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.

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

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 , which speeds up the rate at which we reach the max­imum value of . Note it DOES NOT in­crease the ac­tual max­imum value of it­self. We also have . This in­creases the max­imum value of . However, it HALVES the speed at which this max­imum value is reached. There­fore we must be care­ful to not buy too many at once.

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

The max­imum value of is . Once reaches this max­imum value, dot will be zero and will not grow any­more un­til we buy either more .

Ap­prox­im­ate vari­able strengths on with all mile­stones are as fol­lows:

Think of buy­ing as throt­tling on the bi­cycle faster. Buy­ing is sim­ilar to shift­ing the bi­cycle gear up by 1 gear.

If all one does is buy and never , then they are stuck in gear 1 forever and make no pro­gress. However, if all one does is buy and never , then this is sim­ilar to try­ing to ride from highest gear from 0 speed, which takes a long time and a lot of ef­fort.

There­fore us­ing the bi­cycle ana­logy, buy only when , speed, is enough to sup­port it; not too early and not too late. Fur­ther­more, later in the pub­lic­a­tion, buy only 1 level of at a time. Buy only right after buy­ing (shift­ing up gear).

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

The strengths of each vari­able are as fol­lows:

Note that and have spe­cial in­ter­ac­tions.

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 . If you’re over , start with buy­ing 10x vari­ables at a time. Oth­er­wise buy 1 level of at a time.

You want to buy un­til the value of in­creases slowly. The pat­tern should be:

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

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

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

Idle Permalink: idle#

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

For each pub­lic­a­tion: auto­buy and .

1. For the first 10 seconds, auto­buy everything ex­cept .
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 and dis­able .

For each pub­lic­a­tion: auto­buy and .

1. For the first 10 seconds, auto­buy everything ex­cept .
2. Then we want to manu­ally buy . See Manual Buy­ing c2. Do this un­til it slows down and you’re within ~ un­der last pub­lic­a­tion mark.
3. Then we auto­buy all un­til has reached its pre­vi­ous pub­lic­a­tion value (fin­ished re­cov­ery).
4. Af­ter­wards, de­ac­tiv­ate 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 and . Buy ONLY right after buy­ing from steps 2 on­wards. Prac­tic­ally, everytime level of is bought from steps 2 on­wards, ~5-6 levels of would be bought.

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 .
2. Then we want to manu­ally buy . See Manual Buy­ing c2.
3. Then we auto­buy . Out of these 3 vari­ables, find the one with the cheapest cost. Then buy un­til its cost ex­ceeds 15% of the cheapest vari­able found above. Buy ONLY right after buy­ing a level of .
4. Once the the­ory has re­covered to its pre­vi­ous pub­lic­a­tion mark, slowly put less em­phasis on . When in doubt, have 's cost be sim­ilar to 's cost. Con­tinue do­ing step 2 un­til pub­lish.

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

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 and manu­ally pur­chas­ing 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 Permalink: ad­di­tional-in­form­a­tion#

Pur­chase when . is
the num­ber of mile­stone 3.

be­gins to slow down upon reach­ing .

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 ~ and is the only the­ory that can get to . You should overnight this and T4 after you get your T2 to .

At first, T6 only finds the area un­der the curve of the graph from to , 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 , this new equa­tion is find­ing the volume of the graph within the bounds of planes and from to and to re­spect­ively.

is equal to the sum of all in­stant­an­eous changes in , or equi­val­ently in­stant­an­eous in­creases to the in­teg­ral, caused by buy­ing up­grades*. When a up­grade is pur­chased, the value of the in­teg­ral in­creases by a re­spect­ive amount.

For ex­ample, let the in­teg­ral equal 10, and you buy an up­grade that costs . should de­crease to . If the in­teg­ral in­creases to 20, then to for , in­teg­ral.

acts as the mech­an­ism which de­creases the cor­rect amount after a pur­chase. If did not counter-bal­ance, then would di­verge as every up­grade would in­crease in­stead of de­crease (also the reason it is and not ).

* This is not the same as the sum of the cost of all up­grades as not every up­grade in­creases the in­teg­ral by the same amount, i.e. an up­grade cost­ing will not in­crease the in­teg­ral by , 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 can be seen as a sur­face in (con­sid­er­ing the set of points , see at­tached im­age). is a point on this sur­face. Our goal is to max­im­ize , i.e. to find that max­im­ize . No­tice that the func­tion 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 is to move to­wards the dir­ec­tion of steep­est as­cent. This is what is pre­cisely done by set­ting (which is the dir­ec­tion the point will move to­ward) to (i.e. the gradi­ent of eval­u­ated at , which gives the dir­ec­tion of steep­est as­cent of at the point .

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

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

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 swap­ping as a long held name of the strategy.

1. Wait till stops grow­ing with stu­dents in R9 push­ing .
2. Start ac­cel (prefer­ably keep it between prestiges).
3. Po­ten­tially sit here to stack t for big­ger when you have stu­dents in . Only do this when you are near a gradu­ation mark. This is not use­ful if you will not swap into .
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 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.

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 mile­stone step. It will speed up once you hit and even faster at and etc. un­til ~-. The worst part is the - grind. The grind to will take a good bit of time but is faster with R9.

At the start, manual buy pri­or­it­izes then then rest. Once you get to 0/​0/​0/​2, pri­or­it­ize and then then the rest. Once at 2/​0/​3/​0, you will pri­or­it­ize and then then the rest after. This con­tin­ues to max at 2/​3/​3/​3.