Doubling the number of apples in the war doubles the rate at which
the boy can find and throw them, but also doubles the rate at which
the old man can return them. The two effects cancel out in the ratio.

Such a ratio, which is independent of starting conditions or of
absolute numbers, is called the equilibrium constant, Keq:

If we know the value of this equilibrium constant, either from past
battles or from knowledge of the agility constants,
and , then
after a standoff has been reached, we can find out how many apples
are on the old man's side merely by counting the apples on the boy's
side and doing some simple arithmetic.