Maze-ish Simple Game

Here is yet another (unoriginal) game of mine played on a square grid.
(It takes some explaining, but is actually pretty simple. It seems,
perhaps, like it would be fun to play.)
You start with an n-by-n grid drawn on paper. (I suggest n be at least
6,
but not huge {at least for beginners}.)
The game is for 2 players.
Definition: If a square of the grid has a line-segment drawn from it
or to it, the square is "taken". Otherwise it is "untaken".
The first move is from the upper-left square of the grid.
The players take turns alternately drawing segments (vertical or
horizontal, from the center of a square to the center of an adjacent
square).
Players draw each segment from any taken square to any adjacent
(up/down/left/right) untaken square.
(On the first move, however, the segment starts in the upper-left
square.)
Each player must draw their segment from the last taken square (taken
in
the previous move by the player's opponent) IF possible.
(ie. if possible: if there is an untaken square adjacent to the last
taken square)
When a square is taken, and this square has no adjacent untaken
squares,
the player who took the square must put an X in this square.
(The X's correspond to the dead-end's of a maze whose passageways are
the lines
of the completed game.)
(By the way, the upper-left square does *not* have an X in it.)
After an X is drawn, the next player may draw his/her segment from any
taken square on the grid to any adjacent untaken square.
After each square is taken, which is a total of {n^2 -1} segments
being
drawn, the game is over.
(With a large grid especially, perhaps someone should keep track of
how
many moves are made so as to ensure all squares are eventually taken.)
Player 1 (first player to move) wins if there is an ODD number of X's.
Player 2 wins if there are an EVEN number of X's.
--
When the game is complete, if you or your computer (if game played on
computer) were to take a grid and erase the unit-walls (unit-wall =
wall along one square) which correspond with those crossed in the game
by segments, and then erase a
unit wall in the upper-left and a unit-wall in the lower-right, you
would have a maze.

Thanks,
Leroy Quet