Exponential Idle Wiki
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Introduction[]

Welcome to Exponential Idle, a math-inspired incremental game. Your goal is to stack up money by taking advantage of exponential growth. To do so, you have to step through time by tapping the equation or simply let the time follow its course. You can perform change of variables to accelerate the process, buy upgrades, and unlock achievements while earning virtual money.


Ticks[]

Expect changes regarding variable tick rate.

A tick is automatically generated every tenth of a second, which increases by and update the function value using the equation. A tick can also be generated with a tap, which increases by 1.


Variables & Upgrades[]

You can improve the growth rate by performing changes of variables. The first variable is free. You can improve other variables once you gain enough money. You can also purchase new variables once you have enough star currency. You can buy in bulk by clicking on the Cost label or buy all available variables using the 'Buy All' button, once available, which iteratively buy the cheapest variable while it can afford it. Similarly, you can buy upgrades to boost your progress. Once an upgrade is available, the two small arrows above the variable list will be highlighted. Auto buyer bonuses will automatically buy affordable variables and upgrades using the same logic.


Variable List
Letter Name Cost Letter Name Cost
- - delta 555
- - epsilon 750
- - zeta 975
- - eta 1230
- 15 theta 1515
- 30 iota 1830
- 75 kappa 2175
alpha 150 lambda 2550
beta 255 nu 2955
gamma 390 xi 3390


Cost[]

Buying a variable or an regular upgrade will DIVIDE the current function value by the cost instead of subtracting it. This cost model makes more sense in the context of exponential growth using multiplicative updates. All other costs (, , and stars) follow the usual subtractive cost model.


The cost of these purchases are driven by a double exponential model, which follows the formula , where x is the level you are buying (starts at one and increments by one after each purchase).


For variable purchase, the following table of a and b can be used:

Variable purchase
Variable a b Base Cost
x - - Free
y 0.5 -4.143474048424998 1.04
z 0.625 -3.1709615006770515 1.08
s 0.78125 1.732020845644619 10.00
u 0.8931739295455593 15.973438351860704 ee4.2871
v 1.0211323755774153 35.10114920824737 ee10.045
w 1.16742248509526 62.78631425389338 ee18.379
α 1.3346705004190407 99.28201254141936 ee29.365
β 1.459278106369075 144.78860583366378 ee43.064
γ 1.6315225219592102 199.47274109054294 ee59.526
δ 1.865258729062898 263.4775066411068 ee78.793
ε 2.085422656904751 336.92853801096174 ee100.90
ζ 2.384185791015625 512.6072746595227 ee153.79
η 2.725750517411985 727.2885715481912 ee218.41
θ 3.1162487047650607 849.4650465308255 ee255.19
ι 3.407188146764981 1123.859483302632 ee337.79
κ 3.9832092821269787 1438.7509481751554 ee432.59
λ 4.453363361722031 1794.6202607002178 ee539.71
ν 5.206251464550825 2191.902179662476 ee659.31
ξ 5.820766091346741 2630.993842401342 ee791.49


For regular upgrade purchase, the following table of and can be used:

dt increase a b Base Cost
dt by 0.10 (changes

when dt upgrade purchased)

2 5.316983346365775 1.000e12
Variable (factor increase) a b Base Cost
y 2 0.6992669815829572 3.08
z 2.2 0.7336524096644791 3.16
s 2.42 2.7382112580862277 102.00
u 2.662 16.973438351860707 ee4.5881
v 2.9282 36.10114920785727 ee10.346
w 3.22102 63.786314253893394 ee18.680
α 3.543122 100.28201254141943 ee29.667
β 3.8974342 145.78860583366372 ee43.365
γ 4.28717762 200.47274109054277 ee59.827
δ 4.715895382 264.47750664110686 ee79.094
ε 5.1874849202 337.92853801096135 ee101.21
ζ 5.70623341222 513.6072746595223 ee154.09
η 6.276856753442 728.2885715481901 ee218.72
θ 6.9045424287862 850.4650465308251 ee255.49
ι 7.59499667166483 1124.859483302631 ee338.10
κ 8.354496338831312 1439.7509481751538 ee432.89
λ 9.189945972714444 1795.620260700215 ee540.01
ν 10.108940569985888 2192.9021796624766 ee659.61
ξ 11.119834626984478 2631.993842401341 ee791.79

