Twice the soldiers is four times the army

Sixty against forty sounds like a near thing. It is not remotely close. And the reason is one word in the rules: whether a fighter gets to choose who to shoot at.

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Sixty of yours against forty of theirs, and each of theirs is worth two of yours. That should be a dead heat: sixty ordinary fighters, eighty fighters' worth of skill. Press Fight and you win it, with twenty still standing.

The reason is that shooting is not shared out evenly. Every one of your sixty is aiming at someone. So their forty take twice as much incoming fire per head as your sixty do, and they thin out faster. Which means that a minute later they are firing back with fewer people. Being ahead makes you get further ahead.

That feedback is what turns numbers into numbers squared. Two fighters are not twice one fighter; they are four times one fighter. Sixty against forty is not 60 versus 40, it is 3,600 versus 1,600, and the extra two thousand is what walks off the field.

Now press Rules: everyone picks a target to turn that off. In a press of bodies where only the front rank can reach, or when nobody can see what they are shooting at, the extra people at the back do nothing but wait their turn. Numbers count once. The same two armies fight again and the other side wins, with ten left. Nothing changed but the geometry.

Then set both sides even and press Send theirs in two waves. With a hundred against a hundred and twenty you lose badly. Meet the same hundred and twenty as two sixties, one after the other, and you win with fifty-three left. You did not gain a soldier. They simply spent their advantage in two instalments, and the square law charges interest on being outnumbered. This is the oldest advice in the trade, and the reason it is advice: defeat them in detail.

The rule, exactly. Let A and B be the two strengths and α, β how deadly one fighter on each side is. When every fighter can pick a target, losses depend on how many enemies there are: dA/dt = −βB dB/dt = −αA which conserves αA² − βB² — Lanchester's square law, written down in 1916. When nobody can pick a target, losses depend on the product instead, dA/dt = −βAB, which conserves αA − βB: the linear law. The page integrates the equations with fourth-order Runge–Kutta and prints the closed-form answer beside them. The companion proof (improve/verify/109-lanchester.js, 61 checks) holds each invariant to under one part in a billion while both sides are still fighting, and matches every survivor count to the integration. Six negative controls: each law's formula gives the wrong answer for the other law's battle; a deliberately mis-signed model fails the invariant; an exactly balanced fight annihilates both sides; splitting a force under the linear law changes nothing at all, so concentration is a square-law effect and not a fact about fighting in general; and a hundred still lose to two forces of three hundred however they are met, so this is not a rule that the small side always wins. What is not modelled: morale, terrain, supply, surprise, or anybody deciding to run. This is the arithmetic of attrition alone, which is why it describes a firefight better than a war.

Twice the soldiers is four times the army.

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