A particle moves on the eight vertices of a cube in the following way: at each

step the particle is equally like to move to each of the three adjacent vertices, independently

of its past motion. Let

i

be the initial vertex occupied by the particle,

o

the vertex opposite

i

. Calculate each of the following quantities:

1. the expected number of steps until the particle returns to

i

;

2. the expected number of visits to

o

until the particle returns to

i

;

3. the expected number of steps until the first visit to

o

.