In math­em­at­ics, a re­cur­rence re­la­tion is an equa­tion that re­lies on an ini­tial term and a pre­vi­ous term to change. We start with the cur­rent tick’s term, , and a con­stant add-on to ob­tain the value of the next tick, . This gives us an equa­tion equi­val­ent to , with a chan­ging value and a con­stant that is the ini­tial value of 1. Later when we add the term, this is now say­ing that we are now adding each tick the value of from the pre­vi­ous tick ago with a con­stant put to the power of . This is the same with the next term , with the value of two ticks ago and a mul­ti­plier put to the power . When we mul­tiply the term by the term chan­ging the con­stant ad­di­tion to be­ing based on the value of from the pre­vi­ous tick with the value of . The fi­nal mile­stone up­grade raises the ex­po­nent of from to to to .

This the­ory also has its ad­jus­ted tick­speed cal­cu­lated by . This lengthens the nor­mal tick length of to that value which speeds up the the­ory.

Ini­tial Permalink: ini­tial#

First mile­stone Permalink: first-mile­stone#

Second mile­stone Permalink: second-mile­stone#

Third mile­stone Permalink: third-mile­stone#

Fourth to Sixth mile­stone Permalink: fourth-to-sixth-mile­stone#

The pub­lic­a­tion mul­ti­plier has no op­timal fit, as it fluc­tu­ates a lot, but here is known: 4-6 to start; 3-4 between and ; the pub­lic­a­tion mul­ti­plier os­cil­lates between 2.5 and 5 past . Once you get your first mile­stone, you can turn off and un­til act­ive strat.

The act­ive strat fol­lows but only works when you have all mile­stones past . T1 is the only the­ory where the re­cent value of in­flu­ences the rate of change of there­fore buy­ing a vari­able as soon as you can af­ford it will slow your pro­gress. Lategame, buy­ing up­grades im­me­di­ately will slow you more than the be­ne­fit of the up­grade be­cause and dom­in­ate. If the next level costs and you have , buy­ing that level will re­duce to . This re­duces your by roughly a factor of . There are terms that in­flu­ence the rate of change of , and all are af­fected by the pre­vi­ous state of . The act­ive strategy around this is known as T1Ra­tio. The val­ues in the chart found here are to be only used when you are past and max mile­stones. They rep­res­ent how to pur­chase each vari­able based on the state of the the­ory at the time of pur­chase.

Note: If you are not do­ing the act­ive strat, then simply turn off and after mile­stone 1 () and auto­buy rest un­til ee6k.

The video be­low is only good for early between and .

The­ory 1 Act­ive Strategy Video Ex­ample and Com­ment­ary By Snaeky

This second the­ory is fo­cus­ing on de­riv­at­ives. De­riv­at­ives in math­em­at­ics are the rate of change of the func­tion they are the de­riv­at­ive of. For the case of and , is the de­riv­at­ive of . This fol­lows the power rule for de­riv­at­ives:

In sim­pler terms, it works sim­ilar to how up­grades work for equa­tion with con­tinu­ous ad­di­tion of the pre­vi­ous to the next term, but with con­tinu­ous ad­di­tion of to the term above . These two val­ues of and are mul­ti­plied to pro­duce the de­riv­at­ive of , shown by New­ton’s de­riv­at­ive nota­tion . This would give the equa­tion of to be . The other mile­stones be­sides more and de­riv­at­ives in­crease the ex­po­nent of and re­spect­ively. The reason why and de­riv­at­ives are more power­ful long-term than the ex­po­nents is that they take time to build up and even­tu­ally over­take and keep in­creas­ing and while the ex­po­nents have a never-chan­ging boost.

Ini­tial Permalink: ini­tial-1#

First and Second mile­stones Permalink: first-and-second-mile­stones#

Third and Fourth mile­stones Permalink: third-and-fourth-mile­stones#

Fifth to Sev­enth mile­stones Permalink: fifth-to-sev­enth-mile­stones#

Eight to Tenth mile­stones Permalink: eight-to-tenth-mile­stones#

The op­timal mul­ti­plier is pretty high and is not known be­fore . The the­ory sim will re­com­mend pub­lic­a­tion mul­ti­pli­ers be­low these val­ues, but the sim’s T2MS does not cur­rently have coast­ing. The mul­ti­pli­ers for act­ive play (which do use coast­ing) we know at the mo­ment are:

• is to
• is

For both strategies the mile­stones are lis­ted in the or­der X>Y, where X and Y are the mile­stones as nu­mer­ic­ally ordered top to bot­tom in-game, are to be maxed in or­der from left to right.

