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Subject: Christian Freeling's "Rotary" rss

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David Bush
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If you have an extra set of Ploy pieces, you can add the game "Rotary" to your Ploy box. I used a Dremel tool to grind down the direction pointers for nine pieces of green and nine of coral. These became "pawns." A circular disc was added to the back side of each pawn, to indicate the promoted state, which is a "queen." Besides the nine pawns, an extra "scythe" is needed for each color, which is a piece pointing in two directions at 180 degrees from each other. An image of the cobbled set is available in the images section. The rules are also available at http://www.mindsports.net/CompleteGames/Checkmate/Rotary.htm...

Here is what the initial setup looks like.



The player with the darker pieces (green if you're using Ploy pieces) moves first. Players alternate taking turns. Each turn consists of either rotating, or moving, or moving and then rotating one piece, subject to the following constraints:

* A piece rotates some multiple of 45 degrees so that it points in at least one new direction on the board. Pawns do not rotate.

* A piece moves in one of its indicated directions. Pawns move and capture as in Western chess, although there is no initial double move. The king (referred to as a commander in Ploy) may move only one space at a time. All the other pieces (which, from the edges inward, are called the rook, the scythe, the axe, and the trident) may move as many unblocked spaces as desired. All captures are by replacement. Pawns may promote upon reaching the 7th and 8th ranks, and must promote upon reaching the 9th rank. A pawn promotes only to a queen, which is a rook with all four directions, shown on the reverse side. It may be oriented either way upon promotion.

* If any piece other than a pawn is moved one space, it may be rotated at the end of the move/capture. The scythe is the only piece which may rotate at the end of a move/capture of any length.

You may not pass on your turn. You may not rotate and then move a piece in the same turn. The object is to checkmate the opposing king.

This variant is arguably closer to chess. Pawns are once again "the soul of the game," as they define the shape of the battlefield for most of the middlegame. Strategical planning replaces the wild tactical melee of Ploy. Draws are rare; for example, if only the two kings remain on the board, then one of them will always be able to checkmate the other! It's an interesting puzzle to solve this two-piece endgame completely. This is different from Ploy; for one thing, the commander in Ploy is weaker, since it may not rotate after moving. Also, Ploy has an alternate victory condition of capturing all opposing pieces besides the commander. This implies there are no Ploy endgames with less than four pieces. If there is interest, I will post my solution to the "two kings Rotary endgame" in my blog.
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David Bush
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Here is my solution, posted for posterity, before I forget it again. I do not prove this, but it would probably not be too difficult to do so, using some form of finite induction, where the verdict for larger distances between the two kings is based on results from shorter distances.

First, it helps to "checker" the board in two colors. Here are some patterns that are useful to know.



I claim this is a zugzwang position. Zugzwang means the obligation to move is disadvantageous, for whichever side has the move. Even though the white king appears trapped in the corner, if it is black's move, then white can force checkmate, although it might take dozens of moves to do so on a 9x9 grid. For example, if black retreats without rotating, white pursues in the same direction without rotating, and the + + kings are again diagonally adjacent, which is zugzwang regardless of where they are on the board. If you can produce a zugzwang position with your opponent to move, you have a win.

Another choice for black from the starting position is to rotate in place without moving. Since rotation must result in pointing in a new direction, there is only one way a king can ever rotate, which is to switch from a + orientation to X or vice versa. White can respond to this move by moving without rotating, which delivers check to the black king and forces it to retreat. The resulting pattern is dealt with in the next branch.

If instead from the starting position black retreats and rotates, white moves in the orthogonal direction and rotates, producing another important zugzwang position:



Now if black rotates without moving, white does the same, which is again zugzwang. It might seem that white is not making progress, but if black again rotates in position, white can move and deliver check:



Now black has to retreat. If black does not rotate after retreating, then white pursues by moving and rotating to produce the X X zugzwang pattern shown above, but with the black king pushed back. If instead black retreats and rotates, white pursues to produce diagonally adjacent + + kings. If white continues to pursue in this described manner, eventually black will be pushed to an edge and checkmated.

That's all very well if the kings are close to each other, but how does one determine which side wins if the kings are far apart? There are essentially six cases to deal with. First, the concept of opposition needs to be explained. The two kings are at opposition if and only if:

* they are on the same rank or file and are on the same color squares, OR

* they are at opposite corners of a rectangle which has the same color squares in all four corners.

If the kings are both +, then if they are on the same color squares, it's zugzwang. Whoever has the move loses. Otherwise, the player to move wins.

If one king is + and the other is X, then if they are on the same color squares, whoever has the move wins. Otherwise, the + side wins regardless of who is to move.

If both kings are X, if they are in opposition to each other, it's zugzwang. Otherwise, the player to move wins.

That's it!
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David Bush
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My reasoning was flawed! In general, a bare king will not be able to force checkmate in Rotary. Consider the following position with black to move:



I called this a loss for whomever was to move. But suppose black rotates in place:



White has essentially two choices here. In my previous analysis, I recommended that white also rotate in place. Call this variation A. But if white does this, black can move and rotate, producing the following position:



If white also moves up and rotates, again we have an X X pattern, and white is not making progress. Black can again rotate in place, resulting in the same pattern as before. Black can shuttle back and forth this way.



The second choice for white (from the second diagram) is to approach diagonally and rotate, variation B. But then black moves back and rotates, and we have the same pattern again.



White cannot make progress.

So, in general, the bare kings endgame is indeed a draw due to insufficient material. Sorry about all this todo about nothing!
 
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David Bush
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I should add that checkmate is possible if the kings are closer to the corner.



With black to move, this would still be a draw, but white to move can be checkmated. After white rotates in place, black moves and rotates as follows:



Now the only way for white to avoid immediate checkmate is to retreat. If white retreats and rotates, black pursues orthogonally and rotates:



Checkmate is inevitable. If from the previous position white retreats without rotating, black moves down without rotating, with mate next move. But if the kings were any further from the corner than this, checkmate could not be forced. Three piece and higher Rotary endgames are a completely different story, of course.
 
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christian freeling
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twixter wrote:
If you have an extra set of Ploy pieces, you can add the game "Rotary" to your Ploy box. I used a Dremel tool to grind down the direction pointers for nine pieces of green and nine of coral. These became "pawns." A circular disc was added to the back side of each pawn, to indicate the promoted state, which is a "queen." Besides the nine pawns, an extra "scythe" is needed for each color, which is a piece pointing in two directions at 180 degrees from each other. An image of the cobbled set is available in the images section. The rules are also available at http://www.mindsports.net/CompleteGames/Checkmate/Rotary.htm...


Rotary has been upgraded to the ArenA. An applet will be available shortly.
 
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