).

At first, T6 only finds the area un­der the curve of the graph \(f(q)\) from \(0\) to \(q\), which is es­sen­tially a 2d plane. This is done us­ing a def­in­ite in­teg­ral, an in­teg­ral that is bound between 2 val­ues giv­ing a single out­put. This does the op­pos­ite of what a de­riv­at­ive does, but within a spe­cific bound. With later mile­stones, this will in­clude the vari­able \(r\), this new equa­tion is find­ing the volume of the graph within the bounds of planes \(f(q)\) and \(f(r)\) from \(0\) to \(q\) and \(0\) to \(r\) re­spect­ively.

The \(-C\) vari­able out­side of the in­teg­ral is defined as the sum of the in­stant­an­eous change in \(\rho\) caused by all of the \(c_i\) up­grades that you have pur­chased. It is very closely re­lated to the sum of the costs of up­grades of all \(c_i\) up­grades, but is slightly dif­fer­ent. It is neg­at­ive be­cause pur­chas­ing \(c_i\) up­grades will in­crease the value of the in­teg­ral, thus \(\rho\), by the change in \(c_i\) in­stantly. \(-C\) is used to off­set this so that \(\rho\) does not di­verge. For ex­ample, lets say that the in­teg­ral is equal to \(10\) and that you can buy the first up­grade (cost \(10\)). After buy­ing it, the in­teg­ral will be equal to \(20\). But, at this point, the cur­rency should be \(0\), so \(C\) needs to be equal to \(20\).

Video of T6 at En­dgame

T6 strategy #

The op­timal mul­tipler var­ies between 6-12, but spikes de­pend­ing on what vari­able is dom­in­ant at the time and how close you are to a mile­stone. If you want an ac­cur­ate mul­ti­plier, check with the sim.

T6 mile­stone route #

The­ory 7 (50σ / 11k) #

T7 over­view #

T7 can be sum­mar­ized as a max­im­iz­a­tion prob­lem : given a sur­face in 3-di­men­sional space, you want to find its highest alti­tude by mov­ing along the sur­face, al­ways in the dir­ec­tion of steep­est as­cent (that’s ba­sic­ally a gradi­ent as­cent). The func­tion \(g(x,y)\) can be seen as a sur­face in \(\mathbb{R}^{3}\) (con­sid­er­ing the set of points \((x,y,g(x,y))\), see at­tached im­age). \((\rho_1,\rho_2,g(\rho_1,\rho_2))\) is a point on this sur­face. Our goal is to max­im­ize \(g(\rho_1,\rho_2)\), i.e. to find \((\rho_1,\rho_2)\) that max­im­ize \(g(\rho_1,\rho_2)\). No­tice that the func­tion \(g\) is un­boun­ded, i.e. you can’t find a proper max­imum (we say that the max­im­iz­a­tion prob­lem is ill-con­di­tioned); so one way to max­im­ize \(g(\rho_1,\rho_2)\) is to move \((\rho_1,\rho_2)\) to­wards the dir­ec­tion of steep­est as­cent. This is what is pre­cisely done by set­ting \(\dot{\mathbf{\rho}}\) (which is the dir­ec­tion the point \(\mathbf{\rho}=(\rho_1,\rho_2)\) will move to­ward) to \(\nabla g(\rho_1,\rho_2)\) (i.e. the gradi­ent of \(g\) eval­u­ated at \((\rho_1,\rho_2)\), which gives the dir­ec­tion of steep­est as­cent of \(g\) at the point \((\rho_1,\rho_2)\).

This is the graph of the func­tion \(g\), taken after the first four mile­stones have been un­locked (Note: here, coef­fi­cients like \(c_1,c_2\ldots\) are ig­nored. The ef­fect of those coef­fi­cients is simply mak­ing the graph steeper in \(x\) or \(y\) dir­ec­tion, de­pend­ing on the value of each coef).

T7 strategy #

The op­timal pub­lic­a­tion mul­ti­plier is \(4\)-\(6\). You will swap from 0/​1/​1 → 0/​0/​2 at near the 2/​3 mark of your next mile­stone. The strategy for manual buy be­fore 4 mile­stones is to only manual buy \(q_1\) and \(c_1\) cheap (e1 less \(\rho\)) and the rest full auto. After mile­stone 5, turn it on full auto­buy for idle. For act­ive, you will fol­low the strat de­scribed in the the­ory sim guide or watch the video be­low.

T7 mile­stone route #

The­ory 8 skip­ping #

Tem­por­ar­ily skip­ping T8 is sig­ni­fic­antly faster than buy­ing T8 right away be­cause of how slow T8 is from 0-e60 (don’t worry, you will still buy it, just at 14k, not 12k). T8 is very slow un­til you get to about \(e60\) (it took the sim 16 hours to get that far into the the­ory without R9). We highly re­com­mend buy­ing t8 for the achieve­ment, then selling it right away and us­ing those stu­dents for \(\varphi\). You will need about e1350 \(\tau\) in or­der to get R9 (\(ee14k\)) without T8, which will help you get through T8 faster than be­fore. You will need to buy T8 again to get R9, and you should start R9 right away after that The cur­rent re­com­mend­a­tion for your \(\tau\) dis­tri­bu­tion for T8 skip is as fol­lows:

Idle #

Act­ive #

If your num­bers are dif­fer­ent, that is fine. You just need to reach 1e1350 \(\Pi\tau\) to get to R9 at 14k.

The­ory 8 (55σ / 12k) #

T8 strategy #

The op­timal pub­lic­a­tion mul­ti­plier is 2.5-5 de­pend­ing on how close you are to the next mile­stone. This the­ory is ex­tremely slow at the start which is why we skip un­til we ob­tain R9. It is also the only one with a \(1e20\) mile­stone step. It will speed up once you hit \(1e60\) and even faster at \(1e80\) and \(1e100\) etc. un­til ~\(e250\)-\(e300\). The worst part is the \(1e50\)-\(1e60\) grind. The grind to \(1e60\) will take a good bit of time but is faster with R9.

At the start, manual buy pri­or­it­izes \(c_2\) then \(c_1\) then rest. Once you get to 0/​0/​0/​2, pri­or­it­ize \(c_2\) and \(c_5\) then \(c_1\) then the rest. Once at 2/​0/​3/​0, you will pri­or­it­ize \(c_2\) and \(c_4\) then \(c_1\) then the rest after. This con­tin­ues to max at 2/​3/​3/​3.

T8 mile­stone route #

Note: the 2/​0/​0/​0 → 0/​0/​0/​2 swap hap­pens at e52 tau.

Fi­nal Push to ee20k #

In gen­eral, you will need about e2050 \(\tau\) in or­der to get to 20k eas­ily. Make sure that you are R9 Swap­ping and hold­ing ac­cel when you can. Re­com­men­ded val­ues for your the­or­ies \(\tau\) is as fol­lows. Be­ing above or be­low these val­ues is fine, just as long as you have e2050 tau.

Act­ive #

Idle #