Pencil and paper games: Blocking

Another theme is pencil and paper games is blocking, i.e., as the game advances, less and less squares are available, until the stalemated player loses the game.

A well-known example is 1970's Snort by Simon Norton, aka Cats & Dogs. 

The rules: on an 8x8 grid, players take turns marking an empty square, provided that the square is not orthogonally adjacent to an adversary mark. The first player unable to move, loses the game.

To prevent mirror strategies, in the beginning the first player should play in the central 2x2 area, and the second player must play outside that area.

There's also a faster variant, called Obstruction, that includes diagonal adjacencies as illegal moves.

Slimetrail, designed by Bill Taylor in 1993, is also a pencil and paper game with a blocking theme. To recap, in Slimetrail each player has a home, each one staying in opposite corners. The snail mark starts at the middle of the board. Each player, on his turn, shades the snail mark, and places/marks the snail on an adjacent empty square. Wins the player that moves the snail into his home or stalemates the adversary.

Domineering (aka Stop-Gate, Crosscram) is another example. Designed by John Conway in 1976, the game is also played on a square grid, where each player marks two adjacent empty squares with a domino. One player marks vertical dominoes, while the other player marks horizontal dominoes. The first player to be stalemated loses (i.e., the last player to move wins).

Cram is a variant where both players can mark dominoes horizontally or vertically. It is what's called an impartial game in Combinatorial Game Theory. This means that Cram is just the game of Nim with other clothes (I mean, rules).

Chomp is another example. The game is played on a rectangular grid. Each player on his turn selects an empty square, and shades all squares that form a rectangle between itself and the grid's bottom-right corner. The player that moves last loses the game.

Blue lost the match after move 11

Another example comes from JeuMok Plus. There are 24 shared 1x2x4 blocks. The game comes with a rulebook containing five different games (one is a NIM game, another is a Jenga), where the main game rules that players drop blocks in empty squares on a board, and each tile must be adjacent to at least another already on board. The board center cannot be used. The player that moves last wins the game.

Pencil and paper games: Improving Tic Tac Toe

Among the most well-known pencil and paper abstract games are Tic-Tac-Toe, Gomoku, and Dots'n'Boxes. Another famous game that can be played with a pen is Connect4 (but many families have bought this game anyway). 

Many connection games can be also played with pencil and paper, like Hex, TwixT or Bridgit. Of course, there are other famous games in this genre like Hangman or Battleship but, as usual, we'll stick here with the abstract ones.

In this post let's check some examples that improve the playability of Tic Tac Toe, one of the simplest abstract games.

An interesting version is Ultimate Tic Tac Toe and is played on a 9x9 grid of nine 3x3 areas (like in a Sudoku board). Each player makes a move on one of the nine mini-boards. But the next player must play at the board relative to the previous opponent's move. For eg, when a player plays at the upper-left corner, the next player must play at the upper-left 3x3 board. When a player makes a 3 in-a-row that mini-board belongs to her. If a move goes to again to that finished 3x3 board, the player can choose whatever open board she prefers. Wins the player that makes a 3 in-a-row of mini-boards (play online here). A similar game is 1976's Tri-o Tri-x.

The next variant is SOS. In this game, played on a square grid of size 3x3 or larger, players can write an S or a O on an empty square. Every time a player makes a orthogonal or diagonal row with the S-O-S sequence, he crosses it with his color and scores one point. The player with highest score wins.

Another variant is 3D Tic Tac Toe, played on a stack of four 4x4 boards (a 3x3x3 version is a trivial win for the first player). So, a player wins by making a 4 in-a-row using orthogonal or diagonal lines in all three dimensions. This can be represented in a page by drawing a line of four 4x4 boards, where the leftmost board is the bottom, and the rightmost is the top. This game was published in 1947 as Qubic:

One variant designed by Trevor Truran in 1981 for Games and Puzzles #81 is Knights and Crosses:

Here's another pen and pencil game called Order and Chaos by Stephen Sniderman that appeared on GAMES #25:

[addendum] Another 3D Tic-Tac-Toe is 1969's Stop Dot, published by Mattel.

Here the game is played on five faces of a 3x3x2 cube, and the goal is to make a 5 in-a-row with five different colors (at least over two faces):

These are the rules: 

Pencil and paper games: Dots and Bugs

Pencil and paper games all share the property that each grid cell can only be used a very limited number of times (usually once or twice). Walter Joris is probably the game designer that most contributed to this genre with his 2002 book 100 Strategic Games for Pen and Paper. There's more recent Joris fun on his pinterest page.

