## Sample interview problems

#### Good Will Hunting Problem

Remember the Good Will Hunting movie, where a genius solves an MIT problem? Our artificial intelligence can solve that too! #### Problem

```G is the graph
vertex set V = {1,2,3,4}
edges E = (1,2),(1,4),(2,3),(2,3),(2,4)

Find:

2) the matrix giving the number of 3 step walks.
3) the generating function for walks from point i → j.
4) the generating function for walks from points 1 → 3.```

#### Solution

```1) The adjacency matrix A is:

A = [[0, 1, 0, 1],
[1, 0, 2, 1],
[0, 2, 0, 0],
[1, 1, 0, 0]]

2) The matrix giving the number of 3 step walks is:

A^3 = [[2, 7, 2, 3],
[7, 2, 12, 7],
[2, 12, 0, 2],
[3, 7, 2, 2]]

3) The generating function for walks from point i → j is:

G(i,j) = (-1)^(i+j) * det(I_{ij}-z*A_{ij})/det(I-z*A)

4) The generating function for walks from points 1 → 3 is:

G(1,3) = (-1)^(1+3) * det(I_{13}-z*A_{13})/det(I-z*A)```

A.I. Evaluation of the Solution

The candidate's solution is correct. They have provided the adjacency matrix and have correctly found the number of 3 step walks and the generating function for walks from points 1-3.

Evaluated at: 2022-10-12 03:01:59

#### Community solutions:

Check these solutions from our community and artificial intelligence:
The candidate's solution is correct and demonstrates a level of completeness. The approach is straightforward and easy to follow.
The candidate's solution is complete and solves the problem. The candidate's approach is to first find the adjacency matrix A, then the matrix giving the number of 3 step walks, then the generating function for walks from point i → j, and finally the generating function for walks from points 1 → 3.