All variables have a component that increases as the level of the variable increases. Except for x, which is capped at n = 3, all variables follow these rules for assigning n:

  1. It starts at 0 and increases by one at level 10 and 25 (after buying 10 and 25 times).
  2. Then, it increases every 25 levels until level 1500
  3. Then, it increases every 50 levels until level 6000
  4. Then, it increases every 100 levels until level 10000
  5. Then, it increases every 200 levels until level 24000
  6. Finally, it increases every 400 levels forever.

Equation[]

The equation shows how things are calculated at every tick. The right side of the arrow is calculated using the values shown in the summary bar above the equation. For example, if , then the right hand side will be equal to:

The left hand side represents the new updated value of , and the left arrow represents the update. So after the tick, the new value of will be 4034.29. At the next tick, this calculation is repeated using the new values of , and , which can change if you buy variables or upgrades. The higher these values, the faster you'll progress. So the trick is to buy variables and upgrades, and perform Prestiges to increase (see below).

Summary Bar[]

The summary bar shows the current state of the game. The parameter is the current time, and is how much will increase in one second. is the currency that you can spend for variables and upgrades. is a parameter that will increase after specific milestones (Prestige), and shows how much will change after a Prestige. is the result of all the variables in the list below. Just remember: The higher these numbers are, the faster your progression will be.

Currency Bar[]

The bar below the equation shows the Prestige (, mu) and Supremacy (, psi) currency that you can spend. These currencies are used to buy special upgrades. Prestige upgrades gives bonuses to regular upgrades and Supremacy upgrades improve the equation itself.

For prestige and supremacy upgrades, the cost of purchases are driven by an exponential model, which follows the formula , where x is the level you are buying (starts at one and increments by one after each purchase). The following values of and are used:

Prestige Upgrades (μ)
Upgrade a b
upgrade 15 15
y factor upgrade 20 20


Supremacy Upgrades ()
Upgrade a b
y exponent: 1.2 to 1.8 2 1.5
y exponent: 2.0 to 4.0 3 0.488281
y exponent: 4.2 to 9.0 5 7.94093e-8
to 6 1000
to 9 1000

Note: y exponent and function layers share the same level, despite having different values for a and b. For example, buying y exponent 2.2 will use x = 6.

Notations[]

Since we are dealing with very large numbers, we need mathematical notation to simplify its representation. At first, numbers are displayed normally. When reaching one million, we use scientific notation, a logarithmic representation based on the number's magnitude. For example, . After reaching , we switch to a double logarithmic representation. For example, .


Infinity[]

While Mathematics defines infinity as an abstract and unreachable number, infinity in this game changes over time. At first, the infinity is defined as . Once you reach this number, you'll be offered to perform a Supremacy to increase the infinite value to . Once that is reached as well, you can do a Supremacy or a Graduation to break it to .

Prestige[]

Prestige lets you reset your current progress while giving a bonus to the constant , giving a big boost in progression. It will also allocate a new currency, named μ, with which you can purchase new upgrades. Performing regular Prestige is the only way to progress further. The increase of after Prestige is denoted . To increase , you need to cumulate money, i.e. . In the beginning, it is recommended to wait until the value of b doubles before Prestige. After some time, you can wait until it multiples by 10 or more. For the first prestige, you can expect to Prestige after 15 minutes, when reaches 100%. After some time, you will be able to Prestige within a couple of minutes, multiplying b by 10 or even 100. For the curious, here's how is calculated:

Prestige currency is calculated as:


Star Bonuses[]

You can collect stars by unlocking achievements and at every tick (automatically or by tapping). Each achievement rewards stars depending on its difficulty and each tick has a chance of generating a star. The star chance per second is . When the game is closed, you will get as many stars as if you were in game. Once you reach a certain level (b = 1), you will be able to spend your stars to get permanent upgrades that do not reset after a Prestige. When you collect a star, the star icon will be highlighted.

