# Ex­po­nen­tial Idle Guides

## The­or­ies 1-4

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

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

### The­ory ba­sics #

Pub­lic­a­tions are equi­val­ent to prestiges for $$f(t)$$ so don’t be afraid to use them. However, the best pub­lic­a­tion mul­ti­pli­ers vary from the­ory to the­ory and will de­crease over time. If you are close to a mul­ti­plier you want, turn off auto­buyer and let $$\rho$$ in­crease without buy­ing up­grades for a faster short-term in­crease be­fore the pub­lic­a­tion (turn on after you pub­lish). This is known and ref­er­enced as “cruis­ing”. Total $$τ$$, found in the equa­tion or at the top of the screen, is a mul­ti­plic­at­ive com­bin­a­tion of all $$τ$$ from each the­ory.

Don’t be afraid to skip get­ting all mile­stones to work on the next or a bet­ter the­ory.

###### Note: If you see # → [# → # → #] → # in the mile­stone route of a the­ory, this is the sec­tion that has an act­ive strategy tied to it.

The gradu­ation route for these the­or­ies.

• 5k → 5.2k → 5.6k → 5.8k → 6k (5.8k → 6k is a long stretch)
• 6k → 7k
• 7k → 8k
• 8k → 8.4k →8.6k → 8.8k → 9k

### The­ory 1 (20σ / 5k) #

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, $$ρ_{n}$$, and a con­stant add-on to ob­tain the value of the next tick, $$ρ_{n+1}$$. This gives us an equa­tion equi­val­ent to $$ρ=at+con­stant$$, with a chan­ging value $$a$$ and a con­stant that is the ini­tial value of 1. Later when we add the $$c_{3}ρ_{n-1}^{0.2}$$ term, this is now say­ing that we are now adding each tick the value of $$ρ$$ from the pre­vi­ous tick ago with a con­stant $$c_{3}$$ put to the power of $$0.2$$. This is the same with the next term $$c_{4}ρ_{n-2}^{0.3}$$, with the value of $$ρ$$ two ticks ago and a mul­ti­plier $$c_4$$ put to the power $$0.3$$. When we mul­tiply the $$c_1c_2$$ term by the term $$1+ln(ρ)/​100$$ chan­ging the con­stant ad­di­tion to be­ing based on the value of $$ρ$$ from the pre­vi­ous tick with the value of $$1+ln(ρ)/​100$$. The fi­nal mile­stone up­grade raises the ex­po­nent of $$c_1$$ from $$1.00$$ to $$1.05$$ to $$1.10$$ to $$1.15$$.

This the­ory also has its ad­jus­ted tick­speed cal­cu­lated by $$q_{1}*q_{2}$$. This lengthens the nor­mal tick length of $$0.1/​sec$$ to that value which speeds up the the­ory.

#### T1 for­mula #

##### Ini­tial

$ρ_{n+1} = ρ_n + c_1c_2$

##### First mile­stone

$ρ_{n+1} = ρ_n + c_1c_2 + c_3ρ_{n-1}^{0.2}$

##### Second mile­stone

$ρ_{n+1} = ρ_n + c_1c_2 + c_3ρ_{n-1}^{0.2} + c_4ρ_{n-2}^{0.3}$

##### Third mile­stone

$ρ_{n+1} = ρ_n + c_1c_2 \left( 1+\frac{ln(ρ_n)}{100} \right) \\\ + c_3ρ_{n-1}^{0.2} + c_4ρ_{n-2}^{0.3}$

##### Fourth to Sixth mile­stone

$ρ_{n+1} = ρ_n + c_1^{1.15}c_2 \left( 1+\frac{ln(ρ_n)}{100} \right) \\\ + c_3ρ_{n-1}^{0.2} + c_4ρ_{n-2}^{0.3}$

#### T1 strategy #

The pub­lic­a­tion mul­ti­plier has no op­timal fit, as it fluc­tu­ates a lot, but here is known: 4-6 to start; 3-4 between 1e100 and 1e150; the pub­lic­a­tion mul­ti­plier os­cil­lates between 2.5 and 5 past e150. Once you get your second mile­stone, you can turn off $$c_1$$ and $$c_2$$ un­til e150 act­ive strat.

