Fractional Hex

[From Edward Jackman conversations] Speaking of Hex, here's another idea to address it's first-player advantage that may not have been explored much. I found it in the Mudcrack Y and Poly Y book, but it applies to Hex as well. The first player draws a line connecting the midpoints of the opposite sides of one cell, dividing it half, creating two 5 sided cells. She plays her opening move to one of the two halves. The fewer sides a cell has, the weaker a move there is. You might even require that the line divides the cell into a 4 and a 6 sided cell and the move goes in the smaller cell -- that would be a very weak move, even in the center of the board.

Standard move, filling entire cell:
        +---+    
       /     \
  +---+       +---+
 /     \     /     \ 
+       +---+       +
 \     /xxxxx\     /
  +---+xxxxxxx+---+
 /     \xxxxx/     \ 
+       +---+       +
 \     /     \     /
  +---+       +---+
       \     /
        +---+

Half move:
        +---+    
       /     \
  +---+       +---+
 /     \     /     \ 
+       +-+-+       +
 \     /  |xx\     /
  +---+   |xxx+---+
 /     \  |xx/     \ 
+       +-+-+       +
 \     /     \     /
  +---+       +---+
       \     /
        +---+

One third move:
        +---+    
       /     \
  +---+       +---+
 /     \     /     \ 
+       +---+       +
 \     /    ,+     /
  +---+    /xx+---+
 /     \  |xx/     \ 
+       +-+-+       +
 \     /     \     /
  +---+       +---+
       \     /
        +---+

Loss

Every loss should always teach a new lesson. [T.Sagme, Meditations]

Game Sources

These are useful sources to find new games:

• Mixing known concepts in strange ways (i.e., applying game mutators).
• Choose a good word and squeeze its content into a game.
• Get a board, some stones (of one or more types) and a position that pleases you, and hear the story they have to tell.

Perfection

"We achieve perfection only with a perfect use of imperfect actions" [T.Sagme, Meditations]

The Waves of Fortune

Talking with an old friend, we noticed that if there were no mistakes in some games we were playing, the moves would be always the same, i.e., the perfect match. That remark stayed inside my mind because it annoyed me the feeling that, in perfect information games (i.e., no luck and no hidden information), two players devoting enough time and skill so that eventually are able to completely master the game they love, will destroy it! The game collapses to a single and fixed contest - always win, always lose, always draw...

In games where the random factor exists this does not happen! I'm not talking of 100% random procedures (they are not games) like the Roulette or the Lottery, but something like Backgammon (a game with luck) or Stratego (a game with hidden information). If the balance of randomness is finely tuned, a good player will consistently beat any worse player. Facing two perfect players in a game like that, the final outcome is not decided at all! Using a fair random generator, each will have a certain percentage of chance to win. A statistical average of wins/loses will still apply but not for an individual game - the actual game being played. And if that average is near 50%, the joy cannot be destroyed by sufficient skill or dedication.

The Ocean of Abstract Games it is an area of flat water, where nothing random or secret appears to disturb the surface. With algorithmic evolution (not just speed, speed is useless for Go), this part of the Ocean will continuously grow smaller and smaller. I wonder if, sometime, most of these games will have the need for the Waves of Fortune...

Balance

"Let your mind be open, but not so open or it will fly away." [T.Sagme, The Book of Parrots]

BI-POD

BIPOD (c) 2004 Bill Taylor

It is played on a trapezium-shaped hex board, with two cells on the long side occupied by different coloured pieces, so that the colour-colour and colour-corner empty-cell numbers are equal:
_______________________________
. . . . . @ . . . . Q . . . . . 1
 . . . . . . . . . . . . . . .  2
  . . . . . . . . . . . . . .   3
   . . . . . . . . . . . . .    4
    . . . . . . . . . . . .     5
     . . . . . . . . . . .      6
=`abcdefghijklmnopqrstuvwxyz-';

The two players alternate in placing a single(*) blocking stone in any empty cell.  On any turn, a player may elect instead to "adopt" the running option.  The other player then becomes the Blocker, and continues playing single blocks on each turn. The Runner now plays single stones of whichever starting colour he chooses for that turn.

Runner wins if he can create two paths, one from each of his start cells, in the appropriate colours, to any points on any of the three short sides. Blocker wins if Runner has failed in his goal when all the cells are filled.

(*) It is recommended that for email play, the players agree to play any number of stones they like, between one and (say) three inclusive.

