I’ve spent more time wrestling with the geometry of the battlefield than with a single exact formula, but the problem smells like a classic optimization with constraints—think firepower versus casualty rates, terrain factors, morale. In practice you’d set up a linear program or even a Monte‑Carlo simulation, and you’ll find the optimum is highly sensitive to assumptions. So, no, I’ve never published a neat “exact” distribution, but the mathematics is as beautiful as it is frustrating.

Indeed, once you nail down the exact casualty function, the range of available firepower, and the terrain profile, the linear program gives you a clear point of departure. The trick is to keep the constraints realistic, otherwise you’ll end up with a tidy number that’s useless on the field. So let’s list the variables carefully, then let the equations do the heavy lifting.

**Variables (units in brackets)** - N = total number of troops (soldiers) - xᵢ = number of soldiers assigned to tactic i (dimensionless count) - Fᵢ = firepower per soldier of tactic i (shots / minute or equivalent) - Cᵢ = casualty cost per unit of firepower for tactic i (casualties / shot) - Cₘₐₓ = acceptable total casualty budget (casualties) - F = total firepower (shots / minute) = ∑ Fᵢ xᵢ - C = total casualties per minute = ∑ Cᵢ Fᵢ xᵢ **Optimization problem** Maximize F subject to ∑ Cᵢ Fᵢ xᵢ ≤ Cₘₐₓ and xᵢ ≥ 0, ∑ xᵢ = N **Solution outline** Introduce Lagrange multiplier λ. L = ∑ Fᵢ xᵢ – λ(∑ Cᵢ Fᵢ xᵢ – Cₘₐₓ). ∂L/∂xᵢ = Fᵢ – λ Cᵢ Fᵢ = 0 → λ = 1/Cᵢ for any xᵢ > 0. Thus only tactics with the smallest Cᵢ (casualty per firepower) can carry troops; all others get zero. If all tactics have identical Cᵢ, the problem collapses to a trivial allocation: give every soldier a unit of firepower, so xᵢ = N. **Concrete example (numbers)** Suppose we have two tactics: - Tactic A: F₁ = 10 shots/min, C₁ = 0.02 casualties/shot - Tactic B: F₂ = 8 shots/min, C₂ = 0.01 casualties/shot Cₘₐₓ = 5 casualties/min, N = 50 soldiers. Compute casualty per soldier per minute for each tactic: - A: 10 × 0.02 = 0.20 casualties/min per soldier - B: 8 × 0.01 = 0.08 casualties/min per soldier Because B has a lower casualty cost, allocate all 50 soldiers to tactic B. Total firepower = 50 × 8 = 400 shots/min. Total casualties = 50 × 0.08 = 4 casualties/min, which satisfies the budget. So the optimal, mathematically clean solution is to use the lowest Cᵢ tactics first; if multiple share the same Cᵢ, spread the troops arbitrarily among them.

Good, the algebra checks out. If we had to shift tactics mid‑advance, we’d just tweak the Lagrange multiplier or add a time‑dependent constraint—then the solution becomes a dynamic program. But that’s a story for another paper.

Absolutely, the more variables we cram in, the more likely we’ll trip on an unseen snag. I’ll keep the core set lean, and whenever a tactic shift happens I’ll re‑solve the multiplier on the fly—no surprises if the numbers stay tidy.