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Ideal Values and Practical Values (part 6). A study of the value of the Furlrurlbakking piece.[All Comments] [Add Comment or Rating]Well, I'm not a mathematician either, but I have an interest in combinatorial math, so I decided (on a lark) to figure the number of unidirectional plus nine rooks. (I've named the whole class 'unirooks'.)
The simplest value to figure is that without allowing for bishop moves. The peices are allowed nine moves out of a total of twenty-eight. However, one of the choices for wazir moves is rendered moot by the choive of rook direction, so the number of choices is actually only twenty-seven.
The formula for k selections out of a set of size n is:
C(n,k) = n!/(k!(n-k)!)
Given nine selections out of twenty-seven gives 4,686,825. However, there are also four choies for rook direction, so the total is 18,747,300.
The situation with the bishop moves allowed is more complex, mainly because they count as two selections. While there may be a generic formuala for this sort of selection, I don't know it, so instead, I figured it in an ad hoc manner.
A 'bishopy unirook' can have one, two, three, or four bishop directions. Like the rook move mooting a wazir move, each selected bishop move moots a ferz move. For each number of bishop moves, one can calculate the number of variants by multipling the four rook moves, the combinations of that number of bishop moves, and the combinations of the remaining (non-moot) small moves.
Skipping all the intermediate math, the total is 11,832,396. Adding this to the previous number gives a total of 30,579,696.
The final wrench in the gears of this idea is the idea that such rooks could have capture moves that differ from thier non-capturing moves. This is provided by simple multiplication. (If multipling numbers this large can be considered 'simple'. I used a computer.) One combination can be selected as the normal moves and another as the capturing moves, so the result is the square of the number of simple combinations, which is 935,117,807,452,416.
This, however, assumes that the capturing rook direction is allowed to vary from the non-capturing one. As the article isn't explicit on this point and it may affect the balance, one may wish to disallow this. In that case the total is a quarter of the above number, which is 233,779,451,863,104.
In conclusion we can state two things. Firstly, I have far too much spare time on my hands. Secondly, there are not thousands of possible unirooks, or even millions, but quadrillions!
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