x is their number (don't know, don't care)
s is the seed that you give.

pick the number x, double it and add s
= 2 * x + s

divide by 2

= (2 * x + s)/2 = 2x/2 + s/2 = x + s/2 (at this point they have their number and half of yours)

then subtract the original number that they picked leaving your seed divided by 2 (hence
the reason the seed needs to be even).

x + s/2 - x = s/2

Example - x is 6 and the seed is 8
(2 * 6 + 8)/2 - 6 = (12 + 8)/2 - 6 = 20/2 - 6 = 10 - 6 = 4 (i.e., half of 8)

In the end of course you aren't guessing their number at all (just half of what you gave them)
but it usually doesn't dawn on them (and of course you never said that you would guess THEIR number).

This can be extended to any multiplier such as 3, 4 or 5 etc.. by having them triple, or
quadruple the number and then dividing by 3 or 4 respectively. The only caveat is that
the seed must be an integral multiple of 3 or 4 etc.

(3x + s)/3 - x = 3x/3 + s/3 - x = x + s/3 - x = s/3

in general

(Nx + s)/N - x = Nx/N +s/N - x = x + s/N - x = s/N

As a last extension you can play this game with any number of people simultaneously by
giving each a different seed, a different multiplier or combinations of both.