Can one mate a lone king with Bishop and two knights without being aided by the attacking king?

Question

In order not to overload the answer to this question, I split off the bishop+two knights part. So the question is: Is it possible to mate with a bishop and two knights without the help of the king?

Answer 1

Work in progress

You can convince yourself easily, that mate is only possible in a corner of the board. We distinguish four cases:

• The corner is the same color as the bishop and the bishop mates
• The corner is the same color as the bishop and a knight mates
• The corner is the opposite color as the bishop and the bishop mates
• The corner is the opposite color as the bishop and a knight mates

For the following analysis we only consider unique positions that are not related by symmetry. Also we don't distinguish between similar positions of a bishop along a diagonal, if the same essential square(s) is covered.

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Starting with the first case, bishop same color as the corner and the bishop mates. There are 5 distinct (up to symmetry) mating positions: 1.1a to 1.1e.

Doing a retro-analysis for 1.1a and 1.1b:

• White's last move was Bb7+ and the bishop came from c8 or a6.
• Black's last move was Ka8 and the king came from b8 or a7. The king was not forced to go to the corner and could have escaped to c7 or b6 instead.

Doing a retro-analysis for 1.1c, 1.1d and 1.1e:

• White's last move was Be4+ and the bishop came from somewhere along the b1-h7 diagonal.
• Black's last move was Ka8. If the king came from b8 or a7, it could have gone to b7 instead of the corner. If the king came from b7, it could have gone to (among others) c6 instead of the corner.

So in conclusion, it is not possible to force a mate with the bishop in the corner of the bishop's color.

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The second case, bishop same color as the corner and the knight mates. There are 2 distinct (up to symmetry) mating positions: 1.2a and 1.2b.

• White's last move was Nb6+ and the knight came from a4, c4, d5, d7 or c8.
• Black's last move was Ka8. If the king came from b7, it could have captured Kxc6 instead (note that the c6 knight was not protected then).

If the king came from a7 or b8, there is actually a longer series of forced moves leading to mate! Needs further invetstigation.

2. Corner opposite to the bishop's color

2.1 Bishop mates

You easily see that there is not even a theoretical mating position for this case.

2.2 Knight mates

You easily see that there is not even a theoretical mating position for this case. In order to mate, one knight would have to check the king in the corner, say a8, the other knight would have to cover the light square b7 and the bishop would have to cover both a7 and b8 which it cannot.