Play takes place on the spaces of the board, of which there are two types: half-octagons and squares. The game is for two players, who are called Red and Blue in the original presentation. The board starts off unmarked. Red moves first. The players take turns to color in either one half-octagon or two squares with their colors. Two spaces sharing a common side that are marked with the same color are connected. Red's objective is to join the red-colored North and South sides of the board with an unbroken chain of connected red spaces. Blue must similarly join the blue-colored East and West sides of the board with a chain of blue spaces. A corner space connects to both sides that meet at that corner. As with Hex, Onyx, and other connection games of this type, the swap rule is in effect. In other words, after Red has colored the first space, Blue has the option either to remain Blue and make his first move as Blue or to switch colors and allow his opponent to make the next move also, this time as Blue.
          Octagons falls clearly into the category of pure, Hex-like connection games, along with Hex itself, Twixt and Onyx. As with Hex, drawn, deadlocked positions cannot occur—one player must win. (This is a consequence of the fact that, except on the edges, it is always the case that three spaces meet where the lines of the board intersect.)
          When I first read about this game, I was interested in the tactical potential of the double move. I felt it was a little awkward coloring in the spaces; nevertheless, I could see that round, homogeneous pieces like Go stones would be inappropriate because of the disparity between the sizes of the half-octagons and the squares and the difficulty of visualizing the connections.
          Then one day Larry Back, inventor of Onyx, mentioned to me the amazing fact that the Octagons board and the Onyx board were equivalent except that the sides of the Octagons board were oriented differently. I checked this out by drawing up an equivalent Octagons board in which play is to take place on the points rather than on the spaces, connections between points being marked by lines on the board.

The resulting grid is topologically identical to an Onyx grid rotated through an angle of 45º with respect to the sides of the board. The only difference in this particular representation is that Onyx's equilateral triangles have now become isosceles triangles. The shaded triangles mark out an 8x8 grid of pairs of points that correspond to the old half-octagons. The original Octagons squares now correspond to the centers of the unshaded squares.
          Black moves first with this new board. Each turn, a player now has the option to take either a vertex of one of the shaded triangles or two square centers. Black must connect North and South; White aims to connect East and West. The swap rule is still in effect.
          The new board makes Octagons more convenient to play in the traditional way with Go stones or similar pieces, and I believe it makes the web of connections much easier to visualize. Also, the new board is a very attractive tiling: in addition to a tessellation of irregular pentagons, there are interlocking hexagonal lozenges, and even an interlaced teardrop pattern. It seemed fortuitous that such a good game could be played on this beautiful pattern. Although there are no longer any octagons to be seen, we still refer to the game by its old name.
          With regard to strategy, Octagons seems to share much in common with other games of this type. However, the tactics appropriate to Octagons are quite different from other connection games because of the double move.
          The first thing to be aware of is that a move to the center of a square is an extremely inefficient way of extending your connections across the board. If connecting two adjacent vertices of a square, a move to the square center is clearly a wasted move. If connecting across the diagonal of a square, a move to the square center is likewise superfluous since there already exists a double connection across the square that cannot be broken. For the same reason it is pointless to move to the center of a square to try to break an enemy connection across the diagonal.

The situation to the left of Diagram 4 is a special case because of the characteristic double move of Octagons. Even though there appears to be a double connection between the two white stones, either through the center of the square or through a vertex of the square, it is possible for Black to split the white stones. Assume this local position is repeated elsewhere on the board. If Black uses the double move to take the center of both squares simultaneously, as shown to the right of Diagram 4, then White can only complete his connection in one of them. Black may then take the fourth corner point in the other square, breaking the connection.

In order to break the connection to the left of Diagram 4, it is not necessary for the other local position to be identical; all that matters is that the second move, to another square center, must be answered. I like to think of this as "spending a threat" in one part of the board to get a double move of a square center and a second point in another part of the board.

Because the two white pieces in Diagram 4 can be disconnected by "spending a threat" in another part of the board, I refer to them as semi-connected; the position is a semi-square. Without the presence of the black piece the two white pieces could not be disconnected, even by spending a threat in another part of the board; they can be referred to as fully connected. Diagram 5 shows the possible positions where two pieces separated by one point are fully connected. The terminology borrows largely from the Onyx article in AG6.
          The most important semi-connections are the semi-square, described above, and what I call the dog's leg. There are two types of dog's legs, both of which convert to semi-squares, as shown in Diagram 6.

The addition of an extra black piece converts the semi-connected dog's leg into a fully-connected house, shown to the left of Diagram 7. Lastly, to the right of Diagram 7 is a position that I tentatively refer to as an angle. The two lower black pieces are semi-connected.

So far we have only looked at the use of the double move in order to break a semi-connection, but it has other tactical possibilities. In addition to threatening to break an opposing semi-connection, one may also move to the center of a square to solidify a friendly semi-connection. Thus two friendly semi-connections may be solidified, or one friendly semi-connection solidified and one enemy semi-connection threatened.
           There are doubtless other tactical applications of the double move. All of these possibilities seem to depend on accomplishing specific tactical goals. It is usually pointless to take square centers early in the game, before such positions have had a chance to develop.
          So ends this brief investigation of Octagons. Clearly it has some very interesting possibilities. I think Octagons deserves comparison with the classic connection games Hex and Twixt and the newer game Onyx. These four games all share a similar strategy, but are characterized by radically different tactics.