The act­ive strat fol­lows but only works when you have all mile­stones past e150. T1 is the only the­ory where the re­cent value of $$ρ$$ in­flu­ences the rate of change of $$ρ$$ 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 $$c_3$$ & $$c_4$$ dom­in­ate. If the next level costs $$10ρ$$ and you have $$11ρ$$ buy­ing it will re­duce to $$ρ_{n+1}$$ to $$1$$ you are re­du­cing your $$ρ_{n+1}$$ by roughly a factor of $$10$$. There are $$3$$ terms that in­flu­ence the rate of change of $$ρ$$. All are af­fected by the pre­vi­ous state of $$ρ$$. Let’s ig­nore the first since it has such a small in­flu­ence and con­sider the above case to de­term­ine when an up­grade would be bet­ter. The val­ues be­low are to be only used when you are past $$e150 τ$$ and max mile­stones. Buy each vari­able when $$ρ_1$$ is $$x$$ times lar­ger than that vari­able’s cost. For ex­ample, if $$q_1$$ costs $$2$$, buy it when $$ρ_1$$ is $$2*5.0=10 ρ_1$$.

Vari­able Mul­ti­plier
$$q_1$$ 5.0
$$q_2$$ 1.15
$$c_1$$ 10000
$$c_2$$ 1000
$$c_3$$ 2
$$c_4$$ 1.01
• Note: If you are not do­ing the act­ive strat, then simply turn off $$c_1$$ and $$c_2$$ after mile­stone 2 (e50τ) and auto­buy rest un­til ee6k.

#### T1 mile­stone route #

• 0/​0/​1 → 0/​0/​1/​1 → 0/​1/​1/​1 → 3/​1/​1/​1
• 3 → 4 → 2 → 1 → 1 → 1

### The­ory 2 (25σ / 6k) #

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 $$q_1$$ and $$q_2$$, $$q_2$$ is the de­riv­at­ive of $$q_1$$. This fol­lows the power rule for de­riv­at­ives:

$q=at^n ↔ q’=nat^{n-1}$

In sim­pler terms, it works sim­ilar to how $$x_i$$ up­grades work for $$f(t)$$ equa­tion with con­tinu­ous ad­di­tion of the pre­vi­ous $$term*dt$$ to the next $$x_{i+1}$$ term, but with con­tinu­ous ad­di­tion of $$q_i*dt$$ to the term above $$q_{i-1}$$. These two val­ues of $$r_1$$ and $$q_1$$ are mul­ti­plied to pro­duce the de­riv­at­ive of $$ρ(t)$$, shown by the new­ton de­riv­at­ive form $$\dot{ρ}$$. This would give the equa­tion of $$ρ$$ to be $$ρ(t+dt)=\dot{ρ}+ρ1*dt$$. The other mile­stones be­sides more $$q$$ and $$r$$ de­riv­at­ives in­crease the ex­po­nent of $$q$$ and $$r$$ re­spect­ively. The reason why $$q$$ and $$r$$ de­riv­at­ives are more power­ful long-term than the ex­po­nents is that they take time to build up and even­tu­ally over­take and keep in­creas­ing $$q_1$$ and $$r_1$$ while the ex­po­nents have a never-chan­ging boost.

