# Problem 575: Wandering Robots
It was quite an ordinary day when a mysterious alien vessel appeared as
if from nowhere. After waiting several hours and receiving no response
it is decided to send a team to investigate, of which you are included.
Upon entering the vessel you are met by a friendly holographic figure,
Katharina, who explains the purpose of the vessel, Eulertopia. She
claims that Eulertopia is almost older than time itself. Its mission was
to take advantage of a combination of incredible computational power and
vast periods of time to discover the answer to life, the universe, and
everything. Hence the resident cleaning robot, Leonhard, along with his
housekeeping responsibilities, was built with a powerful computational
matrix to ponder the meaning of life as he wanders through a massive
1000 by 1000 square grid of rooms. She goes on to explain that the rooms
are numbered sequentially from left to right, row by row. So, for
example, if Leonhard was wandering around a 5 by 5 grid then the rooms
would be numbered in the following way. Many millenia ago Leonhard
reported to Katharina to have found the answer and he is willing to
share it with any life form who proves to be worthy of such knowledge.
Katharina further explains that the designers of Leonhard were given
instructions to program him with equal probability of remaining in the
same room or travelling to an adjacent room. However, it was not clear
to them if this meant (i) an equal probability being split equally
between remaining in the room and the number of available routes, or,
(ii) an equal probability (50%) of remaining in the same room and then
the other 50% was to be split equally between the number of available
routes. (i) Probability of remaining related to number of exits (ii)
Fixed 50% probability of remaining The records indicate that they
decided to flip a coin. Heads would mean that the probability of
remaining was dynamically related to the number of exits whereas tails
would mean that they program Leonhard with a fixed 50% probability of
remaining in a particular room. Unfortunately there is no record of the
outcome of the coin, so without further information we would need to
assume that there is equal probability of either of the choices being
implemented. Katharina suggests it should not be too challenging to
determine that the probability of finding him in a square numbered room
in a 5 by 5 grid after unfathomable periods of time would be
approximately 0.177976190476 \[12 d.p.\]. In order to prove yourself
worthy of visiting the great oracle you must calculate the probability
of finding him in a square numbered room in the 1000 by 1000 lair in
which he has been wandering. (Give your answer rounded to 12 decimal
places)