The op­timal pub­lish mul­ti­plier for T5 is between 2-3 and 6-10, please check with The Sim for ac­cur­acy. 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 be­ha­vior of c1 and c2 vari­ables in this the­ory, as the be­ha­vi­ors are quite unique.

$\dot{\rho} = q_1^{1.15}q_2q$

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

The first line states that the rate of change of

$\rho$

$q_1, q_2, q$

$q_1$

$q_2$

$q$

The second line defines this the­ory. It de­scribes the be­ha­vior 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

$c_1$

$q$

$q$

$c_2$

$q$

$c_3$

$c_2$

$q$

$c_2$

The max­imum value of

$q$

$c_2c_3^{1.1}$

$q$

$q$

$q$

$c_2, c_3$

Ap­prox­im­ate vari­able strengths on

$\dot\rho$

Think of buy­ing

$c_1$

$c_2$

If all one does is buy

$c_1$

$c_2$

$c_2$

$c_1$

There­fore us­ing the bi­cycle ana­logy, buy

$c_2$

$\dot{q}$

$c_2$

$c_1$

$c_2$

When de­cid­ing when to buy

$c_1, c_2$

$c_1$

$c_2$

The strengths of each vari­able are as fol­lows:

$c_3 \approx q_2 \geq c_2 \gt q_1 \gt c_1$

Note that

$c_1$

$c_2$

Manual buy­ing c2 - READ THIS BE­FORE DO­ING THE STRATEGIES Permalink: 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

$c_2$

$e150$

$\rho$

$c_2$

You want to buy

$c_2$

$q$

1. Buy

   $c_2$

   .

   $q$

   should in­crease.
2. Once

   $q$

   in­creases slowly (or stops in­creas­ing), buy more

   $c_2$

   .
   

If you buy a

$c_2$

$q$

$c_2$

$q$

$c_2$

Once you’ve reached within

$e5$

$c_2$

$c_2$

Idle Permalink: idle#

It is not re­com­men­ded to idle the­ory 5 (see ex­plan­a­tion on

$c_1$

$c_2$

For each pub­lic­a­tion: auto­buy

$c_3$

$q_2$

1. For the first 10 seconds, auto­buy everything ex­cept

   $c_2$

   .
2. Af­ter­wards, simply auto­buy all un­til pub­lish.

Semi-Idle Permalink: semi-idle#

Semi-idle is sim­ilar to idle, but once the the­ory is re­covered, manual buy

$c_2$

$c_1$

For each pub­lic­a­tion: auto­buy

$c_3$

$q_2$

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 ~

   $e5\ \rho$

   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 Permalink: act­ive#

Here’s a simple yet ef­fect­ive act­ive strategy that can be used right un­til en­dgame. To find more op­tim­ized strategies, please see the List of the­ory strategies.

For each pub­lic­a­tion: auto­buy

$c_3$

$q_2$

$c_1$

$c_2$

$c_2$

$c_1$

Note that for faster speed, dur­ing the first part of step 2, 10 levels can be bought 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 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, 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.

All mile­stones into the 2nd mile­stone to un­lock

$c_3$

$q_1$

$c_3$

Com­ment­ary

(By Snaeky)

The­ory 5 Act­ive Strategy Video Ex­ample and Com­ment­ary By Snaeky

No com­ment­ary

(By Snaeky)

The­ory 5 Act­ive Strategy Video Ex­ample By Snaeky

(By Playspout)

The­ory 5 Act­ive Strategy Video Ex­ample By Playspout

T5 will al­ways give its best res­ults from act­ive play. However, after step 3, you can still get good res­ults while auto­buy­ing

$q_1$

$c_1$

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 Permalink: ad­di­tional-in­form­a­tion#

Pur­chase

$c_2$

$1.5q > c_2*c_3^{m_3}$

$m_3$

$q$

$2q > c_2*c_3^{m_3}$

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

T6 has the low­est de­cay of all the the­or­ies. It will be second place to T5 un­til ~

$e750$

$\gt e1300\tau$

$e350+$

At first, T6 only finds the area un­der the curve of the graph

$f(q)$

$0$

$q$

$r$

$f(q)$

$f®$

$0$

$q$

$0$

$r$

$C$

$\rho$

$c_i$

$c_i$

For ex­ample, let the in­teg­ral equal 10, and you buy an up­grade that costs

$5\rho$

$\rho$

$\rho=10-5=5$

$\rho=5$

$C=$

$-\rho=20-5=15$

$C$

$\rho$

$C$

$\rho$

$\rho$

$-C$

$+C$

* This is not the same as the sum of the cost of all

$c_i$

up­grades as not every up­grade in­creases the in­teg­ral by the same amount, i.e. an up­grade cost­ing

$5\rho$

will not in­crease the in­teg­ral by

$5\rho$

, but more or less.

Video of T6 at En­dgame

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

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

$g(x,y)$

$\mathbb{R}^{3}$

$(x,y,g(x,y))$

$(\rho_1,\rho_2,g(\rho_1,\rho_2))$

$g(\rho_1,\rho_2)$

$(\rho_1,\rho_2)$

$g(\rho_1,\rho_2)$

$g$

$g(\rho_1,\rho_2)$

$(\rho_1,\rho_2)$

$\dot{\mathbf{\rho}}$

$\mathbf{\rho}=(\rho_1,\rho_2)$

$\nabla g(\rho_1,\rho_2)$

$g$

$(\rho_1,\rho_2)$

$g$

$(\rho_1,\rho_2)$

This is the graph of the func­tion

$g$

$c_1,c_2\ldots$

$x$

$y$

The op­timal pub­lic­a­tion mul­ti­plier is

$4$

$6$

$e67$

$q_1$

$c_1$

$e1$

$\rho$

The­ory 7 Act­ive Strategy Video Ex­ample and Com­ment­ary By Snaeky

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

$f(t)$

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

Re­search 9 Swap­ping Pre-ee20k Video Ex­ample And Com­ment­ary By Snaeky

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

$1e20$

$1e60$

$1e80$

$1e100$

$e250$

$e300$

$1e50$

$1e60$

$1e60$

At the start, manual buy pri­or­it­izes

$c_2$

$c_1$

$c_2$

$c_5$

$c_1$

$c_2$

$c_4$

$c_1$