Tile Rotation Overview

Tile Rotation

Tile Rotation is an iOS app played with two players. The game consists of a board made of 64 tiles arranged in 8 columns and 8 rows. Each tile is divided diagonally into a white half and a black half    divided across the diagonal.

Wherever neighboring tiles touch each the touching edges are the same color. Notice that black touches black, and white touches white.

Additionally, the board is actually a torus, so the bottom row "touches" the top row, and the left column "touches" the right column. That will become clear, below.

At the start of the game, the tiles are rotated randomly (with each player seeing his own rotation on his own device), but with the constraint that edges touching each other are the same color. No stones are on any tile at the start of the game. However, there are two blank stones next to the board.

The first player drags a blank stone to anywhere on the board. Whatever color the player chooses remains their color for the remainder of the game. While the player is moving the stone, a 3x3 grid of tiles, centered on the touch location are highlighted yellow to indicate a valid move.

When the player lifts his finger, the stone settles on the tile in the center of the yellow grid. White stones settle on the white half of the tile. Black stones settle on the black half of the tile. The yellow grid then turns orange/brown, indicating an out-of-bounds area for the next player.

The blank stone of the player who just moved turns grey, indicating it is no longer that player's turn.

The first player's orange/brown tiles fade and the tiles around where the second player placed a stone are highlighted orange/brown. Additionally, the white portion of the tile darkens on a tile occupied by a black stone, and the black portion of the tile lightens on a tile occupied by a white stone. This is done to show territorial occupation by that player. Only one stone can occupy a tile. This lightening or darkening of the tile helps to visually indicate the tile is not available to the other player.

Play alternates between the two players, who can place their stones anywhere that is not orange/brown or faded orange/brown. In other words, a player cannot move next to a tile that was played on either of the previous two moves.

You may have noticed, above, that the board has a different pattern when white is playing than when black is playing. That is because each player is using a separate device, and the rotation can be controlled at any time by the player holding the device. Here, consecutive turns are shown side by side. The stones are on the same tiles for each player, but each player sees a different rotation configuration of the board. The tiles may be rotated any time.

The goal of the game is to create chains of connected stones. Stones are considered connected if the tiles on which they sit are touching each other with the same color. In the paired pictures, above, the board on the left is arranged optimally for the white player to form a long chain. Similarly, the board on the right is arranged optimally for the black player. It's possible both players will use the same configuration. However, since they are competing against each other, it's more likely the rotations will differ between the players.

As the chains get longer, green is added to the tiles to indicate, visually, where the chains are. Scoring occurs geometrically, so that one 8-stone chain is worth more than two 4-stone chains. The green visual aid becomes stronger the longer the chains get.

If there are fewer than 18 tiles left without a stone, the orange/brown restrictions are lifted. Players may now place new stones anywhere there is not already a stone occupying a tile. Note in the paired pictures, above, the right picture shows a board with no orange/brown masks.

You may also have noticed that although it is black's turn on both pictures, the board is configured slightly differently. As stated earlier, the board can be rotated at any time. This gives a player something to do while they are waiting for their opponent to place a stone. In this example, it appears that the black player has rotated the board so that more tiles can be used to extend his longest chain. His score is temporarily lower given the new configuration, but if he fills out the blank tiles, the new score will be higher with the new rotation.

When all tiles have stones, the game ends. The final score is determined by the chains of stones each player has in their own rotation configuration. Here is a sample end game with the left picture showing a good rotation for the white player and the right image showing a good rotation for the black player.

At the start of this document it was mentioned that the board is actually a torus. To explore what that means, take a look at this image.

Let's pay special attention to the tiles with white stones. There are two touching each other near the cent at the top. The tiles are colored green, but just barely, because there are only two tiles in that chain. The other two tiles on the top row with white stones are colored bright green, though. They appear to be just as bright as the ones just below. The same color green implies they belong to chains of similar length. Let's follow the chain for the top, right stone. There are 5 stones that obviously connect to each other. However, since the board is a torus, the top connects to the bottom, so the bottom, right stone is also part of the chain. Also, the left edge and right edge join together so the tile three down on the right connects to the tile three down on the very left edge. This extends the chain across again until it hits the right edge again, which connects to the 5th tile down on the left. The total number of tiles in the chain is 27 tiles.

Scoring will be treated in a separate document. Suffice it to say that in the current scoring scheme, the score for the left picture is:

White: 19,699
Black:  1,754

For the right picture, the score is:

White:    935
Black: 19,699

It's possible a different configuration could create a higher score for each player. It is up to the players to find the optimal configuration. In this case, the game would have ended in a tie because white would have chosen the left configuration and black the right.

Note: Rotations are uniquely specified by the rotations of the tiles along a diagonal. Each tile has 4 possible orientations. There are 8 tiles along a diagonal, so there are 4^8 = 65,536 possible different board configurations.

No comments:

Post a Comment