A Argentinian old challenge

I found at the Argentinian magazine El Acertijo #17 (1995) the following challenge: 

Definition: A five-line is a piece that results in joining five segments of equal size, aligned consecutively by right angles. There are a total of 23 different five-lines.

The challenge is to use these 23 five-liners to build a perimeter with an area as large as possible, assuming the five-lines are supported by a ground level.

The first example, presented by Héctor San Segundo, the puzzle inventor, had a total area of 491:

In a subsequent number, the best answers to the challenge were presented:

The best result was 793 by Marcelo Iglesias. But Héctor had found a result with 811 units of area (the one shown above).

Afterwards, there was a new record by Pablo Coll with 814 units of area:

He argues, in the text above, that if the perimeter made by the five-liners were able to exactly follow the semi-circle, the maximum area would be around 830 units. So, 814 units is pretty close to that theoretical maximum.

The magazine does not seem to include more information about this puzzle.

Nowadays this problem might be solvable by searching for all valid possibilities. We cannot enumerate and try all solutions (23 factorial is still pretty big) but by searching with some good heuristics, this seems a doable problem.

Hexagonal Checkers

There are abstract games with hexagonal boards from the 19th century. And Checkers is a much older game. Also, Chinese Checkers, from 1893, while not being a Checkers game, uses its name and applies the same jumping principle (but without captures). So, while using hexagonal boards to play Checkers seems like -- in retrospect -- a rather obvious idea, did it had to wait until the 1970s and 1980s for the first variants to appear?

The best-known variant is Christian Freeling's 1979 HexDame, applying the rules of International Draughts on a hexagonal setting (cf. at BGG),

Less known variants include:

1) Pskov Checkers (undated), which uses the rules of Checkers instead of those from Draughts, and promotions are only possible at the last corner space (instead of the entire last rows of HexDame).

one of the proposed board/setup for Pskov checkers

 

Here's a translated ruleset (from Russian)

2) Damex, that appeared in Jeux & Strategy #13, uses the rules of Dames (the name of Checkers in France) in a squared-liked hex-board:

Damex solves a problem from traditional Checkers, where even three Kings vs one does not force a win in all possible positions. In Damex 2 vs. 1 is a win.

The game is attributed to M. Lavictoire that sent the rules to J&S in 1980. According to Ralf Gering, this is an iteration of an older game from Joseph Boyer*, Les Dames Hexagonales, described in his book (authored together with Vern Parton) Les jeux de dames non orthodoxes et autres jeux à pions from 1956 (referenced at google books but not available, hélas!). 

(*) Boyer also wrote three books about chess variants, Les Jeux d'échecs non orthodoxes (1951), Nouveaux Jeux d'Echecs Non Orthodoxes (1954), and Nouveaux jeux d'échecs intéressants (1956).

In 1925, US Patent 1623881 specifies a Checkers-like game with an hexagonal grid:

The moves and goals, however, make it a distant relative of the Checkers family. But we were able to go back from 1956 to 1925. 

A bit earlier, in 1914, Hervey Dexter Thatcher submitted patent US1106991 for a King Bridge Checker Game, also using a hexagonal grid:

In this game there is a middle three hex bridge, and when a piece crosses it, it is promoted to 'King'. This game was published as we can see in the next pictures,

Kingbridge, AGPI Quarterly volume 7 [4]

 

a modern edition of Kingbridge

We can still step back 20 years, for a game by J. F. Beaman in 1894, described in US Patent 529582:

This is a proper Checkers game. So, our first 19th century hit! 

Another hexagonal checkers was patented by Charles E. Duryea in 1888 (patent US384195):

Can we go further back? Yes, here's US Patent 259695 by John F. Kingwill from 1882:

This (unnamed) game is basically a Checkers for three players. So, we have already some 19th century Checkers games! At BGG it was named Hexadraughts.

For the oldest entry, Ralf Gering mentioned a book by J. G. Lallement, Les Quatre Jeux de Dames from 1802 (which had a 2023 reprint),

If we zoom in the boards:

we notice the triangular/hexagonal nature of the connections between the board spaces!

So, after all, the idea did not have to wait for the Seventies!

 

 

 

The Damex solution for figure 2, from the J&S puzzle:

Spectrum

Spectrum is an uncredited 1975 game published by Intellect Games.

I was unable to find the entire ruleset and tried to reconstruct the game rules from the available information (if anyone have them, please let me know).

The game is played on a 10x10 board, with sets of six red and violet pieces, 16 orange and blue pieces, and (probably) 28 green and yellow pieces, for a total of 100 pieces. One player owns the red, orange and yellow pieces, while the other owns the green, blue and violet pieces.

The game starts with both players placing, in alternate fashion, their six red and six violet pieces in the board. These pieces must not be adjacent to each other.

Then each player, on her turn, drops one friendly piece on an empty square provided that it is only adjacent to pieces of adjacent colors in the color spectrum:

red ↔ orange ↔ yellow ↔ green ↔ blue ↔ violet

I'm assuming that a player passes if she has no legal moves; and the match ends after two consecutive passes.
 
Wins the player with more placed stones; or if both players have an equal number, the match is a draw.

Ralf Gering said about this game: The game appears to be flawed because the final scores are always very close (+/- 1). In fact it seems that the best the second player can achieve is a draw. Maybe the game can be fixed by multiplying the number of tiles of the remaining colors. The player with the smaller product wins.

Brax

Brax is an 1899 game by Frederic Denham (patent 406632), published at Fratelli Fabbri Editori.

Brax is a capture game where the goal is to capture all enemy pieces.

Each player has seven pieces, initially placed in the seven center intersections at the player's first row.