Here are two pencil and paper games played in the Eastern bloc.

Dots. Played on a grid of any size - traditionally on a 39x32 grid typical of a squared copybook in Russia [wikipedia].

Each player draws a dot of his color on an empty intersection. If the player closes a chain of his own dots (including diagonal connections) and the interior includes at least one enemy dot, then the player connects his dots with lines. Those dots, including the inner area, are out of the game (let's say the player captures the area). The player scores one point per surrounded enemy dot. If the inner area does not have an enemy dot, then the player cannot capture that area. Also, if there are more than one option to close chains, the player must select just one of those to close.

When the board is full, the player with more points wins.

Usually, the game starts by playing a limited number of moves in a central area to prevent players moving initially in the borders (since those dots cannot be captured). A nice way to solve this problem would be to make a toroidal board, but at a probably high cost in clarity.

The wikipedia page mentions a variant that allows for knight-like connections, which encourages more offensive play. 

Another variant prevents a chain from scoring if it gets surrounded by a larger enemy chain.

An extra rule is the idea of Houses. A House is a chain that surrounds an area with only empty spaces. If the adversary moves there, and does not close a chain himself, then the House becomes a closed chain. This rule seems to open a cold phase at the endgame, when dots must be placed in the remaining intersections that penalize the players (getting the game closer to a no-pass Go).

§

The second game is Клоподавка (Bug Supression) aka Virus War. Again, the game is played on a large grid. The players decide on a move budget (say N usually between 3 and 6) relative to the size of the grid.

Each player has a home base, each on opposite corners. On his turn, each player marks N squares on the grid. Markings can be on empty squares, or by shading a square marked by the adversary (let's call it a wall). A wall becomes unplayable. However, players can only mark or shade squares that have a connected path from their home bases. 

A player that cannot complete his move, loses the game.

In the following example (with N=6) Blue answered Red's early and over-extended attack by making four walls, cutting red walls from the rest of their army. Now, if Red wants to reconnect, he must spend four markings to get to his three walls.

play this game here

Virus is a very underrated game, I find it quite fun to play and surprisingly tactical. Perhaps the game is too decisive, being hard to counter an early advantage, but who knows what good players could make of it?

Games and their times

All classifications are arbitrary, but some are more useful than others (to paraphrase George Box).

When thinking about the origins of abstract games, I visualize three broad categories:

• Ancient/Traditional Games. These are games within the culture of a particular people or region. Their exact origins are often unknown; we only estimate their age by their first written references. These games usually lack a fixed rule set (if any), having several regional variations. It is often more accurate to think of them as families of related variants rather than single, fixed games.
• Old Games. These appeared after the commercialization of play. They were either published or patented by their creators. While variants of these games may exist, they generally arise due to the game's popularity and how players engage with their core ludemes.
• Modern Games. Most abstract games published online today fall into this category. Although their creators retain authorship, many have relinquished commercial rights, effectively placing these games in the public commons. Much like open-source software, they embody the spirit of freedom and collaboration that characterized the early internet. 
  

These categories are, of course, not rigid. There are still modern commercial abstract games (e.g., the GIPF project) and older games that were always intended to be public (e.g., Laska). However, I find these distinctions useful for understanding games in their historical context.

Even though the boundaries between these categories are not perfectly defined, it is possible to identify their temporal limits.

The era of old games began around the mid-19th century, when commercialization led to the emergence of the first game publishers. This period also saw the introduction of the patent system, which allowed some types of game rules and physical board designs to be removed from the public commons and be privately owned.

In my opinion, the era of modern games began with the rise of the video games industry, a transformation that took place between 1980 and 1990. This shift is particularly evident in board game magazines such as GAMES and Jeux et Stratégie, as well as in the game's sections of popular science magazines like Science et Vie. During this period, columns devoted to computer games gradually occupied more and more space. One consequence of this shift was the decline — and, in some cases, the disappearance — of these magazines in their original formats, as they struggled to compete with publications specialized in video games and software. Another outcome, in my view, was a decrease in the commercial value of abstract games. This reduced the pressures of commercialization and, coupled with the advent of the internet, opened the era of modern abstract games that continue to this day.

