# Math and Logic of Infinite Chess

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 $1/Q + 1/R + 1/B + 1/N$?

• I think you should state in the question that this is the IBM December 2011 'Ponder This' 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. – Chris Taylor Dec 20 '11 at 11:32
• I posted an answer (edited with correction by Sp3000), but decided to delete it in light of the contest. What a fun problem! – JDH Dec 20 '11 at 14:06
• Is it really appropiate to post this kind of question (contests/challenges) here? – leonbloy Dec 23 '11 at 4:24
• A link about paper co-authored by JDH that is out today in arXiv titled mate-in-n of infinite chess is decidiable may illuminate? – Sniper Clown May 18 '12 at 0:41
• 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 domino.research.ibm.com/Comm/wwwr_ponder.nsf/solutions/… – sdcvvc May 18 '12 at 0:46

Conclusion. So we have $Q=2$, $R=3$, $B=6$ and $N=\infty$, giving $$\frac{1}{Q}+\frac1R+\frac1B+\frac1N=\frac12+\frac13+\frac16+\frac1{\infty}=1.$$