Math in Banished – Maximum Population

enfo131 posted a 4000+ population reddit yet I think some of the popular models/conclusions might be wrong so I decide to write this paper.

The majority of this paper is based on theory and is yet to be testified.

The game is so time consuming that I don't even know if I'll be able to test my theory.

Maximum Population - What is the limit

Trade Economy

Food, together with other necessary supplies like tools, clothes and fuel, limited by import.

House number, influenced by space available.

Peasant Economy

Productivity per land area, influenced by production layout, folk's health and happiness.

Trading Post and Economy Models

According to pollackey 2, 5 types of merchants visit randomly every 6 months if dismissed immediately.

Towns with population greater than 150 have fully developed merchants.

A detailed calculation shows firewood economy or ale economy is the preferred type for maximum population.

Firewood Economy

Fully developed general goods merchants carry 200(batch size) * 20 (batch number) = 40003 logs if only log is specified in order.

Fully developed resource merchants carry 200(batch size) * 5 (batch number) = 10003 logs if only log is specified in order.

By probability a maximum of 5000 log can be purchased every 2.5 years, or 2000 per year.

A typical wood cutter produces 500-1100 firewood per season, or consumes about 600*4/3=800 logs per year. To avoid storage full and allow smaller storage space, 3 wood cutters per trading post is recommended. Excessive wood cutters can be transformed into stock pile easily if desired.

When the general goods merchant comes, we need at least storage space for 4000 logs and 2000 firewood assigned to that trading post, which is 4000/22.75+2000/62.5=208 sq.

When the resource merchant comes, we need at least storage space for 1000 logs and 500 firewood assigned to that trading post, which is 1000/22.75+500/62.5=52 sq.

With 5 types of merchants coming randomly,

\operatorname{E}(X) = 0.2 \times 208 + 0.2 \times 52 = 52 sq
\operatorname{Var}(X) = \operatorname{E}\left[(X - \mu)^2 \right] = 0.2\times156^2 + 0.6\times 52^2 = 6489.6 sq^2
\sigma(X) = \sqrt{\operatorname{Var}(X)} = 80.56 sq

Using Central Limit Theorem, with a 95% confidence, 52+1.96\times80.56 = 210 sq is required per trading post, which is a lot of land.

In real production, storage space is best assigned to separate 6x6 spaces (road excluded) such that it can be easily converted into wood cutter or vise versa.

Considering wood cutter output, purchasing power is 4 \times (600 \times 3 - 600/3 \times 2)=5600 for food per year per cutter.

Considering import limit, purchasing power is 5000 \times 3 \times 3 - 5000 \times 2=35000 for food per 2.5 years per post, or 14000 per year.

However, Food merchant comes with at most 2000(batch size) * 15 (batch number) = 300003 food per 2.5 years per post, which eliminates the requirement of other products at firewood trading post.

A typical firewood trading posts can store 9999 firewood, just enough for 30000 food, which is perfect.

The excessive purchasing power can be redirected to food only trading posts for maximum possible food import.

Ale Economy

Orchards Production

For maximum land efficiency a 5 \times 4 orchard produces 195 food per year according to Calculator4.

A tavern produces an estimated 20003 ale per year, with 2000/10\times30 = 6000 fruit consumption.

A constant 6000/195 \times 20=615 sq orchard land, or 30 orchards is required per tavern. Considering orchard trees need regrowth every few years, even more orchards is required. This consumes much more land than wood cutter as only 210/3 = 70sq is required per cutter.

However, purchasing power is 2000 \times 8-6000=10000 for food per tavern per year, much higher than that of firewood economy.

Fruit Trading

Fruit trading exempts the low efficiency orchards.

To maximize specialization, trading post need to specify the fruit it requires for ale production. Take apple as an example.

Fully developed general goods merchants carry 2000(batch size) * 20 (batch number) = 400003 apple if only apple is specified in order.

Fully developed food merchants carry 2000(batch size) * 15 (batch number) = 300003 apple if only apple is specified in order.

When the general goods merchant comes, we need at least storage space for 40000 apples assigned to that trading post, which is 40000/200 = 200sq or 40000/6000 = 6.67 storage barn.

When the resource merchant comes, we need at least storage space for 30000 apples assigned to that trading post, which is 30000/200 = 150sq or 30000/6000 = 5 storage barn.