Most abstract game designers align with the era in which they live. Designers from the old era typically created commercial games — for example, Alex Randolph, Sid Sackson, Eric Solomon, and Robert Abbott. Meanwhile, modern designers tend to create games for the public domain, with early examples including Christian Freeling, Wayne Schmittberger, Fred Horn, and Ralph Betza.

However, there are exceptions. Some designers from the old era (approximately 1850–1980) adopted a modern, non-commercial approach to game design. Notable examples include V.R. Parton, Joseph Boyer, Martin Gardner, Sid Sackson (again!), and George Dekle. Conversely, some modern designers still produce commercial abstract games. Perhaps the most famous are Reiner Knizia and Kris Burm.

The commercialization of abstract games appears to be on the decline, even as the global board game market continues to thrive, with its popularity steadily increasing over the past twenty, thirty years. But every category remains alive today. Interestingly, traditional games still appear to be the strongest link. Cultural traditions are very resilient to change, with few dependencies that could threaten their continuity. It seems unlikely that Chess, Go, or Backgammon will lose their public in the medium, long term. In a way, this reflects the Lindy effect at work, with the longevity of traditional games correlating with their cultural significance and resiliency.

In contrast, commercial abstract games appear to be losing relevance, especially when compared to highly competitive industries like video games and social media. And modern games are deeply dependent on the free flow of communication — a privilege we must not take for granted in the decades to come.

Catch

Catch is a 1995 game by Reiner Knizia, published in Spielbox magazine.

The game is played on an 8x8 board, initially empty.

Each player has twenty pieces, and there are twenty domino-like tiles shared by both players.

Rules:

• One player drops tiles vertically, and other player drops tiles horizontally
• On his turn, the player drops one tile on two empty squares
• For any enclosed area (either by tiles and/or board edges) with less than four squares, the player drops one friendly piece on each one of those squares
• When there is no valid moves left, the game ends. Wins the player with more friendly stones

The next board shows a final position where the game ended in a 13-13 draw.

This is a pencil & paper game. There's no need for a board to enjoy this game.

I would suggest using an odd-size square board, like 9x9. These boards have an odd number of squares. Since any group of tiles only removes an even number of squares, the remaining pieces will amount to an odd number, making draws impossible.

Take it Away

Take it Away is a 1969 game by Sid Sackson, published at Dover.

It is played on a 8x8 board with 34 white, 20 red and 10 blue stones (for a total of 64 stones).

• The game begins by players randomly placing every stone on the board, one per square.
• The first player starts by removing a white (which he keeps)
• On each remaining turn, the current player must pick one stone on the board and execute with it a sequence of checker-like jumps (orthogonal and/or diagonal). The sequence cannot stop while there are jumps to be made. Every jumped stone is captured.
• When there are no more jumps available, the player with highest score wins the game.
• blue stones value three points, red stones two points, and white stones one point
• There is an optional way to end the game: any player at any time can say "Take it Away" meaning he removes himself from playing. The other player continues making jumps while possible. When he ends, the remaining stones on the board will count as negative points for the last player (which the rules call the patsy).
  

The 'take it away' rule is what makes this game interesting. It is an auction-like rule that adds a very nice tension to the game. The rules especially shines if the game is played with three or four players.

The game appeared in Sackson's book A Gamut of Games.

Here's its description on Jeux et Strategie #45, where it was called Butin (booty):

Nevo

Nevo, aka 2-5-8, is a 1979 game by Michael Vonderheid, published by ArtDeco Games.

The game is played in a 7x12 grid, and each player has twelve pieces. 

The initial setup,

The rules:

• On her turn, the player moves one friendly piece to an adjacent (orthogonal or diagonal) empty space.
• Any piece that becomes isolated, i.e., not adjacent to another friendly piece, is removed from the board.
• Pieces that move to the adversary area score points. In the first enemy row, the player gets two points; in the second row, five points; and in the last row, eight points.
• for a piece to score the previous rows in its column (in the adversary area) must be occupied by friendly pieces, 
• for each one of these columns, only the furthest piece scores (eg, three pieces in the same column only score eight points)
• The player that first achieves a position worth sixteen points wins the game, but with three different pieces (so, two pieces scoring eight points each is not enough).
  

