Turn Your Othello Into Binaryon[8]

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Augustine Careeno

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Apr 14, 1997, 3:00:00 AM4/14/97
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For all of you who would like to enable your Othello Game's teeth!
Here's the ticket....

Binaryon[8] --THE BINARY NUMBER GAME

(C)1997 Augustine R. Carreno

{NOTE: Please set your reader's font to FIXED PITCH.}


In its original form this quick-moving game uses a board with LE
switches and theoretically can be played on a grid of any size n with
nxn pieces. We introduce here the "default" version, i.e., one with
board size =8. So, if you own an Othello or Reversi you are already
equipped too! Once you are familiar with it, you'll discover that
Binaryon[8] is more than a game, it's a game generating game.


USING AN OTHELLO

To play Binaryon[8] with an Othello board, simply place all the
pieces on the board as shown in Fig. 1. Then the two players (one
called the ON player and the other the OFF player) take turns
flipping over the pieces vying to make the largest 8-digit binary
number.

The white side is "1" for ON, but "0" for OFF. Conversely, the black
side is "1" for OFF, but "0" for ON. This means that in the starting
position one half of the pieces are 1's and the other half are 0's
from each player's perspective.


+---+---+---+---+---+---+---+---+
| x | x | x | x | x | x | x | x | 8
+---+---+---+---+---+---+---+---+
| x | x | x | x | x | x | x | x | 7
+---+---+---+---+---+---+---+---+
| x | x | x | x | x | x | x | x | 6
+---+---+---+---+---+---+---+---+
| x | x | x | x | x | x | x | x | 5
+---+---+---+---+---+---+---+---+
| o | o | o | o | o | o | o | o | 4
+---+---+---+---+---+---+---+---+
| o | o | o | o | o | o | o | o | 3
+---+---+---+---+---+---+---+---+
| o | o | o | o | o | o | o | o | 2
+---+---+---+---+---+---+---+---+
| o | o | o | o | o | o | o | o | 1
+---+---+---+---+---+---+---+---+

a b c d e f g h

Fig. 1 Starting Position



MOVING

Players move by turning pieces: ON turns the black pieces and OFF
turns the white ones. Play starts with one player turning one of the
opponent’s pieces.
From there on, you may turn only a piece meeting the following two
conditions. One, it is in a straight line, horizontally, vertically
or diagonally with the last piece turned (a line "reflected" at equal
angles(*) off the board edges is considered straight too.)
Two, it is connected to more pieces in its own state than to pieces
in the other.
For instance, an "on" in a midboard position is surrounded by eight
switches. Of these, at least five must be "on" as well, before that
one can be turned to "off". If the "on" is in a corner, you need two
more "on's".

WINNING

A player with no moves must pass; and the game ends when neither
player has a move.
The winner is whoever has the largest 8-digit binary number, which
must be in the form of a string of adjacent pieces in a straight line
formation --or, as before, a line "reflected" at equal angles(*) off
the board edges. In a tie, identical numbers cancel each other and
the largest of the remaining numbers wins.


(*)Except straight angles


EXAMPLE OF A FINAL POSITION

+---+---+---+---+---+---+---+---+
| o | x | o | x | x | o | x | o | 8
+---+---+---+---+---+---+---+---+
| x | o | x | o | o | x | o | x | 7
+---+---+---+---+---+---+---+---+
| o | x | o | o | x | x | o | x | 6
+---+---+---+---+---+---+---+---+
| x | x | o | x | o | o | x | o | 5
+---+---+---+---+---+---+---+---+
| o | x | o | x | x | o | x | o | 4
+---+---+---+---+---+---+---+---+
| x | o | x | o | o | x | o | x | 3
+---+---+---+---+---+---+---+---+
| o | x | o | x | x | o | o | x | 2
+---+---+---+---+---+---+---+---+
| o | x | o | x | o | x | x | o | 1
+---+---+---+---+---+---+---+---+

a b c d e f g h


Fig. 2. In this example, OFF wins with the sequence, d8-a5-e1,
yielding 11111110. (Notice the "reflection" at a5.) The most ON can
come up with is c5-f8-h6-f4, or 11111001.


Q&A

Q. How many numbers that meet the game's conditions can be built on
the board?
A. 204 per side.

Q. What is the maximum number of 8-digit numbers meeting the game's
conditions a piece can belong to?
A. 36

Q. If the piece is in a corner, how many numbers can it belong to?
A. 6

Q. "Theorem": Is the number 11111111 obtainable?

BINARYON AS A GAME GENERATOR

As we know, anything related to numbers has a tendency to having more
than one track, and Binaryon[8] is no exception. To inquisitive
players this game offers creative possibilities by letting them add
refinements of their own within the framework of the rules.

For example, to increase the game's complexity players may add
supplementary conditions. Like, say, that the string of digits be
horizontal or vertical or of any other predetermined characteristic;
or that it contain a particular square. Or some other sorting
criterion.

Further complications can also be achieved by using a poker approach:
you bet that you can identify a certain group of numbers that are
identical, consecutive, or linked in another way and contained in
lines crossing at a common square, which you reveal if your bet is
called. In fact, as they say, the possibilities are limitless. :-)

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