December 2011
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Queens:
One queen cannot checkmate, even if the Black king is cooperating, since it cannot threaten the king's current location and the eight surrounding squares simultaneously. Two queens can force a checkmate by first closing in on the king in. If, for example, the two queens are on a1 and c1 and the king is on, say, b9, the following moves force a checkmate:
1. Qa7 Kb10
2. Qc9 Kb11
3. Qa11 mate
So Q = 2.
Rooks:
Again, two rooks can not checkmate even a cooperating king; but two rooks can close the king into a single column, and a third rook can then checkmate it.
So R = 3.
Bishops:
There are two kinds of bishops: light-squared and dark-squared. If there are five bishops, then in one of the colors there are no more than two bishops. The king can then stay only on that color and keep avoiding checkmate; up to a single situation: B_K_B, where the black king is either in stalemate, or can move to the other color and immediately back (and thanks for Leon Smith for this comment).
On the other hand, six bishops can easily close on the king.
So B = 6.
Knights:
Since the knights move quite slowly, even three knights can not follow the king and three knights cannot checkmate.
So N = infinity.
1/Q + 1/R + 1/B + 1/N = 1/2 + 1/3 + 1/6 + 1/inf = 1
No finite number of knights can checkmate the king. However, as some of you noticed, infinitely many knights (even with density zero, if you ignore the 50-moves rule) can be arranged to ensure checkmate.
For example put a knight in any place where either the X coordinate or the Y coordinate is a perfect square.
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