There is an infinite chessboard, and an ant $A$ is in the middle of one of the squares.
The ant can move in any of the eight directions, from the center of one square to another. If it moves 1 square north, south, east or west; it requires $1$ unit energy. If it moves to one of its diagonal neighbor (NE, NW, SE, SW); it requires $\sqrt 2$ units of energy. It is equally likely to move in any of the eight directions. If it initially has $20$ units of energy, find the probability that, after using the maximum possible energy, the ant will be $2$ units away from its initial position.
Assumption
If in case it doesn't have enough energy to move in a particular set of directions, it will move in any of the other directions with equal probability.
I approached this problem, considering that the case that it finally ends up $2$ units to the east (we can multiply by four to get all the cases).
If it ends up $2$ units to the east, then $\text{Total steps to right}=2+\text{Total steps to left}$.
We will somehow balance these steps, considering that the ant has a total of $20$ units of energy at the start.
I don't know how to effectively calculate the sample space either.
If the ant takes a total of $n$ steps, such that while taking all $n$ steps it is equally likely to move in any of the eight directions, then the sample space would be $8^n$.
But here we do not know $n$. Further, if the energy left after the second-last step is less than $\sqrt 2$ and more than $1$, then the ant will not be able to move diagonally.
I wasn't able to think of much after this. Help is appreciated.