Conference rusure::math

1548.0. "Combinatorial Chess problem" by CIVAGE::LYNN (Lynn Yarbrough @WNP DTN 427-5663) Mon Jan 27 1992 11:28

On the next page is a Chess problem. Non-Chess-players will be forgiven 
for ignoring the rest of this note, but the problem really is interesting 
mathematically, so if you know how the pieces move you may want to read on.

	  a   b   c   d   e   f   g   h
	+---+---+---+---+---+---+---+---+
	|   |///|   |///|   |///|   |///|8	White to play and mate in 2
	+---+---+---+---+---+---+---+---+
	|///|WK |Bp |   |///|   |Bp |   |7	Composed by A. R. Gooderson
	+---+---+---+---+---+---+---+---+
	|   |///|   |///|   |///|   |///|6	*British Chess Magazine*, 1948
	+---+---+---+---+---+---+---+---+
	|///|Bp |WR |   |Bq |   |///|   |5
	+---+---+---+---+---+---+---+---+
	|   |///|   |///|Bk |///|   |///|4	(Bk = Black King, 
	+---+---+---+---+---+---+---+---+	WN = White Knight, etc)
	|///|   |///|   |///|WR |///|WN |3
	+---+---+---+---+---+---+---+---+
	|   |Bp |   |///|   |Bp |Br |WN |2
	+---+---+---+---+---+---+---+---+
	|///|   |Bn |WQ |///|   |///|   |1
	+---+---+---+---+---+---+---+---+
	  a   b   c   d   e   f   g   h

Let me first comment in explaining this problem that for White to have 
multiple checkmates at his disposal is usually regarded as an artistic
defect, a constructive weakness in the problem. The composer of this
problem has turned multiplicity into a constructive asset. 

White's first, *key* move is 1.N[h2]-g4. This threatens RxQ mate. At the 
same time there are four other, secondary, threats, which are defended by 
the presence of the BQ on e5: Q-d5, R-f4, N-g5 are defeated by the BQ 
capturing the checking piece, and if the BQ were to move away allowing the 
WQR to guard the f5 square, R-e3 is a threat as well.

The Black Q has 2**4 = 16 possible actions in response to the threat,
including ignoring the threat. The problem has been constructed so that
each of the possible moves of the Black Q permits a different combination
of the secondary threats to come into play! [Four is also the maximum
number of threats to be involved in this way, since 2**5 = 32 is larger
than the maximum number of options that any piece can have available in one
move.] 

I won't take the space to itemize - you may find it worth checking out the
16 different actions to see how various combinations of threats are defeated.