Learn · Advanced incremental
In some idle games the best generators don't make coins at all: they make more of the generator below. Stack a few tiers like that and growth changes shape, because every tier raises the power of time by one. Here is an hour of coins, made six ways.
Generators make coins, and that's all. The baseline the others are measured against.
Also called flat production
One generator, making a coin a second. Nothing makes more generators, so income never changes and coins pile up in a straight line: twice the time, twice the coins.
Each tier makes the one below it, and nothing is bought: one generator at the top does all the work. Every tier added raises the power of time by one.
Also called makers of makers
A maker builds a generator every second, and each generator makes a coin a second. Income now grows in a straight line, so coins grow with the square of time: ten times as long, a hundred times the coins.
Also called makers of makers of makers
Put a tier above the makers and they multiply too. Each tier adds one to the power of time, so ten times as long now means a thousand times the coins.
Also called dimensions, derivatives
Eight tiers, each making the one below. Divided by 8! (40,320), the top generator's work starts slow, yet by the end of a minute it has made billions of coins, and ten times as long means a hundred million times the coins.
Back to one tier, but every coin goes on more generators. Whether that explodes or stalls depends only on the price.
Also called compounding, reinvesting
One tier again, but every 60 coins buys another generator. Income buys income, so it grows by the same share every second, doubling about every 42 seconds: exponential growth. Slow at first, it overtakes even eight tiers before the hour is out.
Also called the shop, the usual loop
The same loop, but each generator costs 15% more than the last. Prices soon outrun income, so the number of generators grows only with the logarithm of time and the climb keeps flattening: in the long run the coins grow like t × log t, little faster than a single tier.
Take two tiers with one maker at the top. After t seconds it has built t generators, so income has climbed in a straight line from 0 to t coins a second. The coins made are the area under that line, a triangle: t² ÷ 2. Add a third tier and the makers grow as t, the generators as t² ÷ 2, and the coins as the area under that, t³ ÷ 6. Each tier up repeats the step:
coins after t seconds, with k tiers = tk ÷ k!
In calculus terms each tier is the rate of change of the tier below: income is how fast coins grow, tier 2 is how fast income grows, and so on up. The k! (1 × 2 × … × k) is why a new top tier starts so slowly, and the power of t is why it wins in the end.
Now own one generator of each tier, and compare the coins that came down from the tier-k one and from the tier-(k − 1) one by time t:
(tk ÷ k!) ÷ (tk−1 ÷ (k−1)!) = t ÷ k
So tier k draws level with the tier below at t = k seconds, and pulls ahead from then on. Early in a run the low tiers carry it; later, the highest tier does nearly everything. In real games the rates are usually slower than one a second, which stretches the same pattern over minutes or hours.
Own one generator of every tier, with no top, and add up what they all make. The sum is a famous series:
t + t2 ÷ 2! + t3 ÷ 3! + … = et − 1
That's the same curve as reinvesting. In the smooth version of the maths, spending every coin on generators that cost one coin each also makes exactly et − 1: buying generators with coins is a chain that feeds itself, one tier deep but endlessly long.
Take the log of a chain's coins and the power comes down in front:
log(tk ÷ k!) = k × log t − log k!
Plotted against log t, that's a straight line whose slope is k, the number of tiers. Exponential growth has no such line: its log grows like t ÷ 60 × log e, which against log t curves ever more steeply upward, so however many tiers a chain has, reinvesting at a fixed price passes it in the end. Rising prices pull reinvesting back down. When each generator costs a fixed percentage more, the number you own grows only like log t, so the climb keeps flattening: in the long run the coins grow like t × log t, and their slope on the pictures sinks towards one tier's.
Now own one generator of each of eight tiers. Every coin can be traced back to the generator it came down from: by time t, the tier-k one has made tk ÷ k! of them. Pick a moment.
1.72 coins made
One second in, the 1st tier has made the most: 58% of it.
6.39 coins made
Two seconds in, tiers 1 and 2 have made exactly as much: tier 2 is just catching up with tier 1.
52.4 coins made
Four seconds in, tiers 3 and 4 have made exactly as much: tier 4 is just catching up with tier 3.
1.77K coins made
Eight seconds in, tiers 7 and 8 have made exactly as much: tier 8 is just catching up with tier 7.
122K coins made
15 seconds in, the top tier has made 52% of it, and its lead only grows: there's no tier above it left to take over.
4.79B coins made
1 minute in, the top tier has made 87% of it, and its lead only grows: there's no tier above it left to take over.
The busiest tier moves up one every second, and on the exact second t = k, tier k draws level with tier k − 1. With endless tiers the shares would follow the shape of a Poisson distribution with mean t, the one that counts random events. With eight, the top tier takes over after eight seconds and keeps the lead for good.
A three-tier chain as an orchard. One elder tree is all you plant; everything else grows down from it.
Tier 3 The elder tree drops seeds 
Tier 2 Each seed grows a tree of sapling pods 
Tier 1 Each sapling grows into a coin tree Coins made by each way of growing, at three moments.
| Way | Grows like | 1 minute | 10 minutes | 1 hour |
|---|---|---|---|---|
| One tier | t | 60 | 600 | 3.6K |
| Two tiers | t² ÷ 2 | 1.8K | 180K | 6.48M |
| Three tiers | t³ ÷ 6 | 36K | 36M | 7.78B |
| Eight tiers | t⁸ ÷ 40,320 | 4.17B | 4.17e17 | 7e23 |
| Reinvesting, fixed price | e^(t ÷ 60) | 60 | 698K | 2.4e27 |
| Reinvesting, rising price | t × log t | 60 | 6.71K | 109K |