The first line states that the rate of change in is . Ini­tially it’s simply without any ex­po­nent. With mile­stones we add more ex­po­nents.

For the second line, the higher the (spelled ‘chi’, pro­nounced as ‘kai’), the higher the . We want to in­crease by in­creas­ing and . The signs of and will al­ways match, so the frac­tion can’t be neg­at­ive. Ad­di­tion­ally, the vari­able is a mile­stone which is not ini­tially avail­able.

The third line is the most com­plic­ated. Gen­er­ally we can fac­tor­ize an equa­tion when its graph touches the x-axis. For a sine curve, it touches the x-axis start­ing from x = 0, and re­peats every x= . These mul­ti­plied factors form the basis of the Wei­er­strass Sine Product. A sim­pler in­ter­pret­a­tion is that we can see ‘x’ ap­pear­ing both out­side and in­side the products in the nu­mer­ator. Since is ‘x’ here, the higher the , the higher the as stated earlier.

Fi­nally, the ac­tual equa­tion: in­creas­ing and in­creases . Note that from the frac­tion, we don’t want to in­crease only or only . Rather we should in­crease both. Us­ing stand­ard strategies this should be no prob­lem. The part in the de­nom­in­ator is a mile­stone term. This means that is bet­ter than as more mile­stones are ac­cu­mu­lated.

Ap­prox­im­ate vari­able strengths on with all mile­stones are as fol­lows:

Early game the vari­able strengths are ordered as fol­lows:

Late game these be­come:

Idle Permalink: idle#

Be­fore you get e400 for idle, simply auto­buy all.

Once you have e400 , starts to be­come ex­tremely bad. Be­cause of this, the new idle strategy would be to auto­buy all for 20 seconds or so. Then turn OFF. Con­tinue to auto­buy the rest of the vari­ables.

Act­ive Permalink: act­ive#

For a simple act­ive strategy be­fore e400 , simply auto­buy and since they double the rates long term. and give ap­prox­im­ately 60% boost (with be­com­ing more power­ful with mile­stones and vice versa for ). We will buy and when their costs are less than 50% of the min­imum of and .

For , we will buy it when its cost is less than 10% of the min­imum of and . For ex­ample, if costs and costs , we would not buy as it’s ‘too ex­pens­ive’ com­pared to .

For act­ive strategy, starts to be­come more power­ful than . If their costs are sim­ilar, we will pri­or­it­ize first. For ex­ample, if costs and costs , we will buy first. Sim­il­arly to the idle strategy, we will buy only for the first 20 seconds or so. If you want more in­form­a­tion on the dif­fer­ent strategies per­tain­ing to WSP, please see List of the­ory strategies.

All mile­stones into the 3rd/​last mile­stone. Then into 2nd mile­stone, then into 1st mile­stone.

For mile­stone swap­ping, swap all mile­stones from 2nd and 3rd into 1st mile­stone. Usu­ally you only do this when you’re about to pub­lish.

The first line is the main part of the equa­tion. We want to max­im­ize to in­crease . The ‘1.06’ ex­po­nent is from mile­stones. The de­fault is no ex­po­nent. From the equa­tion, we can see that is pro­por­tional to ap­prox­im­ately . This means that if we quad­ruple , we would ap­prox­im­ately double long term. The de­nom­in­ator of the frac­tion has a gamma sym­bol () which looks like the let­ter ‘y’. As our in­creases, our be­comes closer to ‘e’, so the de­nom­in­ator will de­crease, which in­creases . We will ex­plore in the next equa­tion.

The second equa­tion refers to Stirl­ing’s ap­prox­im­a­tion of Euler’s num­ber ‘’. As in­creases, con­verges to Euler’s num­ber. Long term we can ap­prox­im­ate this con­ver­gence as lin­ear. The im­plic­a­tion is if we double , will be twice as close to Euler’s num­ber, so in the first equa­tion will be halved.

The third equa­tion relates with and some up­grades. The most in­ter­est­ing part is the ex­po­nent part con­tain­ing . The neg­at­ive ex­po­nent ac­tu­ally im­plies that as in­creases, DE­CREASES. If is high, does­n’t grow as fast (it still grows). This has im­plic­a­tion on the first equa­tion as well, since de­pends on , which de­pends on .

The fourth equa­tion relates with some up­grades. This one is re­l­at­ively simple; in­crease and to in­crease . The ‘1.04’ ex­po­nents are from mile­stones.

