https://i.stack.imgur.com/mNEBV.png
I'm trying to figure out part a. However, I'm not sure if the set of simultaneous equations I've found is correct. Or at least, I can't solve the set. Any help would be immensely appreciated.
Let Tk be the avg # of moves from square k(white, grey, black):
Tw = 3/8Tw + 3/8Tg +1
Tg = 1/2Tw + 1/4Tg + 1/4Tb + 1
Tb = 3Tw
Let $x=T_w,y=T_g,z=T_b.$ Then your equations are $$x=(3/8)x+(3/8)y+1,\\ y=(1/2)x+(1/4)y+(1/4)z+1,\\ z=3x.$$ When I put this system into a 3 by 4 matrix form and row reduced it, there was a third row of three $0$'s followed by a $1,$ which would indicate no solution exists. But this is one approach to solving such a system, and maybe you have one or more equations down incorrectly, or maybe I did something wrong entering the matrix form.
Added: A look at the link shows that for part(a) if one is on a black square one dies immediately. This means simply $T_b=0$ in your notation. I would also think, since from the grey state one lives a step and then goes onto either the white or black with equal probability, one has $T_g=1+(1/2)T(W),$ since no need here to add the $(1/2)T(b)=(1/2)\cdot 0$ for the transition into the black state. Now for the start in the white state, there is a $(1/4)$ chance of immediate death, so only the $(3/4)\cdot(1+T_g)$ is left, meaning that $T_w=(3/4)(1+T_g).$