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 Clarify please. by Dej Mar)

ENE?
Which of the 8 compass rose positions is that?

Let us get another point clear here as well.  In the 8 direction game Merkle is NOT advising any of the three directions since Merkle could always be 'out of range' a la Chess bishop's move, hence the need for Merkle to move unbeknowns to H's landing position.

There is no compulsion to nominated any of the three directions associated with the designated quadrant.

If I wrote this as a computer program M would be assigned a random location, not (0,0).  H would be randomly told N, E or NE.

If M 's not captured on that H's move, then M moves randomly in one of 8 (unless restricted) directions.  According to the quadrant rules H is advised of one of 3 directions.    Um?  I guess that there can only be one direction if Merkle is on the same horizontal or vertical line.

Unfortunately neither SH or DJ presented comment at review that I recall so that such issues might have been tidied, although I think some questions posed are due to misreading).

(as to review - I indicated that this was not a field in which I had expertise.  I had hoped that important issues had been teased prior to publication.  I do assume that the reviewers did assume what were your stated questions and assumed what might have been my answer.  That said, your belated questions would have assisted in framing this better).

Posted by brianjn on 2012-01-19 06:41:06