Some notes:

• Since isolated pieces are removed, it means that a piece movement must be thought in tandem with the group it belongs
• The original rules state that it is a decision of the adversary to remove these isolated pieces. I'm not sure if there is any board position where it would be advisable to keep these enemy pieces on board.
• If a column (in the adversary area) is not entirely occupied, the more advanced pieces don't score points
  
• Notice that there are no jumps or captures. This is a race and blocking game.

Here are the original rules:

Perhaps it's not needed, but since pieces can move backwards, there should be a rule about repeating positions. A Ko-like rule would not be easy to implement. It could result in a draw after three repetitions, just like in Chess, or end with a win for the player with the current highest score.

 

The game had a brief review on Jeux et Strategie #2:

And an even briefer mention in GAMES magazine #20,

 

 

Oh, and here's Omar Sharif, well-known actor and less-known game tutor:

Lambchop

Lambchop is a 2001 game by Dan Troyka, published online.

The game is played on a square board, say 5x5, initially empty.

Each player has just one piece (the lamb).

Rules:

• Initially, the players drop their lamb on an empty square
• On his turn, the player moves his lamb to an empty square
• if the square is unmarked, the player marks it for himself (thematically, the lamb just eaten the grass of that square)
• Wins the player with more marked squares

In the following position, White's turn to move, having each player marked nine squares. 

If White goes north to capture d5 and e5, he loses the game, since he would leave to Black three of the remaining seven grass (unmarked) squares. Then, White either captures b2 or a5, Black will capture the other.

There's a ZRF to play Lambchop. It includes several variants with different board sizes, and the use of more than one lamb per player.

Conflict

Conflict is a 2000 game by Alexander Stevens, published online.

The game is played on a 10x10 board, initially empty.

Each player has eight soldiers (myrmidons), twelve shields (bucklers) and four jumpers (assassins).

Rules:

• Initially players take turns to drop their pieces in their first four rows
• On his turn, the player moves a friendly piece:
• soldiers and shields move like Chess queens
• soldiers can only capture if they are in moving range from another friendly soldier
• captures are by replacement
  
• shields cannot capture or be captured 
• assassins can move one or two steps in straight line, or jump over a piece landing in the immediate next square that must be empty or occupied by an enemy piece (that is captured)
• Wins who capture all enemy soldiers.
  

The game originality is in the restricted capturing ability of soldiers. On the downside, the initial phase of dropping 48 pieces on board is not a good way to start a game.  It's too long. The game needs a fixed setup, just like Chess or Checkers (the old ones knew a thing or two). This is important to improve the game's interaction.

after grueling 48 turns,
all pieces are finally on board

Also, having twelve shields seems a bit too much. It might give an opportunity for very defensive tactics.

There is a ZRF to play Conflict.

Quadraphage on Winning Ways

Winning Ways for Your Mathematical Plays, from 1982, is a book that marks the beginning of an entire mathematical area, Combinatorial Game Theory, and a new set of numbers, the Surreal Numbers. It was written by Elwyn R. Berlekamp, John H. Conway, and Richard K. Guy. The book contains an impressive number of mathematical techniques and insight and has very hard sections in it (there are more recent books with the goal of introducing the main concepts with a more pedagogical approach). Below, let's call the book just WW (for Winning Ways).

Among the many games explored in the book, some are closer to the idea of abstract games that motivate this blog. This post mentions one of them: Quadraphage.

The rules of Quadraphage (meaning, the square eater) by Richard Epstein in 1973: 

• In a NxN empty board, a King is placed on a square
• One player moves the King (the Mover), the other player drops a stone (e.g., a Go stone) on any empty square (the Placer)
• Turns alternate, as usual. 
• Goal: if the King reaches any square at the edge, the Mover wins; if the Go stones surround the King, the Placer wins

Since moving first is never a disadvantage, there are three possible outcomes: (a) the Eater always wins, (b) the Mover always wins, (c) the first player to move wins. The book calls a fair position every square for the King to begin, where option (c) occurs.

The book includes the use of other pieces besides the King. It calls Chessgo to this family, and it considers Kinggo (the previous rules) and Dukego (using a Duke, ie, a one-step Rook). Other reasonable options include Knightgo and Ferzgo (using a Ferz, ie, a one-step Queen).

One interesting result from WW is that there are only two possible board sizes where fair positions occur, and that are 33x33 and 34x34 boards (!). On a smaller board the Mover always wins, and for bigger boards the Placer always wins (cf. chapter 19).