List of bonuses available:

  • Variables (see Variables & Upgrades)
  • Buy All Button (Single purchase; 15 stars)
  • Acceleration Button (Single purchase; 50 stars)
  • Variable Autobuyer (Single purchase; 200 stars; requires Buy All Button)
  • Upgrade Autobuyer (Single purchase; 200 stars; requires Buy All Button)
  • x2 Automation / s: Multiplies the rate of automation by 2 (3 purchases; prices 500, 1250, 2000 stars respectively; requires Variable Autobuyer OR Upgrade Autobuyer)
  • Auto Prestige (Single purchase; 500 stars; requires Variable Autobuyer AND Upgrade Autobuyer)
  • Permanent variable levels (infinitely buyable, increases with each level bought, requires variable u)
    • Formula is , where l is the level being bought
    • a is based on a table for each variable (see below)
Value of a
Variable a
y 1
z 0.27
s 0.09
u 0.1
v 0.12
w 0.17
α 0.22
β 0.5
γ 0.7
δ 0.9
ε 2
ζ 3
η 4
θ 4.5
ι 6
κ 7
λ 8
ν 9
ξ 10


In the rewards panel, you can use stars to unlock minigames. These stars are permanently used and are not given back during a supremacy (minigames will stay unlocked forever). Current minigames include:

  • 15-Puzzle (10 stars to unlock)
  • Torus Puzzle (75 stars to unlock)
  • Arrow Puzzle (150 stars to unlock)


Supremacy[]

After reaching a certain point in the game, you will be offered to Supremacy (a new button will appear in the Prestige menu). A Supremacy restarts the game at and resets all variables, regular upgrades, and prestige upgrades. It also allocates a new currency, named (psi), with which you can purchase new upgrades. Past infinity, you will be able to purchase an upgrade that modifies the equation, which gives a significant boost in progression.

Supremacy currency is calculated, based on cumulative , as:


Graduation[]

Graduation is the third prestige layer of the game. A Graduation resets all variables and normal, prestige, and supremacy upgrades in exchange for students. Stars and bonuses are not reset. After a Graduation, you will get new students that you can assign to different research projects. These projects introduce a new parameter, , into the main equation. You can access the students panel using the icon in the navigation bar. Don't be afraid to assign students and experiment, you can reassign them whenever you want. The total amount of students depends on your lifetime using the following formula:


Theories[]

Theories can be seen as smaller idle games. Each theory increases , which contributes to the progression of the main game. However, theories are not influenced by the parameters of the main game. Only one theory can be active at any time. You can change the current theory whenever you want and you can resume a theory without losing previous progress.

A theory has its own equation that generates its own currency, , that can be used to buy upgrades. The generated currency is then converted into a value, e.g., . The values from all theories are then multiplied to give the total , which is used in the main equation. In other words, maximize to progress in the main game.

Tap the "Upgrade" label to access a second panel of features.

Upgrade cost is driven by an exponential model, same as for prestige and supremacy upgrades. The formula can be expressed as where x is the level preceding the one to be purchased, or equivalently .
(first) Each listed theory offers the first level of its first upgrade for free, transforming the formula into or equivalently .

Each upgrade displays current value. Values written in scientific notation increase by and are equal to .
(offset) Each listed theory has upgrade start from value 1 instead of 0 unless it is (first), transforming the formula for total value into .

