NOTICE: Mosaic is shareware. If you use it, send US$25 check or money
order to register to:
Lindsey Spratt
633 Indiana
Lawrence, Kansas 66044
USA
You may copy and distribute this program and its documentation as long as
no changes are made to either the program or the documentation. If you use
this program you must pay the registration fee.
This program and/or its documentation may NOT be distributed for profit
(for example, as part of a compilation of shareware or public
domain software) without the express permission of the author.
The author can be reached at the above address, or via email on
Compuserve as: Lindsey Spratt[72027,2566].
The game of Mosaic, the basic concept, the rules, and the graphic design,
were all created by Lindsey Spratt.
SYSTEM REQUIREMENTS
This program runs on any Macintosh with at least 2 megabytes of RAM. It works under Systems 6 and 7. It runs in black&white or color, with any size monitor.
INTRODUCTION
Mosaic is a game of strategy. The players, 2 to 4, take turns building and altering a "board" of square tiles. (Any of these players may be the computer). Each of these tiles has one or two colors on it. Each player attempts to create large regions of her own color. The player who creates large enough regions at the end of her turn, for two of her turns in a row, wins the game.
This game was designed to be challenging to the mind, pleasing to the eye, and surprising to both.
***** Complexity of the Game
In the years I have been musing about this game, I haven't really developed any strategies (multiple-move heuristics), just tactics (one-move heuristics). In the two player game, the number of possible moves at each ply is generally several thousand (but sometimes as few as one), so it makes it very hard to plan in detail for even one's NEXT move, much less several moves in the future (as one routinely does in chess or go). [In chess there are generally some 10 to 20 possible moves at a ply, but never more than 100. In Go there can never be more than 361 possible moves at a ply, with substantially fewer than that as play progresses.]
I conjecture (but haven't tried to calculate it) that the 3 player game is an order of magnitude more complex than the 2 player game, and that the 4 player game is an order of magnitude more complex than the 3 player game. In any event, I find the 3 and 4 player games practically incomprehensible (strategically), although fascinating to watch.
SAMPLE GAME FILES
There are some saved games included with the Mosaic application. There are two 2-person games, and one each 3-person and 4-person games. The 2-person game files show the same game at two different stages. The 3-person and 4-person game files provide a way to see what a completely constructed board looks like, without going through the involved process of playing through the 3 or 4 person building phase.
MENUS
The "File " menu has options to start a new game, open a saved game, save the current game, set the evaluation space (see the discussion on storage at the end of this document), or quit from Mosaic. (The 3 and 4 player versions of Mosaic are MUCH more difficult to play than the 2 player version and they require more memory, so you are encouraged to start out with the 2 player version.)
The "Display" menu has options to affect the appearance of the game: impose a grid (indicating tile boundaries), label each tile with a "name", set the color and pattern for each player, bring the "hand" windows to the front, and redisplay all of the windows.
The "Moves" menu has two options - to undo the step just made, and to specify which players are people and which are the computer.
The "Help" menu has two options for affecting the display of help information: remove the help window or toggle the "automatic" help mode. Also, there is one option for each help topic. The topics are essentially the information in this document.
WINNING
Mosaic is won by a player when she creates large enough regions at the end of two of her alteration initiation turns in a row. A region is a solid area of color. If you consider each tile to be divided diagonally, making four triangle-shaped quarters, each quarter on any tile is of exactly one color. Any two quarter's which share a side and are of the same color are in the same region. A tile is "spanned" by a region if at least one of its quarters is in that region. The points that a region is worth are the square of the number of tiles spanned by that region. The score for a player is the sum of the region points. The score needed to win depends on the number of players. For two players, 100 points are needed. For three players, 200 points are needed. For four players, 300 points are needed.
METHOD OF PLAY
The play of the game is in two phases, the building phase and the alteration phase. In the building phase players take turns placing tiles from their hands in the board, adjacent to already played tiles.
In the building phase, players take turns doing board building routines. A board building routine consists of two steps, first a hand tile selection step ("Step SH: Select hand tile") and then a board placement step ("Step P: Place tile"). There are four rules which must be followed in making a tile placement, these are discussed below in the "Tile Placement Constraints" section. Mosaic only allows valid moves.
