Are you good at math /r9k/? - Actual solution

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This problem intrigued me so much that I decided to ACTUALLY solve it.
The analytical answer, and how to reach it.

The way we will tackle this question is by using recursive logic.
P(n) indicates the probability of extinction given that n aliens are alive.
P(0)=1, if we have 0 aliens
P(1) is what we are looking for, since we start with 1 alien
P(2) is the probability of the aliens going extinct if there are currently two of them
and so on
P(inf)=0 this will help us solve this problem like a boundary value problem

We have this recursive relation
P(n)=0.25*(P(n-1)+P(n)+P(n+1)+P(n+2))
We want to solve this "difference equation" subject to the boundary conditions P(0)=1 and P(inf)=0
We can use characteristic equation method to solve this
https://en.wikipedia.org/wiki/Characteristic_equation_(calculus)

If you understand that method, you'd see that if we name roots of the equation 1-3z+z^2+z^3 as r1, r2 and r3 (slightly more complicated if r1 r2 r3 are not distinc)
ANY function P(n) in the form P(n)=c1*r1^n+c2*r2^n+c3*r3^n will be a solution, where c1 c2 c3 are arbitrary constants
We need to find the values of c1 c2 c3 that will fit the boundary conditions P(0)=1 and P(inf)=0
r1=1
r2=-1-sqrt2~-2.41
r3=-1+sqrt2~0.41

clearly, the coefficients c1 and c2 must be 0, since those roots can never become 0 when raised to the power of infinity
To fit P(0)=1, c3=1
So P(n)=(sqrt2-1)^n
and the answer to our problem is P(1)=sqrt2-1~0.414214