An ant is at a vertex of a cube. Every 10 minutes it moves to an adjacent vertex along an edge. If N is the number of one hour journeys that end at the starting vertex, find the sum of the squares of the digits of N .

Step-by-step solution

We have divided vertices into four categories X Starting vertex Y Adjacent vertex Z Adjacent to Y but not same as XW Adjacent to Z but not same as Y Let a_n= number of ways that after n steps ant is at Xb_n= number of ways that after n steps ant is at Yc_n= number of ways that after n steps ant is at Zd_n= number of ways that after n steps ant is at W We need to find a₆ 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) and d_ n+1 & =3 c_n ...(iv) aligned By eliminating b_n, c_n