Naked Diagonal exists to demonstrate some connection game principles. (And conversely, the principles
themselves comprise a game.) Naked Diagonal may not be a game that anyone would ever want to play, although
there's nothing stopping you.
Naked Diagonal has a sister game, Candy Necklace, which also explores properties of connection games.
In a typical, square-grid connection game, the north and south edges would be colored red, the
east and west edges colored blue. Red's goal would be to connect the red edges with an orthogonally (horizontally
or vertically, or both) interconnected path of red stones. Blue's goal would be to connect the blue edges.
The crosscut is a formation that can bedevil such a game (see the bottom-left formation in the first figure below).
If a crosscut is formed, it could simultaneously prevent both Red and Blue from forming a winning path.
One way to prevent crosscuts is to prohibit "naked diagonals" (see the first figure below). A crosscut is comprised of
two naked diagonals. There can't be a crosscut without a naked diagonal.
You can't just ban naked diagonals. There may come a point in the game where all available placements by both players
form naked diagonals.
A naked diagonal-equivalent area (explored in detail below) is an area that can't be filled in without forming
a naked diagonal.
If a naked diagonal-equivalent area forms, then (if the area fills in) a naked diagonal will form.
If a naked diagonal-equivalent area never forms, then a naked diagonal will never form.
Players are not allowed to form a naked diagonal-equivalent area or a naked diagonal.
One of the two players will always have a placement
available. No naked diagonals will form, no
crosscuts will form, and one of the players will complete a winning path of stones.
What follows is a discussion of some properties of naked diagonal-equivalent areas.
Naked Diagonal is a two-player game. A square board of any size is used. The north and
south board edges are colored red. The west and east edges are colored blue. The board is
initially empty.
The two players, Red and Blue, take turns placing their own stones onto unoccupied points on the
board, one stone per turn, starting with Red. Passing is not allowed, but if you don't have an available
placement, your turn is skipped. See PLACEMENT RESTRICTIONS below.
Red must form an orthogonally interconnected path of red stones connecting
the two red board edges. Blue must connect the two blue edges.
A "diagonal" is comprised of a pair of diagonally adjacent, like-colored stones. If no like-colored stone is
adjacent to both of them, they form a "naked diagonal". In the following figure, there are three red naked diagonals and
two blue naked diagonals.
The following figure shows a "switch" formation. The green dots indicate points that must be unoccupied.
A switch can also be a mirror and/or rotation of this formation.
An "empty area" is a group of orthogonally interconnected, unoccupied points (not necessarily maximally inclusive).
An "empty path" is an empty area in the shape of a single width path.
An empty area is "naked diagonal-equivalent" if there is no possible sequence of placements, by either
or both players, which can fill the area without forming a naked diagonal somewhere along the way.
Players are not allowed to form a naked diagonal or a naked diagonal-equivalent area. [There could also be a
rule for not allowing a switch.
A switch is easy to spot, and doing so can shortcut position evaluations.]
Imagine an empty area in which one player, say Red, has no available placements. Red is "banned" from that
area. Blue may have available placements in the Red-banned area, but there is no possible sequence of blue
placements with which Blue can fill the area without creating a naked diagonal somewhere along the way. Blue
is "disconnected" in the area. Such an area is "banned and disconnected" (BAD). A bi-colored formation which
creates a BAD area is a BAD formation.
Every bi-colored formation considered in this document (including the switch) is a BAD formation,
except for one: the "special" formation, described below.
To construct a BAD formation, first create a monocolored formation of stones. Without loss of generality, red
stones will be used throughout this document to create this initial, monocolored formation. There must be an
empty area within the formation in which every point is an illegal placement for Red because it would form a naked diagonal.
Around the periphery of the red formation, there will be empty paths which Red could use to begin filling the
Red-banned area. Blue must block these paths. In doing so, Blue will complete a BAD
formation.
The staircase is a repeatable, monocolored pattern of stones. Using red stones for the following example, two
staircase-shaped, red stone paths in close proximity sandwich a staircase-shaped empty path between them. Every point
in this empty path is an illegal placement for Red because it would form a naked diagonal.
In the following figure, black dots indicate a Red-banned area.
Assume that one of the players, say Red, has formed a staircase pattern.
In the following figure, the green dots represent peripheral, empty paths which Red could use to begin filling the Red-banned area.
Blue must block these peripheral, empty paths. There are a number of ways Blue can do this. One way is shown in the following
figure. By blocking these paths, Blue has now completed a BAD formation. Red is banned, unable to access the empty area,
and Blue is disconnected in the area. There's no possible sequence of blue stone placements with which Blue can fill in the
Red-banned area without forming a naked diagonal somewhere along the way.
A knight pattern is a monocolored pattern of stones. The only spatial relationship a stone
can have with a neighboring stone in the pattern is the knight move. The following figure is an example of a red knight pattern.
In the following figure, the black dots represent an empty area inaccessible by Red.
In the following figure, the green dots represent peripheral, empty paths which Red could use to begin filling a Red-banned area.
Blue must block these peripheral, empty paths. There are a number of ways Blue can do this. One way is shown in the following
figure. By blocking these paths, Blue has now completed a BAD formation. Red is banned, unable to access an empty area,
and Blue is disconnected in the area. There's no possible sequence of blue stone placements with which Blue can fill in the
Red-banned area without forming a naked diagonal somewhere along the way.
The following figures show two more examples of knight pattern BAD formations.
The staircase pattern and knight pattern can be combined to form hybrid patterns, which can then be blocked. Two examples follow.
The 2x2 empty area in the following figure is not a BAD area, because neither player is banned. But the area is
naked diagonal-equivalent. As soon as a player places makes any available placement within the 2x2 area, the remaining
three unoccupied points become a blocked staircase. So there's no possible sequence of placements, by either or both
players, with which the 2x2 empty area can be filled without forming a naked diagonal somewhere along the
way. This special naked diagonal-equivalent formation may be the only one that's not also a BAD.
With the exception of the special naked diagonal-equivalent formation described in the preceding section,
it seems that every possible naked diagonal-equivalent formation can
formed by creating the staircase pattern or the knight pattern, or both cobbled together, with one player's stones,
and then blocking the pattern with the other player's stones.
Feel free to publish this rule sheet, and to program the
game of Naked Diagonal for online or offline play. No licensing fee or royalties are
expected. However please don’t change the name or the rules, and please
attribute the game to me, Mark Steere. My other games can be found at
Mark Steere Games.