Idle Permalink: idle#

For the idle strategy, you want to pri­or­it­ize buy­ing mile­stone levels of 1>2. If you have more than 4 mile­stones, you will pri­or­it­ize mile­stone 1>2>3>4. You will want to pub­lish at about 10-100 mul­ti­plier be­fore and about a mul­ti­plier after , but lar­ger mul­ti­pli­ers are fine. If pos­sible, swap to mile­stones 3>4>1>2 at the end be­fore pub­lish­ing for an ad­di­tional boost.

Act­ive Permalink: act­ive#

The goal of the act­ive strategy is to grow and as much as pos­sible while be­ing able to take ad­vant­age of the ex­po­nent mile­stones too, yield­ing a large boost from that growth. The act­ive for T2 is on a 50-second cycle between two mile­stone sets: 10 seconds for ex­po­nent pri­or­ity (Mile­stones 3 and 4) and 40 seconds for de­riv­at­ive pri­or­ity (Mile­stones 1 and 2) . You will start a pub­lic­a­tion with ex­po­nent pri­or­ity as the cost of the vari­ables gained from mile­stones 1 and 2 are too large for you to get right away. When you can af­ford them, you will start the cycle. The full cycle is lis­ted be­low:

1-3 Mile­stones

3>4 (10s) → 1 (40s) → 3>4 (10s) → 2 (40s) →
re­peat → coast and pub­lish

4+ Mile­stones

3>4>1>2 (10s) → 1>2>3>4 (40s) →
3>4>1>2 (10s) → 2>1>3>4 (40s) →
re­peat → coast and pub­lish

Past , the act­ive strat will be­come ex­po­nen­tially less ef­fect­ive. At , you would start to idle T2 overnight only. Un­til you have over from the­ory 2, this is the best the­ory to run idle overnight.

When you get to The­ory 3 at ee7k, move on to push­ing The­ory 3 when act­ive and run­ning T2 overnight. The above is simply an op­tion if you rather not work on T3 now.

The­ory 2 Act­ive Strategy Video Ex­ample and Com­ment­ary By Snaeky

The basis of this the­ory and un­der­stand­ing how it works is based on
mat­rix mul­ti­plic­a­tion. The fol­low­ing color-cod­ing helps dis­plays
how mat­rix mul­ti­plic­a­tion works:

Ori­ginal Im­age (April 2021 - May 2025)

This gives the basis for why cer­tain up­grades are more power­ful than oth­ers. The ex­po­nents on , , and all dir­ectly af­fect pro­duc­tion which is used for . An ex­tra di­men­sion roughly gives more pro­duc­tion as it adds an ex­tra term to the pro­duc­tion.

The op­timal pub­lic­a­tion mul­ti­plier is about 2-3 without cruis­ing and 3-4 with cruis­ing. If you de­cide to play act­ively, there is a form of ex­po­nent swap­ping strat to be aware of. This is a dif­fi­cult strategy be­cause it re­quires you to no­tice when a cer­tain threshold hap­pens. It hap­pens when the fol­low­ing oc­curs:

When this hap­pens swap your ex­po­nents from to and you will get a little up­grade boost. It also al­lows for a slight push of for up­grades to and , but this is a lot less im­pact­ful and less no­tice­able. This strategy also works with and but is usu­ally not as com­mon.

If you de­cide to buy manu­ally, the fo­cus areas are buy­ing , , and when their cost is e1 lower than , , and re­spect­ively. These all dir­ectly boost the pro­duc­tion of which is used for . After this, if you are do­ing the act­ive ex­po­nent swap­ping strategy de­scribed in the pre­vi­ous para­graph, your next fo­cus will be on , , and as these boost pro­duc­tion which in­creases the like­li­hood for the ex­po­nent swap to oc­cur. This leaves the , , and up­grades at the low­est pri­or­ity. If you are not us­ing the ex­po­nent swap­ping strategy from the pre­vi­ous para­graph, then all the re­main­ing up­grades should be bought at equi­val­ent pri­or­ity.

At the end of any pub­lic­a­tion, around a 2-3 mul­ti­plier, you should turn off and as they cost . You will cruise un­til you get to a 3-4 mul­ti­plier. Pub­lish and turn back on cost­ing vari­ables and re­peat.

Com­ment­ary

The­ory 3 Act­ive Strategy Video Ex­ample By Snaeky

The­ory 4 is based on Poly­no­mi­als, which con­tain terms of the form etc. In this case, in­stead of ‘x’ it’s ‘q’. The strategies for this the­ory are quite simple com­pared to the pre­vi­ous the­ory, es­pe­cially late game strategies.

The first line states that the rate of change of is the sum of a bunch of poly­no­mial terms. We have a bunch of ‘c’ vari­ables mul­ti­plied by ‘q’. We can in­crease by buy­ing and up­grades. Note that this is with all mile­stones. You’ll not have all of these at the be­gin­ning.