#### T2 for­mula #

##### Ini­tial

$q_1(t+dt)=q_1+q_2*dt$

$r_1(t+dt)=r_1+r_2*dt$

$\dot{ρ}=q_1r_1$

##### First mile­stone

$q_1(t+dt)=q_1+q_2*dt+\frac{1}{2}q_3dt^2$

$r_1(t+dt)=r_1+r_2*dt$

$\dot{ρ}=q_1r_1$

##### Second mile­stone

$q_1(t+dt)=q_1+q_2*dt+\frac{1}{2}q_3dt^2+\frac{1}{6}q_4dt^3$

$r_1(t+dt)=r_1+r_2*dt$

$\dot{ρ}=q_1r$

##### Third and Fourth mile­stones

$q_1(t+dt)=q_1+q_2*dt+\frac{1}{2}q_3dt^2+\frac{1}{6}q_4dt^3$

$r_1(t+dt)=r_1+r_2*dt+\frac{1}{2}r_3dt^2+\frac{1}{6}r_4dt^3$

$\dot{ρ}=q_1r_1$

##### Fifth to Sev­enth mile­stones

$q_1(t+dt)=q_1+q_2*dt+\frac{1}{2}q_3dt^2+\frac{1}{6}q_4dt^3$

$r_1(t+dt)=r_1+r_2*dt+\frac{1}{2}r_3dt^2+\frac{1}{6}r_4dt^3$

$\dot{ρ}=q_1r_1^{1.15}$

##### Eight to Tenth mile­stones

$q_1(t+dt)=q_1+q_2*dt+\frac{1}{2}q_3dt^2+\frac{1}{6}q_4dt^3$

$r_1(t+dt)=r_1+r_2*dt+\frac{1}{2}r_3dt^2+\frac{1}{6}r_4dt^3$

$\dot{ρ}=q_1^{1.15}r_1^{1.1}$

#### T2 strategy #

The op­timal mul­ti­plier is pretty high and is not known be­fore $$e30$$. The mul­ti­pli­ers for act­ive play we know at the mo­ment are:$$e25$$-$$e100$$ is $$1k$$ to $$10k$$; $$e100$$-$$e175$$ $$10k$$-$$100k$$.

##### Idle

For the idle strategy, you want to pri­or­it­ize your mile­stones on x/​x/​0/​0 with $$q_{3}$$ and $$q_{4}$$ be­ing more im­port­ant than $$r_{3}$$ and $$r_{4}$$. If you have more than 5 mile­stones, you will pri­or­it­ize $$q$$ ex­po­nent over the $$r$$ ex­po­nent. You will want to pub­lish at about a $$1000$$ mul­ti­plier, but lar­ger mul­ti­pli­ers are fine.

##### Act­ive

The goal of the act­ive strategy is to grow $$q_1$$ and $$r_1$$ as much as pos­sible while be­ing able to take ad­vant­age of the ex­po­nent mile­stones. The act­ive for T2 is on a 1-minute cycle: 40 seconds on 0/​0/​x/​x mile­stones and 10-20 sec on x/​x/​0/​0 mile­stones. You will start a pub­lic­a­tion on 0/​0/​x/​x as the cost of the x/​x/​0/​0 mile­stone up­grades are too large for you to get right away. When you can af­ford them, you will start the cycle. This is what you will do for the fol­low­ing num­ber of mile­stones:

• 1-2 mile­stones: 0/​0/​1(2)/​0 (40s) → 1(2)/​0/​0/​0 (10s) → 0/​1(2)/​0/​0 (10s) → Re­peat (1 min total)
• 3 mile­stones: 0/​0/​3/​0 (40s) → 2/​1/​0/​0 (20s) → 0/​0/​3/​0 (40s) → 1/​2/​0/​0 (20s) → Re­peat (2min total / 2 1min’s)
• 4 mile­stones: 0/​0/​3/​1 (40s) → 2/​2/​0/​0 (20s) → Re­peat (1 min total)
• 5+ mile­stones: Do the same thing as 4, but with mile­stones in 0/​0/​x/​x when you go to 2/​2/​x/​x. Pri­or­it­ize q ex­po­nents and q3/​q4 with ex­cess dur­ing each swap.