\operatorname{E}(X) = 0.2 \times 40000 + 0.2 \times 30000 = 14000
\operatorname{Var}(X) = \operatorname{E}\left[(X - \mu)^2 \right] = 0.2\times26000^2 + 0.2\times16000^2+ 0.6\times 14000^2 = 304000000
\sigma(X) = \sqrt{\operatorname{Var}(X)} = 17436

Using Central Limit Theorem, with a 95% confidence, 14000+1.96\times17436 = 48175 is required per trading post, or 8 storage barns.

To purchase 70000 apple per 2.5 year, or 28000 apple per year, 3500 ale is required, or 2 taverns.

Excessive food value is 28000 - 3500/10 \times 30 = 17500 per year per post, a little higher than that of a firewood economy.

During the process 10500 apple is actually consumed per year for ale production, out of the average 28000 yearly total import, so it is possible to import 2 food types for diversity. Assume we assign apple and another food (wheat or vegetable as protein can be acquired by fishing) to the food merchant, we will get 21500 apple with 6500 second food per year. If the excessive 17500 food is consumed at the ratio of 21500 apple : 6500 food, 13437.5 apple will be consumed as food, causing not enough apple for ale production. Though, one out of 2 trading post using this scheme may solve the problem, but requires remembering.

It is worth noting that the seemingly higher purchasing power could be the other way around, since local folks can consume ale too, which is not controllable - folks with 5 star happiness 'waste' ale, could be a lot of loss.

Major con of the ale economy is the lack of food diversity.

Wool Coat Economy

Fully developed general goods merchants carry 100(batch size) * 20 (batch number) = 20003 wool if only wool is specified in order.

2000 wool per 2.5 year, or 800 wool per year, gives a maximum 800 wool coat per year, worth 800 \times 15-800 \times 6 = 7200, which is less than firewood or ale economy. Also the storage problem is severe on large scale production.

Other Economy

Other economy types make even less profits and are out of concern.

Economy Summary

Per year per trading postAleFirewood
Purchasing Power1750014000 (18666 for non food)
Storage Req.240sq (8 storage barns)210sq
Production Facility2 taverns3 wood cutter
ConsHard to maintain food diversity

Note that the purchasing power of ale may be misleading. Being not a necessary resource itself, once produced, the ale can be wasted by folks, losing its purchasing power significantly - especially when the population number is high.

Also note that this is not the whole map as there might still be empty lands.

Land Usage

Giving the purchasing power of firewood economy, with 89 trading posts, an estimated 89(60+210+342) = 35244 sq is allocated to production, about 13.4% of the 512x512 map size.

For each folk per year, minimum 100 food + 2.5 tool + 2.5 clothes + 30 firewood = 135 living expenses.

For each folk per year, maximum 120 food + 5 tool + 10 clothes + 5 herb + 60 firewood = 200 living expenses.

A total purchasing power of 89\times14000/135 = 9229 to 89\times14000/200 = 6230 folks can be supported by the fire wood economy.

Take 8000 folks, or 3000 houses, covering 3000*20 = 60000 sq, about 22.9% of the map size.

An estimated 20% of the map is unusable mountain/water.

The remaining area, or the 43.7% of the map, is still available. These lands should be used for self-sufficient production loops, and houses for extra population.

Functional buildings:

  • strict necessary: storage barns, school, hospital
  • necessary: herbalist's, market
  • unnecessary: chapel, graveyard

The land usage shows that it is possible to maintain the food diversity of an ale economy by introducing vegetable and grain farmers, which is preferable.


A best model could be ale economy supplemented by vegetable and grain production, with self-sustained clothes production, assisted with limited firewood production(for self use).

Note that for a 10000 folk town (my target), or 4000 houses, annual firewood consumption is 80000, requiring 100-140 wood cutters, or 33-44 trading posts importing logs merely, so it's still quite a large share of economy.

A best model could also be firewood economy supplemented by self-sustained clothes production. Without ale production, the food usage is 100%, which might turn out even better.

The pure firewood economy is preferred by author.

Misc and Tips

Seed 298 with 89 trading posts available, credit to enfo3.

Market is still required even though all the food is imported and has a considerable diversity, since herbs can only be distributed by markets.

Do not build hospitals near schools - common sense.

Use graveyard cancelling trick.