*************

Sample Game:

1. e13f2   su3
2. a3c3   adopt

Move 3 does not have impact in the position, so in fact it is a pass to see if Second places another stone. But Second adopts.
_______________________________
      x . @ . . . . Q . . . . . 1
       x . . . . . . . . . . .  2
  x x x . . . . . . x x . . .   3
   . . . . . . . . . . . . .    4
    . . . . . . . . . . . .     5
     . . . . . . . . . . .      6
=`abcdefghijklmnopqrstuvwxyz-';

3. v2 o    j4

While First saves the right position, Second tries to block the left sector.
_______________________________
      x . @ . . . . Q . . . . . 1
       x . . . . . . . o . . .  2
  x x x . . . . . . x x . . .   3
   . . . . x . . . . . . . .    4
    . . . . . . . . . . . .     5
     . . . . . . . . . . .      6
=`abcdefghijklmnopqrstuvwxyz-';

4. g3 o    h2
5. j2 o    i3
6. m3 o

First then makes a threat at g3, which provides extra space.
_______________________________
      x . @ . . . . Q . . . . . 1
       x x o . . . . . O . . .  2
  x x x o x . o . . x x . . .   3
   . . . . x . . . . . . . .    4
    . . . . . . . . . . . .     5
     . . . . . . . . . . .      6
=`abcdefghijklmnopqrstuvwxyz-';

 6...     m5
 7. p4 o  y3
 8. w3 O  x4
 9. v4 O  w5
10. u5 O  v6
11. t6 O  p6
12. resigns  

But even that extension is not enough because Second uses the right white pieces to block part of the way of the left white pieces.
_______________________________
      x . @ . . . . Q . . . . .  
       x x o . . . . . O . . .  
  x x x o x . o . . x x O x .    
   . . . . x . . o . . O x .    
    . . . . . x . . . O x .      
     . . . . . . x . O x .      
=`abcdefghijklmnopqrstuvwxyz-';

Players

Even if there are many types of games, there are two main types of serious players: generalist and specialist gamers. Specialists dedicate most of their free time to one game (even if they know and play other games) and dedicate large amounts of time learning openings, tactics and strategies about it. Generalists are meta-gamers curious with new rules, different ideas. Sometimes the former define the later as "people who failed to become good at one game", but this is an unfair critic. Most player are not generalists and so are bound to tradition and mainstream culture, playing games that society (sometimes just by historic reasons) maintains. In a sense, the same happen with religion, scientific models (at some extent), music... But it would be better for many obscure and excellent games if that was not so.

A generalist analysis a scenario with much less prejudice (and much less depth) than a specialist. Specialists are more uncomfortable with the unfamiliar. They like the assurance that practice and study provide. Even with enough intellectual skills, the generalist do not have the patience for the learning required. Performance is not the main goal. What matters are delight and surprise by unusual new thought processes.

There are two opposing problems. A specialist would think: "Why should I lose all my investment in game A to look at game B?". A generalist would think: "Why should I invest time in game A? Why not game B?". These questions do not have easy, general answers. I'm still not able to answer mine. [T.Sagme, Meditations]

Haiku

At each game's end, a
dialog full of untried
possibilities.

Chinese and Japanese Go rules

The following Go example shows a safe set of stones with two eyes:

   . . . . . .
   . . o o . .
   . o o . o .
   . o . o o .
   . . o o . .
   . . . . . .

Would this set be called a single group even though it is composed of two disconnected halves?

Most players would call it a single group. In area rules it doesn't matter what you call it. In territory rules it could conceivably matter what you call it, (though it doesn't in practice), because territory rules, in all their artificial absurdity, have to refer to groups from time to time in order to define what is considered "dead" and what is considered "alive".

In territory rules, this matters(!). In area rules, it doesn't matter - if there is any dispute you just play it out (and with no cost) until a group or groups is removed from the board.

Logically speaking, you should call the above two separate groups, each helping to keep the other alive. But people never speak so precisely in practice.

There is even a worse situation. The following group has only one true "eye"; the other one, in the NW corner, is a so-called "false eye", and can eventually be filled and the whole lot captured.

   . x x . o .
   x o o o o o
   x x x x o .
   x . x x o o
   x x x x o .

Territory rules actually have to define the concept of eyes, false eyes, and the rest. It is lunacy. (Area rules define nothing - you just play it out to the grim end if necessary). Territory rules, with their defined false eyes, come to grief in this famous sort of position:

   . x x x x x
   x o o o o x
   x o o . o x
   x o . o o x
   x o o o o x
   x x x x x .

Here, black has TWO false eyes, and not a single true one! And yet, both separate groups, or parts of a group, are keeping one another alive; rather like your above example. And even the Japanese admit that black is alive, in spite of what their rule books say!

Basically, territory rules are an abortion. Computers cannot handle them because they are essentially logically flawed. [Bill Taylor]

A progressive game of Y

Here follows a Y game with a 1222 progressive mutator and the following restriction: both drops must be on non-adjacent groups.

   1          .
   2         . .
   3        . . .
   4       . . . .
   5      . . . . .
   6     . . . . . .
   7    . . . . . . .
   8   . . . . . . . .
   9  . . . . . . . . .
     /       /       /
    a b c d e f g h i

         X       O
    1. c5      c7 e8
    2. d7 f8   c6 e7
    3. d6 f6   d8 g8
    4. f7 d4   f9 e5
    5. e6 h8   g9 i9
    6. a5 b7   b5 a6
    7. b6 a4   g7 a4
    8. h9 a3   b6 b3
    9. resigns

Final position:

   1          .
   2         . .
   3        O X .
   4       X . . X
   5      X . X . O
   6     O X O X X X
   7    . X O X O X X
   8   . . . O O X O X
   9  . . . . . O O O O
     /       /       /
    a b c d e f g h i

'O' cannot play both b4 and b5 (the two drops would belong to the same group). So 'X' make occupy one of those cells plus another below to connect the bottom edge.

