Unorthobox
Unorthobox is a two-player game best played on paper - play by email or in any other textual medium becomes unwieldy because of the non-orthogonal nature of the game. (Try drawing a line at 30 degrees, in email.)
The idea of the game is to form boxes, and to prevent your opponent doing so. The game is played on a arbitrarily sized grid of dots - 8 by 8 is a sensible starting size.
The players take it in turns to place a piece (or make a mark - I recommend using the classic O and X). The exception to this is that the player who goes second may, on their first turn, choose to replace the opponent's piece with their own in lieu of any other move. (This removes any advantage gained from going first; the Pie Rule.)
A box is formed when points all occupied by the same player mark the corners of a square at any angle. The classic example of this is:
. . . . . . X . X . . . . . . A simple orthogonal square owned by X. . X . X . . . . . .
A less obvious example that displays the more uncommon facet of this game would be:
. . . . . . . . X . . . . . . . X . . X . . . . A simple non-orthogonal square owned by X. . . . X . . . . . . . .
When a valid box exists, the owning player should draw lines denoting its sides, and gains a point.
A completed box is invalid only if it partially intersects a pre-existing box - boxes are still valid if their sides pass through another (non-box) piece, if they are completely contained within another box, if their side touches the side of another box, and if they share a side with another box. Examples:
. . . . . . . X O X O . . . . . . . The last of these pieces to be played does not form a box, . X O X O . because the box would partially intersect the opponent's completed box. . . . . . .
. . . . . . . X O X . . Both of these boxes are valid, because the box of O is . O . O . . contained entirely within the box of X. (Though the sides . X O X . . meet, they do not intersect.) . . . . . .
. . . . . . . X X X . . There is a valid box for X here - the pieces of his own . . . O . . and of the opponent do not form squares, and so cannot . X . X . . block his box. . . . . . .
. . . . . . Several different boxes can be formed here depending on the
. X X X . . order in which the pieces were placed. If the central
. X X X . . piece was last, only two boxes can be formed, one inside
. X X X . . the other at 45 degrees. Otherwise, five boxes can be
. . . . . . formed, all orthogonal; four small, sharing their inner
edges, and one large, sharing its sides with the outer
edges of the small boxes, and containing them all.
. . . . . . . X X X X . While this configuration appears to form many boxes, it can . . . . . . only actually support one - there are two there, but they . X X X X . partially intersect each other, so cannot both be used. . . . . . .
. . . . . . . X . X . . Both X and O have a box, here - they share part of a side, . . . O . O but do not intersect. . X . X . . . . . O . O
It is possible that a player may form a box without noticing, or they may choose to not claim the box when it is formed. If the opponent notices a box being formed, they are not obliged to point it out. If the opponent chooses to point it out, the player who formed the box is obliged to claim it immediately. Other than in this contingency, you can only claim boxes during your turn (though it need not be the same turn in which the box was formed). Thus, if the turn passes to your opponent and you then notice an unclaimed completed box that your opponent blocks with their turn, you cannot retroactively claim the box.
When the board is filled, or when both players agree that the remainder of the board is worthless, whichever player has most points wins the game.
Note, there is no circumstance in which a piece cannot be placed at an unoccupied point - lines and boxes cannot affect the placement of pieces, only the claiming of new boxes.