The following tables list , values exactly and rounded to 6 significant digits:

T1
Recurrence Relations
Upgrade a b α
(first) 2 5 1
10 100 3.32193
(offset) 2 15 1
10 3000 3.32193
104.5 1e4 14.9487
1e8 1e10 26.5754
T2
Differential Calculus
Upgrade a b α
(first) 2 10 1
2 5000 1
3 3e25 1.58496
4 8e50 2
2 2e6 1
2 3e9 1
3 4e25 1.58496
4 5e50 2
T3
Linear Algebra
Upgrade a b α
(first) 1.18099 10 0.239997
1.308 10 0.387363
1.675 3000 0.664446
6.3496 20 2.66667
2.74 10 1.45418
1.965 1000 0.974529
18.8343 500 4.23529
3.65 1e5 1.8679
2.27 1e5 1.18269
1248.27 1e4 10.2857
6.81774 1000 2.7693
2.98 1e5 1.57531
T4
Polynomials
Upgrade a b α
(first) 1.305 5 0.38405
3.75 20 1.90689
2.468 2000 1.30334
4.85 1e4 2.27798
12.5 1e8 3.64386
58 1e10 5.85798
100 1000 6.64386
1000 1e4 13.2877
T5
Logistic Function
Upgrade a b α
(first) 1.61328 10 0.689996
64 15 6
(offset) 1.18099 1e6 0.239997
4.53725 75 2.18181
88550700 1000 26.4
T6
Integral Calculus
Upgrade a b α
(first) 3 15 1.58496
100 500 6.64386
1e5 1e25 16.6096
1e10 1e30 33.2193
(offset) 2 10 1
5 100 2.32193
1.255 1e7 0.327687
5e5 1e25 18.9316
3.9 15 1.96347
T7
Numerical Methods
Upgrade a b α
(first) 1.51572 500 0.6
(offset) 1.275 10 0.350497
8 40 3
63 1e5 5.97728
2.82 10 1.4957
60 1e8 5.90689
2.81 100 1.49057
T8
Chaos Theory
Upgrade a b α
(first) 1.5172 10 0.601411
64 20 6
31.15 100 1.82271
51.15 100 2.67022
71.15 100 3.22846
step x y z
0.02 -6 -8 26
0.002 -10.6 -4.4 28.6
0.00014 -6 15 0


Automation[]

Some automation features are available to simplify the progress, which can be purchased with stars. First, you will get the 'Buy All' button that buys all affordable items. Then, there are the variable and upgrade auto buyers that automatically buy all affordable items every 5 seconds. Then, there is the auto prestige that will perform a prestige as soon at the ratio db/b is above the given ratio in the auto prestige settings. For example, if you want to prestige when is 10x higher than b, you input 10 as the ratio. Accepted format are the same as the one used in the UI. This means that you can use 1e6 or 1.000e6 for 1 million. The check for prestige is done along with the auto buyers. After the 1.4 Upgrade, you can buy an auto Supremacy which can take the same type of input as the auto Prestige.


Rewards[]

You can increase by 50% for 2 hours by watching a short advertisement. You can cumulate rewards by watching more than one ad. For example, watching three ads in a row will increase by 50% for 6 hours. You can also play minigames, unlocked using stars, which rewards you some amount of stars once you succeed, depending on the difficulty.

The formula for stars given by a game is , where gameBaseReward is the number of stars given by a set of game and difficulty initially (updated on 2020-08-05).

is the expansion of .


Accelerator[]

An accelerator button is available after you collect enough stars. The button will appear in the equation section. Maintain your finger on the button to multiply the value of dt by some amount, which provides a boost in progression. For more comfort, you can drag your finger around the screen after pressing the button. While you finger remains on the screen, the accelerator will be active and you will be able to continue purchasing variables.

The formula for acceleration is , where s is the time in seconds since you started holding the button. As of 1.4, the acceleration has a limit of 1 hour (3600 seconds, ~x3.17), before releasing the multiplier back to 1.

Leaderboard[]

After signing in, you can access the leaderboard from the Settings menu. The displayed score corresponds to the current 'ee' income. For example, if you reached , then your leaderboard score will be 400.

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