n – puzzle solvability check with inversion

I have the following matrix [[1,2,6,3],[0,9,5,7],[4,13,10,11],[8,12,14,15]] I am getting inversion 15. So logically it shouldn’t be solvable. But it is solvable and I am getting the following output. Can anybody explain where I am making mistake? ref:

Why is this 8 puzzle unsolvable?

1 2 6 3 0 9 5 7 4 13 10 11 8 12 14 15

1 2 6 3 4 9 5 7 0 13 10 11 8 12 14 15

1 2 6 3 4 9 5 7 8 13 10 11 0 12 14 15

1 2 6 3 4 9 5 7 8 13 10 11 12 0 14 15

1 2 6 3 4 9 5 7 8 0 10 11 12 13 14 15

1 2 6 3 4 0 5 7 8 9 10 11 12 13 14 15

1 2 6 3 4 5 0 7 8 9 10 11 12 13 14 15

1 2 0 3 4 5 6 7 8 9 10 11 12 13 14 15

1 0 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Check subset sum for special array

I was trying to solve the following problem.

We are given N and A[0]

N <= 5000 A[0] <= 10^6 and even  if i is odd then  A[i] >= 3 * A[i-1] if i is even A[i]= 2 * A[i-1] + 3 * A[i-2] element at odd index must be odd and at even it must be odd. 

We need to minimize the sum of the array.

and We are given a Q numbers

 Q <= 1000  X<= 10^18 

We need to determine is it possible to get subset-sum = X from our array.

What I have tried,

Creating a minimum sum array is easy. Just follow the equations and constraints.

The approach that I know for subset-sum is dynamic programming which has time complexity sum*sizeof(Array) but since sum can be as large as 10^18 that approach won’t work.

Is there any equation relation that I am missing ?

How to check security and other updates avaliable on linux distro automaticaly?

Let suppose we have network of 10 plus Linux machines having different distro (debian, Centos , Redhat and Arch etc ).

Can we implement some centralized system where we have update about update on all of these machines(there could be security , application or Linux kernel updates).

So we can patch vulnerabilities timely and mitigate intrusion threat.

Can mathematica check an ansatz?

I’m just a beginner learning Mathematica and I’d like to do something with it but I don’t know if it is possible.

Suppose you want to solve a system of differential equations, maybe a non linear one, and you just want to check whether or not a certain function satisfies the system, is it possible to give your ansatz and let mathematica tell you if you guessed the solution correctly?

Moreover, if there are some parameters, is it possible to know the values of the parameters for which your solution is valid?

If this is possible, can you tell me how to do it or where I should look for (books, pdf or anything) to learn how to do it?