The fi­nal equa­tion simply states the value of . The lower the bet­ter. De­fault without mile­stone is .

Ap­prox­im­ate vari­able strengths on with all mile­stones are as fol­lows:

All vari­ables in SL are about the same in power, ex­cept for and (which are slightly worse than and . Se­lect­ively buy­ing vari­ables at cer­tain times (act­ive) yields very little res­ults. There­fore, we can get away with auto­buy all for idle. Be­fore auto­buy, simply buy the cheapest vari­able. If you want more de­tails on SL strategies, in par­tic­u­lar the ex­e­cu­tion of vari­ous strategies, please see List of the­ory strategies.

For act­ive, there is a mile­stone swap­ping strategy that is sig­ni­fic­antly faster than id­ling (ap­prox­im­ately twice the speed). If we care­fully ex­am­ine the ef­fects of each mile­stone, we can con­clude the fol­low­ing:

1st mile­stone: In­creases ex­po­nent and in­creases straight away. The ac­tual value of does not in­crease.

3rd/​4th mile­stone: In­crease / ex­po­nents, and , and . This also in­creases . However, the ef­fect is long-term and not in­stant­an­eous un­like the ef­fect of the 1st mile­stone.

We have dif­fer­ent mile­stones which af­fect the same thing (), but one is in­stant­an­eous, while the other builds over time. This forms the basis of ‘mile­stone swap­ping’, swap­ping mile­stones at cer­tain times to max­im­ize per hour. If you’ve done T2 mile­stone swap­ping, this should be fa­mil­iar.

We ini­tially put our mile­stones in the 4th and 3rd mile­stones. Once our does­n’t in­crease quickly any­more, we switch mile­stones to the 1st one to gain a burst of . Once our is not in­creas­ing quickly any­more, we switch back to the 4th and 3rd mile­stone!

(Cour­tesy of Gen).

x>x>x>x rep­res­ent the max buy or­der of mile­stones not the amount al­loc­ated.

For ex­ample, 4>3>1>2 means “Al­loc­ate everything into 4th mile­stone, then use leftovers into 3rd mile­stone, then into 1st mile­stone, then into 2nd mile­stone”.

SLMS

4>3>1>2 (60s) → 1>2>4>3 (60s*) → re­peat

* between , 1>2>4>3 should be 20s.

SLMS2

1>2>4>3* (30s) → 2>1>4>3* (60s) → 1>2>4>3** (30s) →
4>3>1>2** (60s) → re­peat

* & turned off.
** & turned off.

SLMS3

2>1>4>3 (20s) → 4>3>1>2 (60s) → re­peat

When to use strategies:

un­til e100: SLMS

e100 - e175: SLMS2

e175 - e200: SLMS3

e200 - e300: SLMS

For a more pre­cise de­scrip­tion of SLMS, check out the the­ory strategy sec­tion.

At this point, the the­ory be­comes very idle. We simply auto­buy all vari­ables. Pub­lish at ap­prox­im­ately 8-10 mul­ti­plier. If you wish to im­prove ef­fi­ciency, dis­able & at about 4.5 pub­lic­a­tion mul­ti­plier and & at 6.0 mul­ti­plier un­til pub­lish.

Idle Permalink: idle-1#

Act­ive Permalink: act­ive-1#

See SL Mile­stone Swap­ping Strategies

The first line is the main equa­tion. We want to max­im­ize . All the terms and their ex­po­nents are ob­tained from mile­stones. Parts of the square root term are also ob­tained from mile­stones. Note that the and the terms are ef­fect­ively re­dund­ant at all stages of this the­ory; but due to them pur­chas­ing and re­spect­ively, they are very im­port­ant.

The second line defines the graph shown. Since is graphed on the com­plex over time, it is pos­sible to have it show as a particle spiral­ing through space.

The third line de­scribes and , which are used to gen­er­ate ‘’ and ‘’ cur­ren­cies. This line by it­self does­n’t do much.

The fourth line simply de­scribes . This is used in the first equa­tion dir­ectly.

The fifth and fi­nal line use the res­ults from the 3rd line, so ef­fect­ively and .

