HEX
Copyright (c) 1942 Piet Hein
"In 1942, Danish poet & mathematician Piet Hein invented a game now called Hex. Piet invented the game while he was contemplating the famous four-color theorem of topology. The theorem, then unproved, is that four colors are sufficient to make any map so that no two countries of the same color have a common boundary. The game was introduced as the Niels Bohr Institute. The game became popular in Denmark under the name Polygon. The game was printed on pads of paper; and sold as a paper & pencil game. For months, Danish newspapers ran a series of "Polygon" problems. Albert Einstein kept the game on a shelf in his study." [Full text]
Hex is a two player strategy game played on a NxN rhombus of hexagons.
|
| An example
Cell [1] is essential to both White and Black, since if one drops a stone there it will cut the other. The two other black stones are can always be connected since there are two possible ways (the cells marked with [2]). If White places a stone in one [2], Black should place in the other [2]. This is called a 2-bridge. |
I include here an idea from Michael Leigh, to equalize the game (he called it Vex): We add an additional neutral stone, say red, which is given to the second player who has the option to place this stone on the board on any empty hex of his choice either before the first player makes his initial move or on any subsequent turn in addition to playing his black stone. Vex is a draw because white can always pretend the neutral stone is an extra white stone (which therefore cannot disadvantage him forcing him to lose) and black may place the neutral stone before the start of the game and then by assuming the stone to be black and playing the 'hex' winning strategy cannot be forced to lose either. By using a neutral stone and giving the second player the option when to play it we no longer need to 'know' which hexes are winning hexes for the first player in order to make the game fair.
In 1995, Michael Howe proposed, on Games Magazine, three variants of Hex:
- Link: in this game after placing a stone a player may move any number of their stones which form a connected line, in of the directions of that line, up to as many spaces as the size of the group. Opposing stones in smaller groups may be pushed, but cannot be pushed off the board. [so, something similar to Abalone without captures]
- Tack: in this game a group of two or three stones in an adjacent line are captured if the other player places a stone to bracket the line on both ends. The captured stones are replaced by the capturing player's stones. [custodian capture Othello-style]
- Fan: in this game after placement, an optional capturing move is made by moving a piece one space so as to end up adjacent to an opposing stone, or to end up one space away from an opposing stone it started adjacent to, or both.
Here's a low-res picture of those pages:
Chameleon is a Hex variant, where both players may use stones of either color to achieve a vertical/horizontal connection, i.e., one player wins if he makes a vertical connection, the other by a horizontal connection. If a move makes both connections, wins the mover.
Words from Cameron Browne: Playing Chameleon is a constant tightrope act. In most connection games, each player can concentrate fully on pushing their connection as hard as possible. However in Chameleon players must keep their connections strong only in their direction or risk having them stolen. Players must consider the implications of each move very carefully. Chameleon has a similar feel to Jade but with clearer goals.
The game was available at gamerz.com, but the link stopped working, so here is the original text:
Rules
Two players, Vert and Horz, take turns placing either an 'x' piece or an 'o' piece on the board.
Vert wins by completing either a chain of 'x' pieces or a chain of 'o' pieces between the top and bottom board edges. Horz wins by completing either a chain of 'x' pieces or a chain of 'o' pieces between the left and right board edges.
If a move results in a connecting chain for both players, then the mover wins. For example:
. . . . . . . . . . o . . . . . . . . . x
. . . . . . . . . . o . . . . . . . . x .
. . . . . . . . . . o . . . . . . . x . .
o o o o o o o . . . . o . . . . . . x . . .
. . . . . . . . . . o . . . . . x . . . .
. . . . . . . . . . o . . . . x . . . . .
. . . . . . . . . . o . . . x . . . . . .
Horz wins Vert wins Last mover winsNotes
Playing Chameleon is a constant tightrope act. In most connection games, each player can concentrate fully on pushing their connection as hard as possible. However in Chameleon players must keep their connections strong only in their direction or risk having them stolen. Players must consider the implications of each move very carefully.
Chameleon has a similar feel to Jade but with clearer goals.
One of the most interesting aspects of Chameleon is that it inherently solves the first move advantage problem which plagues most connection games. While opening in the centre is a winning move in Hex, it is a death sentence in Chameleon. The first player's best opening move is well away from the centre and any opponent's edge.