The second line is more unique. It says that is pro­por­tional to the in­verse of it­self! This means that the more we have, the slower grows, as de­creases. This means that and are not as strong as they first ap­pear. However, we still want to buy them in gen­eral un­less stated oth­er­wise as slow growth is bet­ter than no growth.

For the more math­em­at­ic­ally ob­ser­v­ant reader, we may in­teg­rate the equa­tion and con­clude that is pro­por­tional to the square root of time. This means that even though grows slower with in­creas­ing , there is the­or­et­ic­ally no fi­nite limit on the max­imum value of .

Ap­prox­im­ate vari­able strengths on with all mile­stones are as fol­lows:

The strengths of each vari­able are as fol­lows:

Early game (be­fore 14k f(t)):

   > > > > > > >

From 14k f(t) to mid-late game (about e350+ T4):

   > > > > > everything else

From e350+ T4 to end game:

   > > > everything else

Idle Permalink: idle-1#

T4 is quite idle friendly com­pared to T3 and T1. Here are some simple idle strategies for T4:

Start to e25:

Auto­buy , . DON’T buy , , ! The term is bad early on. Pub­lish at ~2.5-3 if pos­sible.

e25 to e175:

Get the ‘Add the term’ mile­stones. Pri­or­it­ize these ones first un­til max­imum. Now auto­buy , , ONLY. Best pub­lic­a­tion mul­ti­plier is ~6-7.

When and are un­locked, add and to the auto­buy vari­ables. DON’T auto­buy , , ! Pri­or­it­ize the mile­stones over the ex­po­nents. Try to pub­lish between 12-20.

e175 to en­dgame:

Simply auto­buy , , ONLY. Buy 1 level of to start the the­ory. Pub­lish at ~4-5.

Semi-Idle Permalink: semi-idle#

There’s no stra­tegic dif­fer­ence between semi-idle and idle for this the­ory. The main dif­fer­ence is with semi-idle, we would pub­lish more of­ten since we check the game more of­ten. We would­n’t over­shoot the op­timal mul­ti­plier as much.

Act­ive Permalink: act­ive-1#

T4 act­ive is more in­volved. However it is not as de­mand­ing as T3 or T1 act­ive.

Start to e75:

Auto­buy . DON’T buy , , ! The term is bad early on. Buy un­til its cost ex­ceeds about 15% of cost. Pub­lish at about 2.5-3 if pos­sible. When we reach , we get the ex­po­nent mile­stone (note the dif­fer­ence between this strategy and the idle strategy). With the ex­po­nent mile­stone, the term re­mains the strongest term IF we can babysit and pub­lish of­ten (at about 2.5-3). The strategy re­mains the same oth­er­wise. Note that since we’re only buy­ing and (NO , , , , , !), all the ‘q’ re­lated mile­stones are use­less for now.

e75 to e175 OR 14k f(t):

Now here is where we can ap­ply some more ad­vanced strategies. Con­sider that the term is strong early on, but falls off as the value of in­creases. Then we can con­clude that we can start with the same strategy as be­fore. But once we reach our pre­vi­ous pub­lic­a­tion point, we can switch to the fol­low­ing strategy:

1. Do the same strategy as be­fore un­til we reach our pre­vi­ous pub­lic­a­tion point.
2. Take point(s) out of the ex­po­nent mile­stones and un­lock all the terms (the first mile­stone). We should now have ac­cess to .
3. Auto­buy , , , .
4. If you want to op­tim­ize a bit more, you can buy un­til its cost ex­ceeds about 15% of . Oth­er­wise it’s ok to also auto­buy .
5. DO NOT auto­buy , , .
6. Pub­lish at ~10-20. Once pub­lished, re­mem­ber to take out the mile­stone point and put it back into the ex­po­nent to re­peat step 1.

If done right, this strategy is sig­ni­fic­antly faster than the idle strategies above. The lo­gic with this strategy is that the , , terms scale well with ‘q’. However we need enough to buy a lot of q. So in the be­gin­ning we buy only as usual to ac­cu­mu­late enough so that we can buy to stack q. Once we have enough q, the , , terms will outscale. Note that after e14k , we will un­lock cer­tain up­grades that make bet­ter again.

e175 OR 14k f(t) to ~e300 T4

We will do the ex­act same strategy as in the #start to sec­tion above. This is be­cause be­come really strong again and the terms take too long to outscale. Note that we still don’t buy .

~e300 to en­dgame

At this point the term starts to be­come dom­in­ant. There­fore we will pri­or­it­ize buy­ing , , . We will NOT buy any­thing else ex­cept 1 level of to start the the­ory. If you wish, you can buy at about 15% ra­tio to cost. It is also ok to auto­buy . The term will re­main dom­in­ant un­til en­dgame.