Past $$e175$$, the act­ive strat will be­come ex­po­nen­tially less ef­fect­ive. At $$e250$$, you would start to idle T2 overnight only. Un­til you are $$1e350$$+ $$τ$$ for the­ory 2, this is the best the­ory to run idle overnight.

#### T2 mile­stone route #

• Act­ive:
• 0/​0/​0/​0 → [1/​0/​0/​0 → 2/​0/​0/​0 → 2/​2/​0/​0 → 2/​2/​3/​0 → 2/​2/​3/​2] → 2/​2/​3/​3
• [1 → 1 → 2 → 2 → 3 → 3 → 3 → 4 → 4] → 4
• Idle:
• 0/​0/​0/ → 2/​0/​0/​0 → 2/​2/​0/​0 → 2/​2/​3/​0 → 2/​2/​3/​3
• 1 → 1 → 2 → 2 → 3 → 3 → 3 → 4 → 4 → 4

### The­ory 3 (30σ / 7k) #

The basis of this the­ory and un­der­stand­ing how it works is based on mat­rix mul­ti­plic­a­tion. Be­low I have put a color-coded im­age to dis­play how mat­rix mul­ti­plic­a­tion works.

This gives the basis for why cer­tain up­grades are more power­ful than oth­ers. The ex­po­nents on $$b_1$$, $$b_2$$, and $$b_3$$ are all dir­ectly af­fect­ing $$ρ_1$$ pro­duc­tion which is used for $$\tau$$. An ex­tra di­men­sion roughly gives $$50%$$ more $$\tau$$ pro­duc­tion as it adds an ex­tra term to the $$ρ_1$$ pro­duc­tion.

#### T3 strategy #

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:

$c_{11}*b_{1}^{1.05\text{ or }1.1}<c_{12}*b_{2}^{1.05\text{ or }1.1}$

When this hap­pens swap your ex­po­nents from $$b_1$$ to $$b_2$$ and you will get a little up­grade boost. It also al­lows for a slight push of $$ρ_2$$ for up­grades to $$b_2$$ and $$c_{12}$$, but this is a lot less im­pact­ful and less no­tice­able. This strategy also works with $$b_3$$ and $$c_{13}$$ but is of­ten­times not as com­mon and good to note any­ways.

If you de­cide to buy manu­ally, the fo­cus areas are $$b_1$$, $$b_2$$, and $$b_3$$ when e1 lower than $$c_{11}$$, $$c_{12}$$, and $$c_{13}$$. These all dir­ectly boost the pro­duc­tion of $$ρ_1$$ which is used for $$\tau$$. After this, if do­ing the act­ive ex­po­nent swap­ping strat in the pre­vi­ous para­graph, the next fo­cus will be on $$c_{21}$$, $$c_{22}$$, and $$c_{23}$$ as these boost $$b_2$$ pro­duc­tion which is the more likely cause for the ex­po­nent swap to oc­cur. This leaves the $$c_{31}$$, $$c_{32}$$, and $$c_{33}$$ up­grades as the last pri­or­ity. If you are not us­ing the ex­po­nent swap­ping strat in the pre­vi­ous para­graph, then all the re­main­ing up­grades are at equi­val­ent pri­or­ity.

At the end of any pub­lic­a­tion, around a 2-3 mul­ti­plier, you should turn off b1 and c31 as they cost $$ρ_1$$. You will cruise un­til you get to a 3-4 mul­ti­plier. Pub­lish and turn back on $$ρ_1$$ cost­ing vari­ables and re­peat.

