A Not-So-Simple Graph (Part I)

Let $x_{n}$ denote the number of ways the ant can move from $t_{n}$ to $t_{1}$ . Clearly, $x_{1}=x_{2}=1$ .

We can define $x_{n}$ recursively. For every $k<n$ , the number of ways to move from $t_{n}$ to $t_{k}$ is $k$ . Therefore, we can deduce that

$x_{n+1}=\sum_{i=1}^{n}ix_{i}$

We also know that

$x_{n}=\sum_{i=1}^{n-1}ix_{i}=x_{n+1}-nx_{n}$

So $x_{n+1}=(n+1)x_{n}$

We can see that $x_{n}=(n)(n-1)(n-2)...(3)x_{2}=\frac{n!}{2}$

$A=x_{15}=\frac{15!}{2}=653837184000$ . The sum of digits of $A$ is therefore $\boxed{45}$ .