Also, the author mentions the game in his 2009's book The Theory of Gambling and Statistical Logic:

 

This game is an offspring of the medieval Tafl games, and a member of the Fox Games' family.

Quintus

Quintus, or Quintun, is a 2003 game by Martin Windischer.

The game is played on a 10*10 hexagonal board. 

There are enough white and black pieces, and enough neutral stones of five different colors.

Definitions:

• the 54 hexes on the edges are the ring
• the remaining hexes are the playing area

Rules:

• On her turn, the player either:
• drops a friendly piece on an empty hex inside the playing area
• drops, on the ring, one neutral piece of a new color
  
• this piece must be at least five empty hexes from other neutral stones in both directions of the ring
• drops, on the ring, two neutral pieces of a color already played
• these pieces must be adjacent to pieces of the same color
• Wins the player that either:
• connects a friendly group with four different neutral colors
• connects two friendly groups, each with three different neutral colors

In the next diagram (from a real match), White resigns: there's no possibility for his large group to connect to a third neutral color

  

This is imho a very good game, a connection game that does not have a fixed connection goal. The only point where this could be improved is in the condition of dropping the first neutral stone of a new color (so to remove the arbitrary distance of five empty hexes). The suggested new rule: 

• drops, on the ring, one neutral piece of a new color in a side with no neutral pieces already in it (including corners) if this is possible, otherwise any ring hex

There are six edges for five colors. If a small number of neutral colors get to have large groups, a new tactical avenue would open by dropping the new neutral color anywhere in the ring.

Diagonal

Diagonal is a 2004 game by Luca Cerrato, published at Fogliaccio degli Astratti #23.

The game is played on an 8x8 board, initially empty.

Rules:

• On her turn, the player drops a friendly piece on an empty square
• After her move, the player scores as many points as the number of their pieces in the completed diagonals where the current piece belongs
• a single piece color in a diagonal does not score
• When the board is full, wins the player with highest score 

in this position, if Black plays [1] the score is 3 points
while if it was White playing at [1], the score is 3+3=6 points

Cerrato also proposes a variant, Diagonal Plus:

• Initially, players drop a number of neutral pieces on board
• These pieces are labeled with positive (bonus points) or negative (malus points) numbers
• Bonus and malus points cannot be placed on the same diagonal.
  
• For the 8x8 board, it is suggested to use: two pieces +2, one piece +3, one piece -2
• It is forbidden to complete a diagonal using only neutral pieces (besides the dropped piece)
• The scores should include the bonus/malus points besides the standard diagonal points

Fox Games: part 2

Fox games are not only from Europe. There are board games with asymmetrical forces in other places.

A well-known traditional game is  Baghchal from Nepal,

As BGG reviews:

At the start of the game, there are four tigers on the board while there are no goats. The goat player places his/her pieces on the board one by one wherever they choose, with the tiger player getting a move between each placing. Once all the goats are on the board, the two players take turns moving one of their pieces one space. A tiger can alternatively capture a single goat by jumping over it in a line to an empty space. The tigers win if they can capture five goats. The goal for the goats (who cannot make captures) is to hem the tigers in, giving them no opportunity to move or jump.

Other traditional examples are

• Diviyan Keliya (Leopards and Cows) played in India and Sri Lanka
  
• The medieval Cercar la Liebre (Catch the Hare) played in Spain, which derived to Coyote played in Mexico c.19th century

• Len Choa (Len Cua Kin Ngoa) from Thailand [1]
• Komikan (El Leoncito) from Chile with a first known reference from 1787 [2]
  

This ludeme is just too simple and attractive and continues to be used in modern games.

One example is 1975's Buffalo Chess (aka Bison) from Alex Randolph.

Ludii description:

Buffalos move one step forward to a free space. Dogs move like a chess queen but cannot capture. The villager moves like a chess king, and can capture buffalos. Dogs and the villager cannot enter the top or bottom rows of the board. Buffalos start, and win by reaching the top row. They lose if they cannot move anymore.

Another (less well-known) example is 1970 Carapace (aka Revanche):

The board is, in essence, hexagonal. The asymmetry here is more pronounced, where one player will win for sure. The adversary goal is resistance. A game consists of two matches, where players switch sides. Some spaces have extra value (check the number of dots in the previous board) that are used for scoring. Wins the player with the better score.

