NAKED DIAGONAL

by Mark Steere


PREFACE

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.

INTRODUCTION

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.

PLAY

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.

OBJECT OF THE GAME

Red must form an orthogonally interconnected path of red stones connecting the two red board edges. Blue must connect the two blue edges.

NAKED DIAGONAL

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.




SWITCH

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.




EMPTY AREA

An "empty area" is a group of orthogonally interconnected, unoccupied points (not necessarily maximally inclusive).

EMPTY PATH

An "empty path" is an empty area in the shape of a single width path.

NAKED DIAGONAL-EQUIVALENT

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.

PLACEMENT RESTRICTIONS

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.]

BAD - BANNED AND DISCONNECTED

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.

CONSTRUCTING A BAD FORMATION

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.

STAIRCASE PATTERN

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.




BLOCKED STAIRCASE PATTERN

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.




KNIGHT PATTERN

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.




BLOCKED KNIGHT PATTERN

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.




KNIGHT PATTERN EXAMPLES

The following figures show two more examples of knight pattern BAD formations.













HYBRID PATTERNS

The staircase pattern and knight pattern can be combined to form hybrid patterns, which can then be blocked. Two examples follow.













SPECIAL FORMATION

The following special formation is the only known formation that's naked diagonal-equivalent but not also a BAD.




CONCLUSION

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.

AUTHOR'S NOTE

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 donít 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.

Copyright (c) 2024 by Mark Steere