Yet aNOTHER boring article on the quad-connection games.  :)
Google may stuff up the fixed-font pix, if so I'll repost.

Quad connection games have (up to now) 4 recognised   . o x .
ways of dealing with the problem of local stymies.    . x o .

There are...

Quax style - diagonal connections may be played, as a whole move;
Quadrex style - the 4th point of a stymie is automatically filled;
Crossway style - the 4th point may never be filled, diagonals count;
Swapway style - stymies may be resolved by swapping 2 pieces
                in them, using up a whole move.

A fifth occurs to me, suggested by the close similarity of Quax
and Crossway styles.

Auto-quax - when two diagonal stones are played, the diagonal
            connection between them automatically appears if poss.

(OC, in all cases diagonal connections are not allowed to cross.)

At first, I thought that autoquax would be isomorphic to Crossway,
but then I managed to construct this counter-example...

| . . . x . . . .   At Crossway, X has already won this battle.
| . . . x . . . .
| . . . x o . . .   At auto-Quax, the player with the move
| o o o . x . . .   wins it - O can play in a central cell.
| . . x . o o o o
| . . o x . . . .
| . . . x . . . .
| . . . x . . . .

This is also an example of a tiny glitch in the Crossway rule.
.........

All of the 5 styles of stymie-resolution have tiny (or large)
glitches in them, at least from the PoV of "game purity".
[They are all equally acceptable "socially", I would say.]

Quax and auto-Quax have the problem that they need a new sort
of piece, to play the game on a physical board.  This problem
is not insuperable, however, and can easily be resolved for
physical-board play, (though not ascii-email play), by making
the stones a little smaller than usual.  Then regular stones
can be used to put in the diagonal positions.

|___|___|___|___|___|___|_   This would be good for many checker
|   |   |   |   |   |   |    boards and pieces, though not so
|___|___|___|___|___|___|_   good with Go stones, as these are
|   | o | x | x |   |   |    already a little oversized.
|___|___O___|___X___|___|_
|   |   | o | o | x |   |
|___|___|___|___|___|___|_
|   |   |   |   |   |   |
|___|___|___|___|___|___|_
|   |   |   |   |   |   |

Quadrex suffers from the slight impurity that there are
certain constructible positions, (which would never be seen
in a real game), whereby the placement of a piece triggers
a chain of automatic placements, leading to a cell requiring
a placement of BOTH sorts simultaneously.  It can easily
be dealt with by a special rule (mover decides), and would
never happen, but it is still an impurity.  (It is an exercise
for the reader to construct such positions - not too hard.)

Swapway suffers from the glitch that certain "KO" positions
can arise, perhaps not so rarely.

| . . . . . . . .  Here, with O to move, he can swap the two
| . . x . . . . .  central pieces, hoping to swap the two
| . . x x o o o .  right-hand ones, (e3 e4), next time,
| . . x x o o . .  to make an unbreakable connection.
| . o o o x . . .
| . . . x x . . .  Similarly, as it is, X is hoping to swap
| . . . . . . . .  d5 and e5 for an unbreakable connection.
|         e

So the game might procede with the two players continually
re-swapping the same two pieces!

Obviously a "ko" rule is needed to resolve this; and it may well
provide a charming addition for those Go players and others who
like the ko aspect of the game; but it remains a glitch.

Finally, we come to the tiny glitch in Crossway; which I had
previously thought to be irreproachable.  Here is the 1st
example again...

| . . . x . . . .   At Crossway, X has already won this battle.
| . . . x . . . .
| . . . x O . . .   Neither player may play in the centre,
| o o o . x . . .   according to the rules as currently
| . . x . o o o o   strictly worded.  However, they can easily
| . . o x . . . .   be reworded to allow X to play there later
| . . . x . . . .   and win, at his leisure.
| . . . x . . . .

BUT!  Imagine the capital letter O stone was not there!   Now,
O to move could win this battle by playing in the upper space,
as the Crossway prohibition no longer applies!

So for O to play, now, at the O cell, is actually a NEGATIVE MOVE.

He is worse off after the play than he was before it!

One of the great features of connection games in general is
that there can never be "negative moves" - moves may be worthless,
but can never actually harm you.  This is indeed the basis
of many strategy-stealing theorems, and is in any event a nice
idea for a player to keep in the bacl of his mind.

The appearance of negative moves, unlikely though they may be,
in Crossway, is perhaps a serious glitch against its "purity"
as a connection game.
.......

But in any event, all 4 or 5 games are great to play, similar,
but each has its own particular flavour.  If anyone can think
of any other stymie-resolution plan, we would love to hear of it!

-- Boardly Bill