MathDB

Let

p

be a prime number. A flea is staying at point

0

of the real line. At each minute, the flea has three possibilities: to stay at its position, or to move by

1

to the left or to the right. After

p-1

minutes, it wants to be at

0

again. Denote by

f(p)

the number of its strategies to do this (for example,

f(3) = 3

: it may either stay at

0

for the entire time, or go to the left and then to the right, or go to the right and then to the left). Find

f(p)

modulo

p

3