[Retros] Happy New Year!

[Joost de Heer]

There are probably easier solutions, but here are mine:


> >  +-----------------+

> >  | . n b q . r k _ |   Noam D. Elkies, 12/2005

> >  | _ p p p n p p p |

> >  | . _ . _ r _ . _ |

> >  | p . b . p . _ . |

> >  | . _ . _ . P . _ |

> >  | _ . _ P _ N _ . |

> >  | P P P B P _ P P |   16+16

> >  | R N K Q . B R . |   SPG-7.5: How many solutions?

> >  |_________________|


There are two groups of white moves: d3, B out (e3-h6), Kd2-c1, Bd2 and f4, Sf3, Rg1.
When the bishop goes to f4, g5 or h6 the move order is exactly defined, and when the bishop goes to e3, there are binomial(8,5) move-orders
There are two groups of black moves: a5, Ra6, Re6 and e5, Bc5, Se7, OO. Re6 must be played after e5, so e5 is the 1st, 2nd or 3rd move.
When e5 is the first move, there are binomial(6,3) move-orders. When e5 is the second, there are binomial(5,3) move-orders (a5 must be the first), and when e5 is the third, there are binomial(4,3) move-orders (a5/Ra6 must be the first two)

So in total there are (3+binomial(8,5))*(binomial(6,3)+binomial(5,3)+binomial(4,3))=2006 solutions.


> >  +-----------------+

> >  | _ . _ . _ . _ . |   Noam D. Elkies, 12/2005

> >  | . _ . _ . _ . _ |

> >  | _ . _ . _ . _ . |

> >  | . K . _ . _ . _ |

> >  | _ . P . _ R _ P |

> >  | . _ . _ . P . _ |

> >  | P P Q P P . P . |   16+0

> >  | R _ B N . B N _ |   OSPG-14: How many solutions?

> >  |_________________|


There are 2 groups of moves: c4/Sc3/Qc2/Sd1 (with the 2nd and 3rd move exchangable), and h4/Rh3/Rf3/Rf4/f3/K-b5.
The white king can go via d3/c4 to b5, then c4 must be played after Kb5, so there are 2 solutions in this case.
The white king can go via d4/c4 to b5, then c4 must be played after Kb5, so there are 2 solutions in this case.
The white king can go via d4/c5 to b5, then there are 2*binomial(14,10) solutions in this case.
So in total there are 4+2*binomial(14,10)=2006 solutions.

Joost

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