Chameleon should be played on larger boards. Games smaller than 10x10 tend to degenerate into a race after only a few moves.References and History
Chameleon was discovered by Randy Cox in early November 2003, then independently rediscovered mid November 2003 by Bill Taylor after an idea by Cameron Browne. Interestingly, there is a good reason for the proximity of these independent discoveries, as both were motivated by the upcoming deadline for the 2003 Shared Pieces game design competition.
The game was originally called Goofy Hex then Funky Hex by Randy, but was first made public under the name Chameleon and that has stuck. This name refers to the fact that players tend to change colours based on their environment; the fact that Bill's eyes pop out when he sees a good move has nothing to do with it.
Implementation and help file by Cameron Browne, November 2003.
Cameron Browne, in 2003, designed the following connection game for the Shared Pieces design competition:
Imnyss
- Players alternate taking turns, where each turn involves:
- Placing a piece of the nominated colour on the board, then
- Nominating what colour the opponent must play next turn.
- The opening piece may be of either colour.
- Goal
- H wins by completing an 'x' chain or an 'o' chain left to right.
- V wins by completing an 'x' chain or an 'o' chain top to bottom.
- If a move results in winning chains for both players then the mover loses.
A sample match:
Imnyss (game 1)
====== CamV BilH
C ==== ====
1 . . . . . . . . . 1. o f3 x e6
2 . . . . . . . . . 2. o d6 o g4
3 . . . o . o x . x 3. x d8 o d4
4 . . . o . o o o . 4. x g3 x c7
5 . . . . . . . x . 5. o f6 x f7
B 6 . . . o x o o x . B 6. o b7 o f4
7 . o x x . x . . . 7. o b8 x g8
8 . o x x . x x . . 8. x i3 x b9
9 o x . . . . . . . 9. x h5 o g6
10. o a9 x f8
a b C d e f g h i 11. x h6 o h4
12. o d3 x d7
13. x c8 o resigns
Bill Taylor and João Neto proposed Dozer Hex in 2008:
- Like Hex, but lines of friendly stones may push, abalone-wise, smaller lines of adversary stones by exactly their length difference.
- if any stones block the full length of a push, the push stops there.
- A line may push its own length into empty space.
- Stones pushed off the edge disappear from the game.
- No friendly stones may be pushed.
- Swap option exists.
A sample match:
__abcdefghijklmnopqrstuvwxyzAB
___________OO_________ _Jo____Bx_ _Jo____Bx_
\ . . . . . . . . . o \ 1. p4 p6 14. p4v4 u7
\ . . . . . . . . o o \ 2. j6 l6 15. s7 u5
\ . . . . . . . o x . \ X 3. r6 n6 16. q3 v8
\ . . . . . . o o o x \ X 4. q5 o7 17. p4 k9
\ . . . o o o x o x . \ 5. r4 n8 18. n4 j10
X\ . . o , , x x x o . \ 6. m9 n8s3 19. s1 w7
X\ . . . x . x o x x . \ 7. o5 m7 20. n4v4: s9
\ . . x . . o . x . . \ 8. m5 s5 21. r8 t6v4:
\ . x o . o x . . . . \ 9. q3 m8 22. q9 l6v6
\ x . . . . . . . . . \ 10. n4 q7 23. resigns
---------------------- 11. k5 u5
OO 12. q3t6 o7s3
12* Hex Playing 19x19 Hex can be very time-consuming. In 2016 Taylor and Neto tried the following version:
- 19x19 board
- 12* moves with move restriction (ie, both stones cannot belong to the same group)
A sample match (in 17 turns):
B ooo
abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ-=_
. . . . . . . . . . . . . . . . . . . 1
. . . . . . . . . . . . . . . . . . . 2
. . . . . . . . . . . . . . . . . . . 3
. . . . . . . . . . . . . . . . . . . 4
. . . . . . . . . . . . . . o . . x . 5
. . . . . . . . . . . . . x x o o o . 6
. . . . . . . . . . . o x o X x x x o 7
. . . . . . . . . . . o x o o x o . X 8
. . . . . . . . . . . x x o x o . . . 9 x
x . . . . . . . . o o x o x o o . . . . 10 x J
J x . . . . . . o x x x o o o . . . . . . 11 x
x . . . . . x x . . . x x o . . . . . . 12
. . . . x o . . x . . x o . . . . . . 13
. . . . x o x . . x o o . . . . . . . 14
. . . . o . . . o o x x . o x . . . . 15
. . . . . . . o . . . x . x . . . . . 16
. . . . . . . . . . . . . . . . . . . 17
. . . . . . . . . . . . . . . . . . . 18
. . . . . . . . . . . . . . . . . . . 19
abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ-=_
B oooBill--O Joao--X
------- --- -------
1. --- z10 z14 C13
2. w12 w15 y11 v14
3. x14 M9 v12 Q15
4. E11 J14 A11 F12
5. G11 J12 K7 I13
6. K13 C7 D10 L16
7. H8 D16 H12 I15
8. I11 O15 K15 P16
9. H14 E15 F14 H10
10 B10 G15 G9 C11
11 J8 L10 F6 F8
12 G5 G7 H6 K9
13 D8 w11 E7 x12
14 J6 F10 L8 E9
15 N6 N8 M5 M7
16 I9 L6 u13 O7
17 J10 Q7 I7 R6
18 resign
A more interesting variant is 2016's 3-CONNECTION also proposed by Taylor and Neto:
- The game consists of three different connection games (games and board size may vary)
- In each turn a player plays two moves in two different games; not the same two as in the previous turn, unless one game is finished by a formal connection.