When all of the hand tiles have been played, the alteration phase starts. In the alteration phase, players take turns initiating board alteration routines. The purpose of the alteration routine is to change a selected edge (the adjacent sides of two tiles) to the initiating player's color. In the course of this change, various other parts of the board may be altered as well.
An alteration routine may involve only the initiating player, or it may involve the initiating player and one or more of the other players. A single-player routine consists of six steps: One step to select an edge and move the tiles that make this edge from the board to the initiating player's hand ("Step SE: Select edge"), two steps to replace each of these "edge" tiles with other tiles from the board to change the color of the selected edge ("Step R: Replace tile"), two steps to select and place one of the hand tiles back into the board ("Step SH: Select hand tile" and "Step P: Place tile"), and one final step to place the remaining hand tile into the board ("Step P: Place tile") .
A multiple-player routine starts with the same six steps as the single-player routine. However, one or both of the final placements by the initiating player creates one or more mismatches (see below). The multiple-player alteration routine continues with one or more mismatch resolution subroutines to restore the board to a completely color-matched appearance. A mismatch resolution subroutine consists of two steps per mismatched tile: one step to replace the mismatched tile ("Step R: Replace tile") and one step to place the mismatched tile back in the board ("Step P: Place tile").
The first step in the alteration routine is for the initiating player to select an adjacent pair of tiles ("Step SE: Select edge"). The shared edges of these tiles are NOT the initiating player's color.
The second and third steps in the routine are for the initiating player to select two other tiles in the board to replace the edge tiles ("Step R: Replace tile"). The replaced tiles must be matched in color exactly by the replacement tile, except for the adjacent sides which must become the initiating player's color in the replacement tiles. The "Replacement Constraint" describes how to handle a special case which can arise when there are 3 or 4 players. Mosaic marks the tiles that are possible replacements.
The final steps in the edge replacement subroutine are for the initiating player to put the initially removed tiles back into the board. The placement of these tiles is constrained in the same way as the addition of hand tiles was constrained in the building phase of the game (excepting the "Building Phase Constraint"). If there is a way to place the two hand tiles back in the board such that the shape and color constraints are satisfied, then the tiles must be placed in that way.
However, it may not be possible to satisfy both shape constraints and the color constraint for both tiles. In this case, there is some 3 or 4 sided hole in the board which must be filled to satisfy shape constraint 2, but neither of the hand tiles can be placed in this hole such that the sides of the hole match the sides of the tile. When this case arises, the hand tile that can be placed in the hole with the fewer mismatched sides MUST be placed there. If there are two holes, then the hole/hand tile combination with the fewer mismatches must be made, leaving one to make the other hole/hand tile combination regardless of the number of mismatches.
If no color mismatches were forced, then the alteration routine is over. However, if there are any color mismatches in the board, the alteration routine must continue so that these mismatches are resolved. Mismatches are resolved by repeating the mismatch resolution subroutine until the board does not contain any mismatches.
The mismatch resolution routine is done by the next player after the one whose play caused the mismatches being resolved (the "resolving" player). For each mismatched tile, the resolving player replaces that tile with one from the board which matches the mismatched tile except for the mismatched sides of that tile, the resolving tile having the appropriate colors on these sides to match the adjacent tiles ("Step R: Replace tile"). If the mismatch can be resolved by a one or two color tile, then this replacement is the same as the replacement discussed above. However, in the 3 and 4 person games it is possible for the mismatch to require a 3 or 4 color tile to resolve the mismatch. Since there are only 1 and 2 color tiles, a different approach is required to resolve such a mismatch. The tile which is used to replace the mismatched tile will cause one or more additional mismatches. The replacement tile must be one with a color pattern which resolves the mismatched edge, and has as few differences in color pattern from the mismatched tile as possible.
After fixing the mismatches, the resolving player places the originally mismatched tiles back in the board, subject to the same constraints as the placement of hand tiles by the initiating player. This placement of tiles may create new mismatches, which the next player must resolve.
