StarCraft scouting calculator

The short version

The enemy is what you saw, plus what you did not see. Both halves are real.
The second half is not a mystery. It is a budget: whatever they earned since you looked, they could have spent on something.
That budget grows every second. Information does not go stale slowly, it goes stale at the rate of their income.
The frontier is a straight line, not a circle. Minerals and gas add; they do not combine under a square root.
The cheapest thing you can do about all of this is look again.

What they could have built since you looked

The enemy is what you saw plus what you did not. This is the second half, priced.

3:00

Since you looked

1800

Minerals they earned

600

Gas they earned

2400resources

The budget line

Minerals plus gas, because that is what a budget is.

You last looked 3:00 ago and saw: a few workers and one production building. Since then they have earned 1800 minerals and 600 gas. Everything below is inside that budget, and nothing outside it is possible. Scout again and the whole list collapses to what you see.

They could have	How many	Supply	What runs out first
Zergling	72	36	Minerals
Drone	36	36	Minerals
Hydralisk	24	24	Minerals
Overlord	18	0	Minerals
Scourge	15	7.5	Gas
Devourer	12	24	Minerals
Infested Terran	12	12	Gas
Lurker	6	12	Gas

One line at a time, not all of them at once: the budget buys any one of these, or a mixture that adds up to the same money.

Everything they could have bought, in resources

The model, in our words

Lesson 7 writes the enemy's true state as the state you observed plus an unknown vector:

R_{\text{true}} = R_{\text{obs}} + |R_u|\,\hat{u}

(1)

The direction $\hat{u}$ is what scouting tells you, and the magnitude $|R_u|$ is what time tells you. The magnitude is the integral of their income since you last looked:

R_u(t) = R_u(t_0) + \int_{t_0}^{t} f(\tau)\,d\tau

(2)

Which, when their income is roughly steady, is the straight line the lab draws: income times the minutes you spent not looking.

A line, not a circle

It is tempting to draw the set of possible enemies as a circle, with radius

|R_u| = \sqrt{c_1^2 + c_2^2}

(3)

Careful: that is wrong. Minerals and gas are not the legs of a right triangle. If they have 600 minerals and 300 gas, they do not have 670 of anything. They have 600 and 300, and every possible army they could field sits under the straight line

|R_u| = c_1 + c_2

(4)

This is not a technicality: the circle would let them afford a mixture that they cannot, and the whole use of this model is knowing what they cannot afford.

Where the budget line is not the whole story

Two things bend it. The tech tree zeroes out whole columns, so "twelve mutalisks" is not inside the budget of a zerg without a spire, and seeing no gas early rules out most of the list on its own. And production capacity caps the rate, so a player with one gateway cannot convert a bank into zealots as fast as a player with six, whatever the bank says.

Both of those make the real set of possibilities smaller than the line. That is the right direction to be wrong in: the lab's answer is the widest honest answer, and every extra thing you know narrows it.

How to use this at the keyboard

Look at the list once, then stop looking at the list. What you want from it is the habit:

Put a number on the delay. Four minutes without a scout against a zerg on two bases is well over two thousand resources of army you have never seen.
Ask what the worst entry on the list does to you. Not the most likely one. The one you have no answer to.
Price the scout against the list. An overlord, a scan, one zergling: all of them cost less than being wrong about the top row.

What this lab assumes

Assumption	Value	Why, and how sure we are
Their income	you type it	Nobody can measure the opponent's income from inside a game. The defaults are a rough two base economy; use what you actually saw.
Income is steady	linear	It is not: they are expanding too, so the real curve bends upward. The line is the floor, not the ceiling.
No tech tree	not modelled	The list prices everything the race can build, not everything they have unlocked. Cross off what you know they cannot have yet.
No production limit	not modelled	A bank is not an army. Building the whole row takes buildings and time the lab does not count.

Scouting Lab