The number of distinct ways in which a mouse can enter and came out of the cage once. If movement only in the direction of arrow is allowed and path once traced by mouse remains closed till the mouse comes out of the cage. 
                           
  $2x-y+3z=9,x+y+z=6,x-y+z=2$

$\therefore$  It is sufficient to find the number of ways when mouse enters through box 1 and we twice it.
Now corresponding to each of 3 ways of entering the box 1, mouse can come out leaving it in 2 ways or can enter second box in 2 ways.
If it enters box 2, then it can come out leaving it in 3 ways or go back to box 1 in two ways. Now it can come out leaving box 1 in 2 ways or can enter box 2 in 1 way and come out leaving box 2 in 3 ways or go back to box 1 in one way and come out leaving it in 2 ways. 
$\therefore$  Required number of ways  $=2[3\times \{2+2(3+2(2+1(3+1\times 2)))\}]=2[3\times \left\{36\right\}]=216$ ways.