CASCADE

Copyright (c) 2009 Torben Ægidius Mogensen

The game is played on the following hex board:

The game is played also with neutral stones (here, we use red):

DROP - drop a stone of either color (white, black or red) on an empty cell There's a possible chain-reaction after each drop:  - if an empty cell is directly below two cells with stones of the same colour, it receives a stone of that colour.  - if an empty cell is directly below two cells with stones of different colours, it receives a stone of the third colour GOAL - When the board is full, wins the player with more stones.

An example White played at b4, which produced a chain-reaction of more four stones (the marked ones).

Here's the complete match from the previous example:

abcdefghijklmno   __Black____________White_____
       .         1. x h13            o g6
      x .        2. x i6 =           x j7 o
     = o o       3. x l7 x=          o e6 ox==   
    x x o o      4. x n7 xxox        o o8 =oo=  
   . x = o =     5. x c6 =xxoxxo=    = d9 oo=         
  x o o x X .    6. = b9 =x==ox      o h3
 . = o = x x x   7. x g2 =x          o j3 o=
. . x x o x x o  8. x e4 x           o k4 o
 = = x = = x =   9. = l5 x           resigns
  = o o = o o   10. 
   x o x x o    11. 
    = = x =     12. 
     = x o      13. 
      o =       14. 
       x        15. 
abcdefghijklmno    

The game was proposed in a post at r.g.a. Here are some words from the author:

> Unusual! What was the inspiration for this one?

A puzzle I once saw:

Given a line of +s and -s, write a new line under this but staggered
half a position. The symbols in the new line (which will be one
shorter than that above) will be made by the multiplication rule:

+ x + = +
+ x - = -
- x + = -
- x - = +

This is continued until the last line has only one symbol. Examples:

- + - - + + - + - +
  - - + - - - - -
    + - - + + +
      - + + +
        - +

The puzzle is if there are arbitrarily large such constructed
triangles with an equal number of +s and -s (and provide a proof or
disproof of this).

The asymmetry of the multiplication rule makes it unsuitable for a
game, so I wanted a symmetric variant. Using three colours, this was
possible. My original game only allowed placing in the top line, but
I found the game more interesting in the general version. This also
allowed using a normal Checker board.

Another way of making it symmetric is to use three predecessors and
use a majority rule. This can be done in 3D by placing stones in a
triangular grid and then place new stones on top of three lower
stones, forming a triangular pyramid in the end. Obviously, placement
is only at the bottom, and counting requires breaking down the pyramid
(or counting unused stones). I tried this with marbles, but it proved
too mechanically unstable. You could use coloured disks instead.

However, I like the idea of using a third colour and that it may
sometimes be advantageous to place a stone of the opponent's colour.