- Overall winner is the player that wins two out of three.
In the following match, there is a Y board, a Gale (Bridg-It) board and a Hex board:
Y Gale Hex
abcdefghijk a b c d e abcdefghijklm
1 . . . . B 1
2 . . , , , , . . . . . 2
3 . . . x . . . . . . . 3
4 . o . . J , x ooooo , J . . x . . J 4
5 . x o . . x . . . . o . . 5
6 . . . . . . , , , , . . . . . 6
. . . B
Bilx Joao
1. --- Hg4 Gc4 Yf5
2. Yd5 Ga4 Hf5 Ye4(continuing...)
Y Gale Hex
abcdefghijk a b c d e abcdefghijklm
1 . x . . B 1
2 . . , x , ooooo . x . . . 2
3 . o . x xxxxx . o x . . 3
4 o o . x J , x ooooo x , J . . x . . J 4
5 . x o . . x o x x . . o x . 5
6 . . o x . . ooooo x , , . . . o x 6
. x . B
Bilx Joao
1. --- Hg4 Gc4 Yf5
2. Yd5 Ga4 Hf5 Ye4
3. Hh5 Ga2 Ga6 Yc4
4. Yg6 Gc6 Hg6 Gb5
5. Hi6 Gi4 Ye6 Hf3
6. Hh3 Yi4 Ge2 Yf3
7. Hg2 Gd3 resigns 1-2
Sente is very important in connection games, but player's must decide where they must let the opponent gain terrain. A constant tension that adds to the game's playability.
Another 2016 game by Taylor and Neto implements a race idea. There are two matches played at the same time, starting with asymmetric positions. Black is favored, and White starts. The first one to win one match, wins both! Here's n implementation using the connection theme, Quickwin Hex:
Here's a starting position favoring Black:
abcdefghijklmnopqrs abcdefghijklmnopqrs
1: . . . . . x x 1 . . . . . x x
2: . . . . . . . 2 . . . . . . .
3: . . . . . . . 3 . . . . . . .
4: . . . o . . . 4 . . . . . . .
5: . . . . . . . 5 . . . . . . .
6: . . . . . . . 6 . . . . . . .
7: x x . . . . . 7 x x . . . . .
abcdefghijklmnopqrs abcdefghijklmnopqrs____Bo___Jx_______________|___Jo___Bx_____________________
1. j4 l4 | j4 l4
2. l2 n2 | l2 h4
3. l6 o5 | i3 g3
4. n4 resigns | end of gameabcdefghijklmnopqrs abcdefghijklmnopqrs
1: . . . . . x x 1 . . . . . x x
2: . . . . . o x 2 . . . . . o .
3: . . . . . . . 3 . . X o . . .
4: . . . o x O . 4 . . x o x . .
5: . . . . . x . 5 . . . . . . .
6: . . . o . . . 6 . . . . . . .
7: x x . . . . . 7 x x . . . . .
abcdefghijklmnopqrs abcdefghijklmnopqrs
Neto and Taylor playtested this mutator with Y and Chess with nice results.