Rule 9

Rule 9 of Coalition Chess stipulates that the Yellow player may place his pieces in a position he deems advantageous, within his setup zone. We show that this allows over 12 billion combinations.

Denote P_x the number of possible places that the piece "x" can take in the initial configuration of the Yellow army. We assign squares to the pieces, removing possible square to place other pieces. We do not include in the count the permutations between pieces of the same nature. Then for 16 possible squares at the beginning we have
P_king = 16; P_artillery = 15; P_engineer = 14; P_moto = 13; P_tanks = (¹²₂) = (12!/(10!2!));
P_{machine guns} = (¹⁰₃) = (10!/(7!3!)); P_guards = (⁷₃) = (7!/(4!3!))

And the total number of combinations is
P_king * P_artillery * P_engineer * P_moto * P_tanks * P_{machine guns} * P_guards
= 16 * 15 * 14 * 13 * (¹²₂) * (¹⁰₃) * (⁷₃)
= 16 * 15 * 14 * 13 * (12!/(10!2!)) * (10!/(7!3!)) * (7!/(4!3!))
= 16!/(4!3!3!2!)
= 12,108,096,000

This means that before any player makes a single move, there are 12,108,096,000 different ways in which the Yellow player can configure his army.

Rule 12

Similar to Rule 9 of Coalition Chess, Rule 12 stipulates that a player, other than Yellow, has the right to position his pieces in his field as he deems advantageous, within his setup zone. Rule 9 provides the possibility of 12,108,096,000 combinations for the Yellow player. We quickly show that this, multiplied with the possibilities of Rule 12 allows for a possibility of over 698 x 10²⁴ combinations.

We use the same analysis as for the analysis of Rule 9. Since Black has one more Machine-gun and one fewer Guard, we find a total number of combinations
= 16 * 15 * 14 * 13 * (12!/(10!2!)) * (10!/(6!4!)) * (6!/(4!2!))
= 16!/(4!4!2!2!)
= 9,081,072,000

Both Green and Red have 3 varieties of 2 pieces to place in 8 squares. Hence, we find a total number of combinations
= (8!/(6!2!)) * (6!/(4!2!)) * (4!/(2!2!))
= 8!/(2!2!2!2!)
= 2,520

Because of Rules 9 and 12, a minimum of at least 12,108,096,000 x 9,081,072,000 x 2,520 x 2,520
= 698,255,003,195,714,764,800,000,000 different games can be played from the first round of turns.

Conclusions about complexity

To scale the complexity of the game, a 100 years is equal to 3,155,760,000 seconds. To see all possible Coalition Chess starting combinations within a 100 years, the pieces should be reconfigured over 221x10¹⁵ times per second.

Moreover, not unlike chess and checkers, the number of games which can be played within these starting configurations is exponential in the time of game play. That is, for every turn played, there is a new number of possibilities equal to the moveable pieces, say X. After t turns, something like X^t combinations can be played.

Hence, however huge the number of possible starting configurations, te total number of games which can be played is much larger. This brings a few important question:

1. Will it be possible, within the next 100 years, to have all Schoenberg Coalition Chess games played?
2. Would it be possible to, at the least, simulate these games on computer?
3. Would that be too much of a drain on (today's) planetary computation ressources?
4. Will IBM be able to build a computer which beats humans at this game?

Copyright 2007, Michel Paquette. License for "Schoenberg Coalition Chess" delivered by BELMONT MUSIC PUBLISHERS.