# Minority Breakthrough

Consider the following well-known motif:

``````[FEN "8/1k6/5pp1/7p/7P/6P1/8/7K b - - 0 1"]
``````

Black wins even with Ka8, but being true Patzer: 1...g5?? 2.g4! Kmoch's crosslever, 1-0.

Now in the critical position after 1...g5, pawns are hanging. Can you conceive a position with the following conditions:

• White, a pawn down, to move wins by a breakthrough
• Everything beyond a pawn-free file isn't counted in "pawn down"
• No free White pawns
• If a White pawn has passed a Black one, the latter isn't counted for "pawn down" either
• Black doubled pawns are OK
• No captures possible in the position
• Kings far away, play no role

(Note: After thinking ten seconds, I know the answer, but I still leave it as a puzzle for you. Feel free to think about how many pawns White can be down for a sufficient breakthrough.)

• Your diagram seems to be upside down.
– bof
Oct 5, 2022 at 2:15
• @bof: Yup, that seems to be the default when BTM. I'd wish the replayer had a little black or white dot or arrow to indicate the right to move and the orientation of the board. Oct 5, 2022 at 7:25

The g6-break comes to mind, as in

``````[fen "8/5ppp/8/5PPP/8/8/8/8 w - - 1 0"]

1.g6 hxg6 2.f6 gxf6 3.h6
``````

We can adapt this. Here, White wins starting with the break 1.g6

``````[fen "8/pppppppp/8/4PPPP/8/8/8/k1K5 w - - 0 1"]

1.g6 hxg6 2.f6 exf6 3.exf6 gxf6 4.h6
``````

Same idea in the center, now five pawns are needed, but we get a minority attack on either side.

``````[fen "8/pppppppp/8/2PPPPP1/8/8/8/k1K5 w - - 0 1"]

1.e6
``````

This breaks works on fifth row as well, given Black can neither check (wK on c3) nor escort his own pawn in due time (wK on c1).

``````[fen "8/8/pppppppp/8/2PPPPP1/1K6/8/k7 w - - 0 1"]

1.e5 dxe5 2. f5 exf5 3. gxf5 gxf5 4. d5 cxd5 5. cxd5 f4 6. Kc3
``````
• Amusingly, on an infinite board White can win with a constant 2/3 of Blacks pawns. (I doubt this value can be improved.) Oct 5, 2022 at 19:44
• @HaukeReddmann not sure what this means: as the second diagram shows, five connected pawns on the 5th rank beat arbitrarily many pawns on the 7th (assuming that Queening first by a few tempi is a winning advantage). Oct 6, 2022 at 13:10
• @NoamD.Elkies: Ah, I see. Any pawn two rows away from the last white obviously has no relevance in the fight. My 2/3 assumes these pawns don't count. Oct 6, 2022 at 19:03