Pieces can move up to two lines of their color, and only one line of opposite color (notice that half the lines are red, and the other half are blue). Pieces cannot move over intersections occupied by pieces, or onto intersections occupied by friendly pieces.

Capture is by replacement; a piece that moves into an enemy-occupied intersection captures that piece. Captures are not mandatory.

The game includes another rule named braxing. After moving, if a player threatens a capture in the next move, she can optionally call a «brax» (the threat does need to come from the last moved piece). When a brax is called, the adversary must move that piece in his next turn (if several are threatened, the player can choose which to move). If two pieces of one color and one piece of the opposing color are left on the board, the player with the single piece can no longer brax.

If both players only have a single piece, after five turns without capturing, the game is a draw.

The game includes extra star pieces to allow matches with three or four players,

Brax is also featured in Roger Millington's Games and Puzzles for Addicts,

Here's an older board design:

The game also has a Wikipedia entry.

Atrio

Atrio is a 2000 game from Christian Gibon, published at LM Créations and Murmel Spielwerkstatt und Verlag AG.

The game is played on the following board (I was unable to find the initial setup),

Each player starts with three marbles, two regular and one special.

• The regular marbles can move one space in each direction (orthogonal or diagonal)
• The special marbles can move one or two spaces in the same directions

The goal is to place your marbles inside the center 9x9 area (inside the interior circle), or else to push out one enemy marble out of the board (outside the exterior circle).

If the moving marble is adjacent, in the moving direction, to an enemy marble, it will push the enemy marble. This is valid only if all spaces where the pieces move thru are previously unoccupied. The Ko rule applies, the next player cannot use the same pushed marble on the opposite direction of the previous move.

Domain

Domain, aka Boomerang and Chameleon, is a 1982 game by Claude Duvernay, and published in several companies.

There are two sets of Tetris pieces. Here is the blue set:

And these are the rules:

Domain was reviewed by Sid Sackson at GAMES #37: 

§

Poliminoes are a much-explored subject in Recreational Mathematics. A good book about the subject is 1965's Polyominoes: Puzzles, Patterns, Problems, and Packings by Solomon Golomb.

 

 

Poliminoes (especially trominoes, tetrominoes and pentominoes) appear in many abstract games. BGG lists more than 100 abstract games using poliminoes.

 

Probably the most successful one is Blokus,

 

which also provides a very nice material set to design and experiment with new tiling games.

Canyon

Canyon is a 1972 game by Frank Ullmann and Tom Werneck, published at Heyne.

Each player has 15 pieces that must be placed on the spaces inside the triangles marked by cells 1, 5 and 11.

Rules:

A piece can move to an adjacent space in any direction: forward, backward, diagonally, or sideways.

It is also possible to jump over one or more friendly or opponent pieces in any direction, provided the next space is free. You can jump as many pieces as desired.

Each opposing piece that is jumped is captured.

The central zone, the canyon, is impassable. Players must go around it to the left or right. Numbers on the edges of the board indicate the value of each row based on each player's perspective.

The players’ goal is to move as many pieces as possible to spaces with high values. 

The game ends when one player has moved all their remaining pieces to the rows with the highest values. Both players add up the scores of the spaces occupied by their pieces. Each piece scores the value of its row. The player with the highest score wins the game.

Brainline

Brainline is an uncredited 1970s game published by Berliner Spielkarten and Palitoy.

 BGG has this description of the rules:

A two player abstract game played on a 7 by 9 array of hexes. Each player has four pegs of his colour on the specified starting hexes. On his turn a player may move any one of his pegs any distance in a straight line but may not jump any other peg. The object of the game is to be the first to get all four pegs of your colour in a straight line. They do not have to be adjacent but there must be no opponent pegs between them.

I've also found this review in Games & Puzzles,

Babylon

Babylon is a 2003 game by Bruno Faidutti, published at Interlude (among others).

The game does not have a board, only twelve stackable tiles (three tiles for each one of four colors).

Initially, all tiles are placed in the gaming area. Let's denote a single tile as a stack of size one.

Rules:

• On her turn, the player must place an entire stack on top of another.
• A move is valid if either:
• both stacks are of the same height
• both stacks have the top-most piece with the same color 
• Wins the player making the last move

This is a game that seems very amenable for analysis using Combinatorial Game Theory. But I'm not aware of any article studying it.

Peace

DIY Draughts

Do-it-yourself draughts, or Dames en Kit, is a c.1890 game by Walter Sanders,

diagrams from Jeux Stratègie's page

Rules. Each player has two pieces and 20 tiles (one light and one dark set).

First phase: On each turn, each player places (a) a friendly piece on an empty same-color tile, (b) a friendly tile at the side of another already placed tile, (c) or both.

Second phase: When all pieces and tiles are placed, pieces are allowed to move to any adjacent empty tile of its color (including diagonals). 

However, in both phases pieces also have the option to capture. Pieces capture by jumping one or more consecutive enemy pieces, assuming there is an immediate empty tile of its color to end its move.

Captures are multiple, i.e., after a capture, the player may continue capturing with the same piece. And, like Checkers, captures are mandatory (if there are several options, the player can choose which). Captures are mandatory in both phases.

• Notice that it is possible to place a tile, opening a capture option, that must be executed in that same turn. However, even if it is possible to place a tile in these conditions, players might choose not to.
  

Wins the player that captures or blocks all enemy pieces. If the interactions between pieces are no longer possible, wins the player with more moving pieces (blocked pieces do not count). Otherwise, the game is a draw.

Example: in this diagram, players cannot interact with each other. So, Black wins since he has two moving pieces against just one moving white piece.

The game appeared at Roger Millington's Games and Puzzles for Addicts (1979),

The game also appeared in Jeux et Stratègie #42 in 1986: