Blatantly Stolen from SomethingAwful

The Law of Diminishing Marginal Returns is an economics theorem that states that as you consume more of a single product, the value of that product to you goes down. That is, the marginal return decreases. This is an intuitive concept that an economics teacher described with Big Macs: The first Big Mac you eat is delicious, but the third is disgusting and greasy.

We can apply this concept to the idea of a ninja battle rather simply. Let us define a Law of Diminishing Marginal Ninja Power: "As the number of ninjas on a side increases (past one), the power of that force decreases." By this law, the most powerful ninja force would be a single ninja. An curve can be set up for this law of the form:


The constant n is the yet unknown ?ninja? coefficient that actually specifies the curve, and N(x) is the power of a force of a certain number of ninja.

We can find proof of this statement in nearly every ninja film, but a powerful example is the recent film Kill Bill Vol. 1. For those that have not seen the film yet, please skip the remainder of this paragraph and go to the next.  
The Law of Diminishing Marginal Returns has an opposite, of course, with the Law of Increasing Marginal Returns, which I believe is much rarer in economics, if it exists at all in that field. It has the same effect, except that with each additional unit, the marginal return actually becomes greater. This law we can apply to the number of zombies in a horror film. As the number of zombies increases, the power of the zombie force increases by a greater and greater amount. This is another graphable curve, of the form


The constant z is in this case the ?zombie? coefficient, and Z(x) is the power of a force of a certain number of zombies.

This law is another intuitive concept ? a single zombie poses no threat to even a weakling. Even a second is no great threat. The fifth, however, makes a dangerous group; a tenth makes a deadly brigade; a fiftieth creates an undead horde; and once you get a hundred zombies together, you?re just sporked.

Now that we have our definitions down, we can consider how these two laws interact. There are some questions we can ask and explore simply through discussion, but others may require actual experimentation. For example, at what point do the two curves intersect? That is, how many zombies, X, would it take to over power a certain number of ninjas, Y? How do these curves change when you use ?fast type? zombies? In what way can we even measure N(x) and Z(x)? I guess we could use a comparable number of Delta Force Strike Operatives, but even this force faces a curve of its own. Additionally, how do the implications of these curves affect other battles of attrition, such as the one between the peaceful Dinosaurs and monstrous Bunnies?

In conclusion, the mathematics of battle forces opens up a completely new field of study for Something Awful. I am sure that with time and effort we can answer these questions, and pave the way to the ultimate simulator of great battles: the ?GBS Zombie/Robot/Pirate/Ninja/Barbarian Simulator 9000!?

Yes, but the zombie horde becomes much stronger with each new zombie.

The following calculations take into account the fact that the zombie and ninja coefficients are unknown, and are assuming we are attempting to find the break-even point, where the battle leads ultimately to both sides being annihilated. It, however, assumes the battle is instantaneous, as it is impossible to find the RATE at which zombie/ninjas are killed, which would be necessary to adjust for the ever changing forces. Furthermore, it assumes each ninja commits hari-kari, as the prospect of ninja zombies introduces a third variable that would be nigh impossible to account for, especially without taking death rates into account.

In all equations, n is the number of ninjas, N is the unknown ninja coefficient, z is the number of zombies, and Z is the unknown zombie coefficient.

It can be shown mathematically that for y number of ninjas, the number of zombies, x, for total annihilation is the solution to the equation

And it can also be shown mathematically that for x number of zombies, the number of ninjas, y, for total annihilation is the solution to the equation

And the two forces cancel out (assuming equal numbers of each) only at ln( N / Z).


EDIT: The equations have been corrected.

Unless you gib the Zombies, they will come back.