MathDB

A drunken horse moves on an infinite board whose squares are numbered in pairs

(a, b) \in \mathbb{Z}^2

. In each movement, the 8 possibilities

(a, b) \rightarrow (a \pm 1, b \pm 2),

(a, b) \rightarrow (a \pm 2, b \pm 1)

are equally likely. Knowing that the knight starts at

(0, 0)

, calculate the probability that, after

2023

moves, it is in a square

(a, b)

with

a \equiv 4 \pmod 8

and

b \equiv 5 \pmod 8

5