Mating with Insufficient Material

Aug 15, 2009, 12:00 AM |
22 | Endgames

As all the members here surely know, any game that comes down to K+B vs K or K+N vs K is immediately declared "drawn due to insufficient material".   With just those three pieces left on the board, a mating position cannot be created even if both players cooperate.  However, while there are no winning chances when the player with insufficient material is facing a bare king, forced mates are possible when his opponent still has some pieces on the board.  For example, take a look at the following position:

Black has just a bishop left, but he has a forced mate in 6: 1...Bc8! 2. h4 Kf3 3. h5 Ke4 4. h6 Kd5 5. h7 Kc6 6. h8=Q Bb7#

Now, admittedly this is a pretty contrived position, considering the awkward placement of White's king and bishop.  With a bishop able to control the squares of one color only, the forced mates available to a bare bishop are quite limited.  A knight can cover all 64 squares, however, and it may surprise you just what a bare knight can achieve. These are long and relatively complex mates from positions that look like they can occur in an actual game.  Here is the first position for your perusal:

Black to play wins in seven moves.  The only winning first move is 1...Kc8! - not 1...Kc7?, which throws away the victory.  (Those of you who automatically played 1...Kc7 to get “the opposition,” that's just for pawn endings, silly.)  After 1...Kc7? 2. Ka8 Nb6+ 3. Ka7 Nc8+ 4. Ka8 Nb6+ 5. Ka7 (Nuts, all these checks obviously aren't doing anything, so let's try something else:) Nd5 6. Ka8 Kc8 7. Ka7 Kc7 8. Ka8.  Nope, no progress there either.  And, since 8...Kc6 9. Kb8 or even 9. a7 draws, there is no win.  

Black's problems in the above drawing line essentially stem from the knight's inability to “lose a move,” or at least that's what some books call this liability.  To give a different explanation, one that an engineer would appreciate: the knight's problem is that he changes color every time he moves, always toggling between a light square and a dark square.  Since the Black king is also limited to toggling between two squares, Black can never get the toggle right once he has committed himself with 1...Kc7?  Hence, that is why he must start with 1...Kc8!  

OK, let's see what happens next.  After 1...Kc8! 2. Ka8 Kc7, the first thing we notice is that 3. a7?? allows 3...Nb6# - already we are seeing a difference by getting the toggling right.  So, White must play 3. Ka7, and now the knight maneuvers to reach c8 while avoiding stalemate: 3...Nc3 (3...Nb2 also works, planning to reach c8 via the route b2-c4-e6-c8) 4. Ka8 Nd5 5. Ka7 Ne7 6. Ka8 Nc8.  This forces 7. a7, and Black finishes the game with 7...Nb6#

The above position illustrates a few other winning ideas with a bare knight.  Again, Black to play has a win in seven: 1...Nb5! (not 1...Nc6? 2. a6, and stalemate is reached after 2...Kc7 3. a7 or 2...Nd4 3. a7) 2. a6 Nc7+! (again, 2...Kc7 3. a7 draws) 3. Ka7 Nd5! 4. Ka8 Kc7, and the rest of this we saw in the previous example: 5. Ka7 Ne7 6. Ka8 Nc8 7. a7 Nb6#

Now, for anyone thinking the above two examples still don't seem like positions that can be reached with reasonable play, since in both cases the White king is trapped in front of his own pawn, well, actually they can be reached quite easily from more complicated positions.  For an example of this, check out the following position.  (I'll present this one as a puzzle for anyone who wants to try solving it:)

The win starts with 1...Kd7!  Since your opponent knows your B+N is enough to mate (you do know how to mate with B+N, right?), he figures his only chance to force a draw is to push his last pawn through with 2. a6.  And, hmm, at first glance it may appear he has pulled this off.   But not so - your winning move is 2...Ba7!!, giving up one of the pieces you need for the B+N mate but reaching a position where the bare knight is enough to win.  After 3. Kxa7 Kc8! 4. Ka8 Kc7, and we are back at a position from  example 1.  Even ignoring the bishop is futile, as 3. Ka8 Kc7 4. Kxa7 Ne7, and again we are back in example 1.

So far we have seen two positions where a bare knight is able to force checkmate in seven moves.  Can we do better than that, i.e. find positions where even more moves are required for a bare knight to mate?  Yes, we certainly can.  In fact, playing around with an Nalimov endgame database, I've found that adding a second pawn to the position can produce much longer forced wins.  The longest wins I have been able to find so far are 17-movers.  Here is one example:

Black has only one move to win: 1...Kc8!  If he starts with 1...Kc7?, White draws by rushing his c-pawn forward before the knight can get into position: 1...Kc7? 2. c4! Ne3 3. c5! Nc4 4. c6! Kc8 5. Ka7! Kc7 6. Ka8 and draws - note we have another one of those toggling dilemmas where the knight can't stop checking.   The toggling is correct after 1...Kc8!, when the pawn blitz leads to 2. c4 Ne3 3. c5 Nc4 4.c6 Kc7 5. Ka7 Nd6 6. Ka8 Nc8 7. a7 Nb6#.

White can delay the mate with 2. Ka7.  At this point, Black has several knight maneuvers to win, although the endgame database shows the mate cannot be done in less than 17 moves if White drags it out as long as possible.  To play out the position, I'll give one possible continuation which also includes a final drawing trap: 2. Ka7 Kc7 3. Ka8 Ne3 4. Ka7 Nc4 5. Ka8 Nb6+ 6. Ka7 Nd7 7. c3 Nb6 8. c4 Nd7 9. Ka8 Kb6 10. c5+ Kc6! (not 10...Kc7? 11. c6 - the toggling issue strikes again) 11. Ka7 Kc7 12. Ka8 Ne5 13. Ka7 Nc6+ 14. Ka8 Kc8 15. a7 Nd8 16.c6 Ne6 17. c7 Nxc7#

So, how ‘bout that... a 17-move win with a king and a bare knight!  If anyone can beat that, be sure to post the position in the comments below.  I for one would love to see it.

I suppose there is just one problem I should mention before I close.  In Live Chess here on, White can still obtain a draw in any of the positions just presented by simply refusing to move - when his clock runs out, the game will be called a draw due to insufficient material!  So just remember, all the positions I just claimed as wins are actually draws on this and any other chess-playing site that will not give a win on time to an opponent with "insufficient material" (heh heh). 

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