An ant is at a vertex of a cube. Every 10 minutes it moves to an...

We have divided vertices into four categories
$X \rightarrow$ Starting vertex $Y \rightarrow$ Adjacent vertex $Z \rightarrow$ Adjacent to $Y$ but not same as $X$ $W \rightarrow$ Adjacent to $Z$ but not same as $Y$ Let $a_n=$ number of ways that after $n$ steps ant is at $X$ $b_n=$ number of ways that after $n$ steps ant is at $Y$ $c_n=$ number of ways that after $n$ steps ant is at $Z$ $d_n=$ number of ways that after $n$ steps ant is at $W$ We need to find $a_6$ $$\begin{aligned} a_{n+1} & =3 b_n ...(i)\\ b_{n+1} & =a_n+2 c_n ...(ii)\\ c_{n+1} & =2 b_n+d_n ...(iii)\\ \text { and } \quad d_{n+1} & =3 c_n ...(iv) \end{aligned}$$ By eliminating $b_n, c_n$ and $d_n$ we get $$\begin{array}{ll} & a_{n+3}=10 a_{n+1}-9 a_{n-1} \\ \because & a_1=0, a_2=3, a_3=0 \text { and } a_4=21 \\ \therefore & a_6=10 a_4-9 a_2=210-27=183 \\ \therefore & N=183 \end{array}$$ Sum of square of digits of $N=74$