Progressive Mutators

One of most general mutators is [Progressive] - players get an increasing set of moves per turn. The most common forms are 1234... and 1222... Most games do quite well with 1234... but most have to be restricted in some way to keep the flavour of the original game as much as possible, and prevent it becoming a mere bludgeoning race: Chess retains check-stop; Go has atari-stop; 1222 Hex is often played by dropping non-adjacent stones, and so on.

Similarities

In the main desert of Konn'ex homeworld I found a bonding game. The rules are: "In a hexagonal board (of size 5 per edge), 11 stones for each player are placed in the two initial rows (players stand on opposite sides). For each moved stone, the player must also move another friendly stone in the same direction and an enemy stone in the opposite direction (e.g., if you move northwest, the enemy moves southeast). Captures are by replacement and mandatory (with a max-capture rule). Wins the player that stalemates the adversary."

The natives explained me that in the desert, strong bonds (be friend or foe) are essential as water, and so their games reflect that concern. How alike are we all... [T.Sagme, Travels]

Non-reversible moves

In the Orion sector, a traveller showed me a book of games concerning reversible and non-reversible moves. In most of these games, pieces were judged by the notion of piece gradient. For example, a stone could move thru a friend if it was moving forward, but it could not jump again backwards. Transparent in one side, opaque in the other.

Earth has games with piece gradient (like Chess and its pawns) and games without gradient (like Go). I personally like games without gradient but it was fascinating to find a complete family of games using this specific concept. [T.Sagme, Travels]

Further biographical remarks about our patron saint, Trabsact Sagme.

Trabsact Sagme was a mystic and game player of the late BC years, Tibetan. Widely regarded as the mother of abstract games, and in particular Go. She had an affinity with Parrots, Snails and Spiders (qv). Her writings on abstract games were preserved for posterity by Megas Bactras, (or Bacttras), a central Asian of mixed Bactrian/Greek, who transmitted them to the West soon after Alexander's conquests. Though eventually lost sight of, they were rediscovered by Bart MacStages, a British adventurer of late C19, (who incidentally helped Col Younghusband's expedition to Tibet).

Trabsact Sagme was the Earth's 1st serious ExoLudologist. She was born around Ulan Bator, in 2293.

Yes; widely regarded as being the same Sagme!

The mystics and physicists at both ends of this great journey, as well as we here in the middle, have all para-simultaneously been groping toward this same conclusion. Trabsact Sagme of BC 454 and Tabsact Sagme of AD 2293 are in fact THE SAME ONE. Apparently this can be achieved by some sort of Quantum gravity effect involving trivalent logic, snail-shell spiral symmetry, and other physico-mystical effects. I'm a bit hazy on the details; it's the sort of thing you chaps would know more about than me anyway. I gather it has something to do with the timelessness and non-locality of abstract games in particular and the abstract world in general; (OC, as a mathie I am more familiar with these concepts.)

The fact that we play almost all our games with o and x symbols these days is in honour of Sagme's fondness for snails and spiders. That she was also fond of parrots is not so much reflected in our games, though we often quote from the Book of Parrots, one of Sagme's most popular!

"The novice squawks loudly, but the wise parrot plays her eggs silently on the board." - The Book of Parrots (Trabsact Annals)

Sente/Gote

It is difficult to balance offense/defense. Many games are flawed because offensive or defense is too strong. The initiative cannot always be good or always be bad. It must depend on the context created by the best player. [T.Sagme, Meditations]

Initio

Trabsact Sagme was (will be) the Earth's first Exoludologist. She was born around Ulan Bator in 2293. Since her infancy, she was very interested by all knowledge created and gathered in the 20th and 21st centuries about abstract board games, a subject almost forgotten on those exciting days. Alien civilizations were found across the big black sky, and Humanity finally read Encyclopedia Galactica. After collecting the most important games on Earth, she decided that her life goal would be to study games from alien civilizations. She was one of the few people with the right to enter into some alien homeworlds. It was no coincidence the ones allowing her to enter were exactly the ones which had more respect about abstract board games. Some of the civilizations she visited:

• Civ-3: They use trilogic as a foundation for all mathematical activities, i.e., instead of true/false logic, they use true/false/unknown.
• Konn'ex: They see all objects and concepts as connections of others, and their games reflect these way of seeing things.
• Zet: Everything is a part of something. Their games, usually create complex pieces from atomic ones.

She found many resemblances with Human games, especially in connection and pattern games, namely, Hex and Gomoku were found in almost all home worlds (however in different board sizes and move equalizers). Curiously, no game similar to Chess was ever found!