TILE PLACEMENT CONSTRAINTS
There are three fundamental tile placement constraints, one constraint which applies only during the Building Phase, and one constraint which applies only during the Replacement Step ("Step R: Replace tile"). These constraints must be honored when doing either a tile placement or replacement ("Step P: Place tile" or "Step R: Replace tile").
• Shape Constraint 1:
A tile being added to the board must have at least two neighbors. A neighbor is a tile in any of the 8 positions (orthogonal and diagonal) surrounding the tile being added. (Exceptions: the first tile placed doesn't have any neighbors and the second tile placed has only one neighbor.)
• Shape Constraint 2:
There cannot be any 3 or 4 sided, one-tile-sized holes in the board.
• Color Constraint:
If two tiles share an edge, they must have the same color along that edge.
• Building Phase Constraint:
If the tile to be placed can be placed next to (orthogonally adjacent) the previously placed tile (in accordance with the Shape and Color constraints), then it must be.
• Replacement Step Constraint:
In the 3 and 4 player games, the color matching constraint can require that a replacement tile have 3 or 4 colors. Since tiles have 2 colors at most, any choice of replacement tile in such a situation will cause at least one mismatch. The tile which is used to replace a tile in such a case must be one with a color pattern which resolves the mismatched or selected edge, and has as few differences in color pattern from the original tile as possible.
ALTERATION PHASE STRATEGY AND TACTICS
A simple strategy is to try and build a region which has your solid and chevron tiles next to each other in a two-by-two square. Such a region is hard for your opponent to break up.
The basic tactical steps of the alteration phase of the game are the "edge" selections. The simplest tactic to follow is to make edge selections which extend your largest region as much as possible, if that edge could be converted to your color without altering anything else in that largest region. This tactic is not usable when making such an "extending" edge selection leads unavoidably to significant alterations to the region you are trying to extend.
The best "extending" edge selection is one where:
1) Converting the selected edge to your color extends your largest region,
2) Converting the selected edge reduces your opponent's largest region,
3) The replacement tiles for the selected edge can be drawn from your opponent's largest region,
4) Extracting these replacement tiles from the board won't leave holes which you must fill, and
5) The extracted edge tiles can be placed in the board such that they add to your largest region.
The "no holes" criterion (4) allows you to take advantage of the "add to largest region" criterion (5). Usually, you won't be able to select an edge such that all of the above criterion are fulfilled, but you may be able to meet several of the criteria.
Other uses of the edge selection move include making your largest region harder to cut up (but not adding any to it's size), and cutting up your opponent's largest region by forcing a mismatch (this is a dangerous technique as your opponent is involved in the mismatch resolution).
THE MOVEMENT STEPS
The phases are composed of players taking turns executing subroutines of the routine for the phase. Players take turns initiating these routines. A subroutine is composed of one or more steps executed by one player. The steps of the subroutines are from the following set (constrained in various ways), "Step SH: Select hand tile", "Step P: Place tile", "Step SE: Select Edge", and "Step R: Replace tile". These steps are described below. This summary of the play of the game is given below in "The Mosaic Algorithm" section.
*****Step SH: Select hand tile.
It is your turn to add a tile to the board from your "hand" of tiles.
Each player has a hand of 8 tiles at the beginning of the game (for 2 players), which are displayed in the Player Hand windows.
You select a tile from your hand in your Hand window by clicking the mouse when the cursor is over the desired tile. (If there are tiles marked in your Hand window with a small circle, you must select from among these tiles.) This selected tile is marked with a small square and the valid possible board locations for that tile are marked in the Board window.
Then, you select a marked location in the Board window where the selected tile is to be placed.
After you have placed the tile from your hand in the Board window, it is your opponent's turn to select a hand tile.
After all of the hand tiles have been placed in the Board window, the second phase of the game begins.
*****Step P: Place tile.
You have chosen a tile to be placed in the board from your "hand". You select a location in the Board window from among the locations which are outlined. These outlined locations are all of the locations at which the selected Hand tile can legally be placed.
Several factors determine what the valid possible locations are, including: the colors of the edges of the selected Hand tile, the shape of the Board tiles, and the colors of the (free) edges of the Board tiles
*****Step SE: Select edge.
It is your turn to begin an "alteration" of the board, by changing an "edge" to your color.
The joint between a pair of adjacent tiles is an "edge". You select an edge in the Board window by clicking the mouse with the cursor near the desired edge. The pair of tiles which share this edge are removed from the Board window and placed in your Hand window.
Then, you choose two other tiles from the Board window to replace the "edge tiles" which were removed to the Hand window. These replacement tiles must replace the removed tiles such that only the selected edge changes color, and the new edge color is yours.
Then, you replace the Hand tiles back into the Board window, satisfying the shape and color placement constraints. It may not be possible satisfy the color placement constraint if you are forced to fill a hole left by one of the replacement tiles, thus your placement of the closest matching Hand tile into that hole will leave a color mismatch in the Board window. If this happens, then the players take turns replacing the mismatched tile with a minimally different tile, and then placing the mismatched tile back into the board, until no mismatches are present. Then it is the next player's turn to select an edge.
*****Step R: Replace tile.
There is a position in the board which must be replaced by another tile in the board, and you must choose among the possible replacements.
This situation has occurred either because you removed an edge (the two tiles of which must now be replaced with the shared edge changed to your color), or because the previous player caused a mismatch by her placement of a tile.
You select a replacement tile (as part of an edge alteration or color-mismatch resolution) by clicking the mouse when the cursor is over one of the circle-marked tiles in the Board window. These circle-marked tiles are all of the possible legal replacement tiles. A Board tile is a legal possible replacement tile if it has the appropriate coloration to replace one of the removed tiles. Appropriate coloration is the same as a removed tile with that tile's changing edge altered to your color. The changing edge of a tile is the shared edge if the tile was removed by an edge selection, and it's the mismatched edge if the tile was removed due to a color-mismatch.
THE MOSAIC GAME ALGORITHM
The play of the game is outlined below in pseudo-code.
Mosaic:
Building Phase,
Alteration Phase.
Building Phase:
CurrentPlayer is First Player(),
while (there is a tile in a player's hand)
Building Routine(CurrentPlayer),
CurrentPlayer is Next Player(CurrentPlayer).
Building Routine(InitiatingPlayer):
Building Subroutine(InitiatingPlayer).
Building Subroutine(Player):
Step SH(Player),
Step P(Player, building constraints).
Alteration Phase:
CurrentPlayer is First Player(),
until (a player has won):
Alteration Routine(CurrentPlayer),
CurrentPlayer is Next Player(CurrentPlayer).
Alteration Routine(InitiatingPlayer):
Alteration Subroutine(InitiatingPlayer),
CurrentPlayer is Next Player(InitiatingPlayer),
while (there is a mismatch):
Mismatch Subroutine(CurrentPlayer),
CurrentPlayer is Next Player(CurrentPlayer).
Alteration Subroutine(Player):
Step SE(Player),
Step R(Player),
Step R(Player),
Step SH(Player),
Step P(Player, alteration constraints),
Step SH(Player), /* This step is trivial, as there is only one tile to select. */
Step P(Player, alteration constraints).
Mismatch Subroutine(Player):
while (there is a possible replacement):
Step R(Player);
while (there is a hand tile for Player):
Step SH(Player),
Step P(Player, alteration constraints).
First Player():
return(1).
Next Player(Player):
if (Player < NumberOfPlayers):
then return(Player + 1),
else return(1).
STORAGE
Mosaic needs more storage ("evaluation space") for more players. So, you may have enough space to play a 2-person game but not enough space to play a 3- or 4-person game. Mosaic will attempt to set the evaluation space to the amount it estimates is necessary. You can sometimes improve Mosaic's performance by increasing the size of the evaluation space, using the 'Eval Space...' option on the 'File ' menu.
If you don't have enough space, you need to quit from Mosaic and increase the "size" in the GetInfo window, then re-open Mosaic. If you are involved in a long game, it is a good idea to save it once in a while. This is because Mosaic can "lose" available memory (bad garbage collection), so that even though you have enough memory to start out with, eventually you run out. When you run out, you may need to quit from Mosaic and then re-open it.