Ap­prox­im­ate vari­able strengths on with all mile­stones are as fol­lows:

Ini­tially, you only have , , and un­locked. Buy at about 1/​8th cost of , and buy when it’s avail­able. At e20 when auto­buy­ers are un­locked, for idle, simply auto­buy all. For act­ive, con­tinue to do what you were do­ing (buy­ing at 1/​8th cost of ). There are also more ad­vanced strategies, in par­tic­u­lar EFAI. For its de­scrip­tion and ex­e­cu­tion, please see List of the­ory strategies.

The first 2 mile­stones are re­dund­ant by them­selves. The term and the term are in­sig­ni­fic­ant com­pared to the term.
Once you un­lock the 3rd mile­stone ( term) however, we can buy at 1/​4th of cost.

The first line is self ex­plan­at­ory. The ex­po­nents on are from mile­stones. ‘’ will in­crease dur­ing the pub­lic­a­tion.

For the second line, both the vari­able and its ex­po­nents are from mile­stones. The ab­so­lute value sec­tion on the right de­scribes the ap­prox­im­a­tion of / to . As / get closer to , the en­tire right sec­tion gets lar­ger and lar­ger (be­cause of the -1 power).

The third and fourth lines are re­cur­rence re­la­tions on and . This means that the cur­rent value of and de­pend on their pre­vi­ous val­ues. We start with = 1, = 3. The equa­tion will then read as:

.

Then .

Sim­ilar lo­gic is ap­plied to equa­tions.

This oc­curs un­til we reach and reach whatever ‘m’ val­ues we have. This is shown in the next equa­tion:

The fourth equa­tion relates ‘m’ as de­scribed above. We can see that as we buy and , our will in­crease, so the 2 re­cur­rence equa­tions above will ‘re­peat’ more of­ten and , will in­crease. From how and val­ues are cal­cu­lated, buy­ing 1 level of or will in­crease by 1.

Ap­prox­im­ate vari­able strengths on with all mile­stones are as fol­lows:

Idle Permalink: idle-2#

For idle, auto­buy all. The idle strategy does­n’t change much. If you would like to be more ef­fi­cient while still be­ing idle, re­move mile­stones and stack them into the ex­po­nent mile­stones when you are about to pub­lish (from around e80 to e500). Don’t for­get to change mile­stones back after pub­lish­ing!

Once all mile­stones are un­locked, auto­buy all!

Act­ive Permalink: act­ive-2#

The act­ive strategies are sig­ni­fic­antly more in­volved. De­pend­ing on how act­ive you would like to be, there are sev­eral po­ten­tial strategies. There’s the stand­ard doub­ling chas­ing CSRd, which is just auto­buy all ex­cept and , where you buy them when they are less than 10% cost of min­imum(, , and ).

For the mile­stone swap­ping strategy, the gen­eral idea is to switch mile­stones from and its ex­po­nents, to ex­po­nent mile­stones whenever we are ‘close’ to a power­ful up­grade. Please see the The­ory Strategies sec­tion of the guide for how to per­form mile­stone swap­ping.

CSR2 Mile­stone Swap­ping Ex­plan­a­tion Permalink: csr2-mile­stone-swap­ping-ex­plan­a­tion#

This the­ory has a mile­stone swap­ping strategy be­fore full mile­stones. We have ex­po­nent mile­stones, which in­crease straight away. We also have re­lated mile­stones, which in­creases the vari­able, which in­creases .

The reason mile­stone swap­ping works is be­cause the be­ne­fits of us­ing re­lated mile­stones (hav­ing high ) re­main when you switch to ex­po­nent mile­stones. If we only use ex­po­nent, then we have really low . If we only use re­lated mile­stones, then we have high , but low . If we reg­u­larly swap them, we can in­crease through re­lated mile­stones, then take ad­vant­age of the ex­po­nent mile­stones, while keep­ing the high value of we’ve ac­cu­mu­lated earlier!

For a more de­tailed ex­plan­a­tion on how to ac­tu­ally do the strategy, please see the The­ory Strategies sec­tion of the guide.

Base Equa­tion Permalink: base-equa­tion#

With and equa­tions be­com­ing:

Note that the up­per bound of the in­teg­ral changes from after pur­chas­ing mile­stone 3.

The first equa­tion is for , which starts off simple, but gets more com­plic­ated as more mile­stones are reached and perma-up­grades are pur­chased. Ini­tially, is fairly simple to cal­cu­late as is just , is just the vari­able, and the rad­ical is just / where is just . However, once is ad­ded to the equa­tion, the rad­ical be­comes which can be es­tim­ated by rais­ing to the highest power of by 1 and upon max­ing out the mile­stone, it be­comes . The vari­ables and are simple mul­ti­pli­ers that do not change over time without pur­chas­ing them with .

The second equa­tion is for , which seems simple at first, but gets more dif­fi­cult to un­der­stand once we get to the frac­tional in­teg­ral. The nota­tion in game is rarely used, but it is used to save space. Tap­ping and hold­ing the equa­tion will give the full equa­tion. When in­creases, the frac­tional in­teg­ral ap­proaches 1, which makes the frac­tional in­teg­ral get closer to, yet still smal­ler than, the full in­teg­ral. By sub­tract­ing the two, then di­vid­ing 1 by the dif­fer­ence, we get a very large num­ber.

Ap­prox­im­ate vari­able strengths on their re­spect­ive var­dots with all mile­stones are as fol­lows:

Idle Permalink: idle-3#

For idle, auto­buy all. The idle strategy does­n’t change much other than not do­ing Mile­stone Swap. If you are able to check in every 30 minutes or so, manu­ally buy and . Make sure that to auto­buy when close to get­ting a boost.

Act­ive Permalink: act­ive-3#

The act­ive strategies are a bit more in­volved. De­pend­ing on how act­ive you would like to be, there are sev­eral po­ten­tial strategies. There’s the stand­ard doub­ling chas­ing FId, which is just auto­buy all ex­cept and , where you buy them when they are less than 10% cost of min­imum(, , and ).

For the mile­stone swap­ping strategy, the gen­eral idea is to switch mile­stones from , to / mile­stones whenever we gain 3x to after pur­chas­ing , or some gain ad­jus­ted for from pur­chas­ing . Please see the The­ory Strategies sec­tion of the guide for how to per­form mile­stone swap­ping.

FI Mile­stone Swap­ping Ex­plan­a­tion Permalink: fi-mile­stone-swap­ping-ex­plan­a­tion#

This the­ory has a mile­stone swap­ping strategy be­fore full mile­stones. We have ex­po­nent mile­stones, which in­creases .

The reason mile­stone swap­ping works is be­cause the be­ne­fits of us­ing re­lated mile­stones (hav­ing high ) re­main when you switch to and mile­stones. If we only use ex­po­nent, then we have really high , however, we don’t have the be­ne­fits to that and provide. If we only use and mile­stones, then we have low , but have nor­mal . If we reg­u­larly swap them, we can in­crease through the mile­stone, then take ad­vant­age of the and mile­stones to gain , while keep­ing the high value of we’ve ac­cu­mu­lated earlier!

For a more de­tailed ex­plan­a­tion on how to ac­tu­ally do the strategy, please see the The­ory Strategies sec­tion of the guide.

In FI, you can un­lock mile­stones in 2 ways:

1. by gain­ing like nor­mal, or
2. by pur­chas­ing the mile­stone up­grades for and in the per­man­ent up­grades tab where you would nor­mally buy pub­lish­ing, buy all, and auto­buy.

Buy­ing the mile­stone up­grades will not give you a mile­stone, but will in­stead in­crease the max level of the mile­stone that you pur­chased the up­grade for. For ex­ample, if you buy the perma-up­grade for lvl 1, you will per­man­ently un­lock the first lvl of the mile­stone. Mov­ing mile­stones into these are al­most al­ways the best thing you can do mid pub­lish, even if you need to sac­ri­fice a vari­able to do so, with one ex­cep­tion.

It is im­port­ant to note, however, is that buy­ing or re­fund­ing mile­stones will re­set your , level and will change the cost func­tion. Sim­il­arly, buy­ing or re­fund­ing mile­stones will re­set your and change the cost func­tion.

FI perma-up­grades are at 1e100, 1e450, and 1e1050 for the mile­stone and 1e350 and 1e750 for the mile­stone. Upon buy­ing these mile­stone, im­me­di­ately put a mile­stone from or into them de­pend­ing on how many mile­stone you have, ex­cept for the 3rd level of the mile­stone.

The 3rd level of the mile­stone is bad early on, and is only worth buy­ing at e1076. Swap­ping to the 3rd level of the mile­stone mid-pub is known as Per­maSwap, check the the­ory sim­u­lator to know if you should do this strategy.

Colored mile­stones are perma-up­grade mile­stones that move into that up­grade.

Main Equa­tions Permalink: main-equa­tions#