#### T3 mile­stone route #

• Act­ive:
• 0/​0/​0 → [0/​0/​2 → 1/​0/​2/​0 → 1/​2/​2/​0 → 1/​2/​2/​1] → 1/​2/​2/​2
• [3 → 3 → 1 → 2 → 2 → 4] → 4
• Idle:
• 0/​0/​0 → 0/​0/​2 → 0/​2/​2 → 1/​2/​2/​0 → 1/​2/​2/​2
• 3 → 3 → 2 → 2 → 1 → 4 → 4

### The­ory 4 (35σ / 8k) #

We start with just one term of con­stants $$c_1c_2$$ and a chan­ging term $$c_3q$$ with $$q$$ be­ing equal to $$q(t+dt)=q+\dot{q}*dt$$ with $$dt=0.1$$ for each tick. $$\dot{q}$$ is equal to an in­verse equa­tion of $$\dot{q}=q_1q_2/(​1+q)$$ with $$q$$ be­ing the cur­rent value. The first 3 mile­stones we grab add more terms to the $$ρ$$ equa­tion with $$c_4q_2$$, $$c_5q_3$$, and $$c_6q_4$$. Next, we in­crease $$\dot{q}$$ by a factor of $$2^x$$ up to $$2^3$$ or $$8$$. Fi­nally, we in­crease the power of $$c_1$$ from $$1.00$$ to $$1.15$$.

#### T4 for­mula #

##### Ini­tial

$\dot{ρ}=c_1c_2+c_3q$

$q(t+dt)=q+\frac{q_1q_2}{1+q}*dt$

##### First mile­stone

$\dot{ρ}=c_1c_2+c_3q+c_4q^2$

$q(t+dt)=q+\frac{q_1q_2}{1+q}*dt$

##### Second mile­stone

$\dot{ρ}=c_1c_2+c_3q+c_4q^2+c_5q^3$

$q(t+dt)=q+\frac{q_1q_2}{1+q}*dt$

##### Third mile­stone

$\dot{ρ}=c_1c_2+c_3q+c_4q^2+c_5q^3+c_6q^4$

$q(t+dt)=q+\frac{q_1q_2}{1+q}*dt$

##### Fourth to Sixth mile­stones

$\dot{ρ}=c_1c_2+c_3q+c_4q^2+c_5q^3+c_6q^4$

$q(t+dt)=q+2^3\frac{q_1q_2}{1+q}*dt$

##### Sev­enth mile­stone

$\dot{ρ}=c_1^{1.15}c_2+c_3q+c_4q^2+c_5q^3+c_6q^4$

$q(t+dt)=q+2^3\frac{q_1q_2}{1+q}*dt$

#### T4 strategy #

The op­timal pub­lic­a­tion mul­ti­plier is 4-6. Dur­ing pub­lic­a­tions, start with x/​1/​3, then you will switch to 3/​0/​x. This will be re­peated back and forth throughout the pub­lic­a­tion. If you de­cide to manu­ally buy and don’t have max mile­stones, fo­cus on $$q_1$$ and $$q_2$$. The next pri­or­ity is go­ing from the highest $$c_x$$ up­grade down to $$c_1$$. Each lower pri­or­ity should be bought $$e1$$ cheaper than the pri­or­ity tier above. If you de­cide to manu­ally buy at max mile­stones, at the be­gin­ning of pub­lic­a­tions, buy $$c_1$$, $$c_2$$, $$c_3$$, $$q_1$$, and $$q_2$$. Once you are within $$e1$$-$$e2$$ of your pub­lic­a­tion mark, swap to only buy­ing $$c_3$$, $$q_1$$, and $$q_2$$.

#### T4 mile­stone route #

• 0/​0/​0 → [3/​0/​0 → 3/​0/​2] → 3/​0/​3 → 3/​1/​3
• 1 → 1 → 1 → 3 → 3 → 3 → 2

### The­ory tier list (Pre-9k+) #

Be­fore you reach 9k, these are the re­com­men­ded val­ues for each the­ory. You may not hit the val­ues but work on get­ting these the­or­ies up to these val­ues later. This list is in or­der of pri­or­ity.

1. The­ory 2 - up to e300-e350
2. The­ory 1 - up to e205-e215
3. The­ory 3 and The­ory 4 - up to e100-150 each