An "x-y" grid game that I know as "Find Merkle" [M] requires a player [H] to begin at (0,0) and zero in on a hidden co-ordinate location of the creature by nominating one of the 4 cardinal directions and an integer distance. Upon failure to land on that location you are given just one cardinal direction towards that site.
Supposing "Merkle" is hiding at (5,5) and you are at (3,8) after your second play, which was either E3 or N8, you are told E or S, nothing more.

Let us allow two changes to this.
Firstly the player is told to move in one of 8-point compass rose directions.
Secondly, upon failure to capture, "Merkle", having no knowledge of the hunter's location, randomly relocates to any of his immediately adjacent 8 locations except for one if already occupied by the hunter.
- This is exemplified if "H" has been told "SE" and has relocated to (6,5).

Oh, and the hunter only knows "Merkle's" location upon capture. Given that the hunter is astute and multiple games are played, what is the most likely number of moves to capture "Merkle" within an NxN grid?

(In reply to Those "Many questions" by brianjn)

I trust that my responses do not come as being trite, as I mentioned I have a dinner engagement.

Understandings:
1. True.
2. Supposing we said "relatively random", like Merkel would like to be trying to prolong the game length.
Now in the 4 direstion game I'd consider a Merkel position of (0,x) or (x,0) as being somewhat ludicrous, but in the 8 directions, having the chance to relocate, would that necessarily be a bad choice?
3. True.
4. I thought that was stated.
5. The direction is One of Eight cardinal rose directions see second bold phrase.
6. H is given 1 or 3 card directions in relation to the quadrant Merkel is in in relation to H.
7. Merkel's relocation is one of eight other than H occupies such a position.
8. H is to zero in on M by way of valid information.
9. Good point. 
Consider integer values from (x,y) to (x+z,y) or x,y+z) or (x+z,y+z), the latter being (I*√2).

 (The correction is noted but HTML in the queue is unavailable to scholars upon publication, I'll need to contact levik).

Edited on January 19, 2012, 3:27 am

Edited on January 19, 2012, 3:28 am

Posted by brianjn on 2012-01-19 03:25:56