Steve Sisk wrote the next rules:

A modern variant -- in fact, more a puzzle than a game -- is the 2001's Wild Goose Chase by Dan Troyka:

 

The tame geese move one step orthogonally, while the wild goose is unable to capture, but moves like a Queen and is also able to fly to the adversary's first row. The geese have 15 turns to capture it. There's a ZRF to play it.

[1] Captain James Low, On Siamese Literature, Asiatic Researches vol.20[2] (1839)
[2] Thierry Depaulis, Inca Dice and Board Games, Board Games Studies 1 (1998)

Rosette

Rosette is a 1975 game by Mark Berger, published at Games & Puzzles #34:

Mindsports has more information about this early Go variant. Also check this BGG forum discussion.

Turning Point

Turning Point is a 1969 game by Phyllis Frederick and Peter H. Justin, published by Mattel.

Each player starts with 27 regular pieces, two double pieces and one 'stop' piece. Each stop piece is given initially to the adversary. There are also two scoring pegs that are placed at the zeros of the scoring columns.

The game uses custodian capture, i.e., when a player 'sandwiches' a line of enemy pieces by two of his own pieces, he captures (here, flips) those sandwiched pieces. Captures can occur orthogonally and/or diagonally.

• First, the players take turns placing two pieces each at the 4x4 center area.
• Then, each player, on his turn, drops a friendly piece on an empty space.
• If he captures/flips one or more enemy pieces, the player gets one point for each piece in the line(s) he just formed (that includes not only the flipped pieces, but his own pieces).
  
• For each scoring line that includes N double pieces (or either army) the score of that line is multiplied by 2N (eg, a line with three doubles, gets a multiplier of six). 
• A double piece just dropped cannot be used as a multiplier (it only counts at subsequent turns).
• A player can play the adversary stop piece, but side down, whenever he wants.
• When the stop piece is turned back, the line it belongs to is not scored.
  
• When the board is full, the game ends. 
• Each player adds an extra point for each friendly piece on the board.
  
• Wins the player with the highest score.

The game is a variant of Othello. The interesting feature is the introduction of special pieces that change the dynamics of the game and might produce sudden differences at the final score. Of course, why stop at just the 'double' and 'stop' pieces? Imagination, as usual in design space, is boundless. Some possibilities:

• The switch: when flipped, it forces all eight neighbors to turn to the other color
• The pass: when flipped, the player must pass his next turn
• The negative: when flipped, the total current score turns negative
  

Also, the final 'one point per piece' rule seems an unnecessary vestige of Othello's ruleset and could probably be removed to make Turning Point a bit more different than its predecessor.

Ko-an

Ko-an is a 1994 game by David Welch and Paul Whitehorn, published at Image Games.

Each player has two types of pieces: five squared pieces and six octagonal pieces. The board is composed of a grid of yellow squares and green octagons.

This is the initial setup:

• Pieces only move forwards, not backwards nor sideways, to an empty space. There's a further move restriction: a piece cannot move between squares [in a better designed board, like the one below, it's easy to see why: squares do not touch on a 4.8.8 grid].
• Pieces can also move forwards to a space occupied by an enemy piece, which is then captured (captures are by replacement).
  
• However, captures also depend on piece type: octagonal pieces can only capture pieces on octagons, and squared pieces can only capture pieces on squares.
  
• Capturing is not mandatory.
  
• Wins the player that moves a friendly piece to the last row or stalemates the adversary (which includes capturing all the enemy army).

Here's a board (by r0cka) to play the game:

From a 1994 review by Richard Breese:

[...] Considering its simplicity, the game can develop in a surprising variety of ways. For example, pieces of both players may bunch up on one side of the board, several 'skirmishes' may occur all over the board, or a large stand off may arise across the center of the board. This variety adds to the attractiveness of the play.

Of the two types of playing pieces the octagonal piece is stronger and this factor probably explains why in most games it is a octagonal shaped piece which makes the winning break through. This strength arises from the situation where an octagonal shaped piece confronts a squared shaped piece. As the square spaces do not join each other it is necessary for all pieces to move onto the octagonal shaped spaces - the spaces on which the octagonal shaped pieces can capture. Consequently an octagonal shaped piece is much more likely to be able to capture a square shaped piece than vice versa.

Some pics from the original game package: