5
$\begingroup$

Two players (White and Black) are playing on an infinite chess board (extending infinitely in all directions).

First, White places a certain number of queens (and no other pieces) on the board.

Second, Black places a king on any unoccupied, unattacked square of the board.

Then, both players take turns moving until Black is checkmated.

What is the minimum number of queens White needs to force a checkmate?

Answer the same problem if White starts with rooks instead of queens.

Do the same for bishops and knights.

Let $Q, R, B$, and $N$ be the minimum number of queens, rooks, bishops, and knights, respectively. What is the sum $1/Q + 1/R + 1/B + 1/N$?

  • 26
    I think you should state in the question that this is the [IBM December 2011 'Ponder This'](http://domino.research.ibm.com/Comm/wwwr_ponder.nsf/challenges/December2011.html) problem. You should also reconsider the 'very intresting' tag in the title, since interestingness is rather subjective. Finally, I don't think that any of the tags 'logic', 'probability-theory' or 'probability-distributions' are appropriate for this question.2011-12-20
  • 8
    I posted an answer (edited with correction by Sp3000), but decided to delete it in light of the contest. What a fun problem!2011-12-20
  • 2
    Is it really appropiate to post this kind of question (contests/challenges) here?2011-12-23
  • 1
    A link about paper co-authored by JDH that is out today in arXiv titled [mate-in-n of infinite chess is decidiable](http://arxiv.org/pdf/1201.5597v4.pdf) may illuminate?2012-05-18
  • 2
    Mahmud: The paper is about a general decision problem, you could (almost) use it to solve automatically this problem. The solution is now available at http://domino.research.ibm.com/Comm/wwwr_ponder.nsf/solutions/December2011.html2012-05-18

1 Answers 1