# Rook and bishop vs. opposite color bishop

I drew a game with rook and bishop vs an opposite color bishop due to the 50-move rule.

[FEN "4b3/k1B5/8/2K5/8/8/R7/8 w - - 0 1"]

Is this a theoretical draw? If not, what is the technique to win?

Definitely not a theoretical draw, this should be easily winning for White. The strategy to win would be:

• Advance with your king and rook to push back Black's king. If Black's light-squared bishop blocks your rook's check, then this means the Black king is on a dark square. Thus, you can use your dark-squared bishop to check the king and push it back.
• Eventually you'll have pushed Black's king to either a light-squared or dark-squared corner.

• If it's in a light-squared corner like a8, the light-squared bishop isn't able to act as a blockader from your rook. In the diagram you posted, here's one example of how to win: 1...Kb7 2.Bb6 Bh5 3.Ra7+ Kb8 4.Kd6 Bf3 5.Rc7 Bg2 6.Ba7+ Ka8 7.Kc5 Bf3 8.Kb6. Now all you need to do is maneuver your rook to the 8th rank to checkmate, which can be accomplished in as little as 2 moves.

• If it's in a dark-squared corner like h8: Put your king on h6 and move your rook to the 8th rank, checking the Black king. If Black manages to block the check with ...Bg8, then you can checkmate immediately by moving your bishop onto the a1-h8 diagonal.

I don't know about the technique beyond trying to keep all your pieces on the dark squares and the enemy king cornered, but the position above is not a draw. According to the endgame tablebase it's mate in 10 moves. Here's one line:

[FEN "4b3/k1B5/8/2K5/8/8/R7/8 b - - 0 1"]

1... Kb7 2. Ba5 Bg6 3. Rb2+ Ka7 4. Rb6 Bc2 5. Kd6
Bd1 6. Kc7 Bc2 7. Bc3 Be4 8. Bd4 Bb7 9. Rd6+
Ka8 10. Rd8+ Bc8 11. Rxc8#

If you want statistics more generally, see http://www.maths.usyd.edu.au/u/thiel/misc/endgames/ . According to the table, for KRB vs KB, with white to move, 98.2% of positions are won by white and 1.8% are drawn; with black to move, 31.1% are drawn. Note that this table doesn't make the distinction between same-colored and opposite-colored bishops. The draws are probably positions where white loses the rook in the next move or two.