Who would win in a fight between a lion and a bear? (endgame fortresses)

My young colleague Leif enjoys endgames, and has an endless curiosity about unbalanced endgames. So he asks:
can you win with three pawns against a Bishop?
how about a Knight?
what about two Knights?

A pattern we kept running into was:
united pawns on their start squares don't win against a piece (a)
united and *advanced* pawns win against a piece (b)
but
there are some special cases where the side with the piece can set up a blockade or fortressagainst advanced pawns (c)

[Event "Fortress (a)"]
[Site "?"]
[Date "2015.11.07"]
[Round "?"]
[White "KPPP"]
[Black "KB"]
[Result "1/2-1/2"]
[SetUp "1"]
[FEN "2b1k3/8/8/8/8/8/2PPP3/4K3 w - - 0 1"]
[PlyCount "0"]

1/2-1/2

[Event "Fortress (b) (not) "]
[Site "?"]
[Date "2015.11.07"]
[Round "?"]
[White "KPPP"]
[Black "KB"]
[Result "1-0"]
[SetUp "1"]
[FEN "2b1k3/8/5K2/2PPP3/8/8/8/8 w - - 0 1"]
[PlyCount "0"]
[EventDate "2015.??.??"]

1-0

[Event "Fortress (c)"]
[Site "?"]
[Date "2015.11.07"]
[Round "?"]
[White "KPPP"]
[Black "KB"]
[Result "1/2-1/2"]
[SetUp "1"]
[FEN "8/8/2bPk3/2P1P3/4K3/8/8/8 w - - 0 1"]
[PlyCount "0"]

1/2-1/2

[Event "Fortresses"]
[Site "?"]
[Date "????.??.??"]
[Round "?"]
[White "N"]
[Black "P"]
[Result "1/2-1/2"]
[SetUp "1"]
[FEN "8/8/8/1p6/pNp5/8/2K1k3/8 w - - 0 1"]
[PlyCount "0"]

1/2-1/2

So it's less like that famous Internet question
who would win in a fight between a lion and a bear?
but
who would win in a fight between a fox and a hedgehog?

Anyhow, we ended up looking at a few fortresses, some I knew, some I had
to look up, and some I found out more about.

There are cases where a lone King can be a 'toad in a stone', wedged in and impossible to shift.

[Event "Fortresses"]
[Site "?"]
[Date "????.??.??"]
[Round "?"]
[White "K"]
[Black "NP"]
[Result "1/2-1/2"]
[SetUp "1"]
[FEN "8/6k1/8/8/6n1/8/6Kp/8 w - - 0 1"]
[PlyCount "0"]

1/2-1/2

[Event "Fortresses"]
[Site "?"]
[Date "????.??.??"]
[Round "?"]
[White "BP"]
[Black "K"]
[Result "1/2-1/2"]
[SetUp "1"]
[FEN "7k/8/7P/7K/8/8/2B5/8 w - - 0 1"]
[PlyCount "0"]

1/2-1/2

The Japanese might say a 'bear in the hole'... Notice that these cases
are all in the corner of the board. Edges and corners are friends of the defending side.

With a pawn to help, the King can resist near a corner without being in
it:

[Event "Fortresses"]
[Site "?"]
[Date "????.??.??"]
[Round "?"]
[White "BP"]
[Black "KP"]
[Result "1/2-1/2"]
[SetUp "1"]
[FEN "2k5/1p6/1P2K3/8/5B2/8/8/8 w - - 0 1"]
[PlyCount "0"]

1/2-1/2

It's perfectly possible to draw when you are the exchange behind, if you
can keep out the enemy King. That's another theme: keeping out the
attacking King.

[Event "Fortresses"]
[Site "?"]
[Date "????.??.??"]
[Round "?"]
[White "RP"]
[Black "BP"]
[Result "1/2-1/2"]
[SetUp "1"]
[FEN "8/1k6/1pb5/p3K3/8/6R1/PP6/8 w - - 0 1"]
[PlyCount "0"]

1/2-1/2

[Event "Fortresses"]
[Site "?"]
[Date "????.??.??"]
[Round "?"]
[White "RP"]
[Black "BP"]
[Result "1/2-1/2"]
[SetUp "1"]
[FEN "8/1k6/p5R1/1p2K3/8/4b3/PP6/8 w - - 0 1"]
[PlyCount "0"]

1/2-1/2

[Event "Fortresses"]
[Site "?"]
[Date "????.??.??"]
[Round "?"]
[White "RP"]
[Black "B"]
[Result "1/2-1/2"]
[SetUp "1"]
[FEN "8/8/2kP2R1/4K3/1b6/8/8/8 w - - 0 1"]
[PlyCount "0"]

1/2-1/2

[Event "Fortresses"]
[Site "?"]
[Date "????.??.??"]
[Round "?"]
[White "RP"]
[Black "B"]
[Result "1/2-1/2"]
[SetUp "1"]
[FEN "8/8/2kP2R1/4K3/1b6/8/8/8 w - - 0 1"]
[PlyCount "0"]

1/2-1/2

The mighty Queen can be kept at bay, as long as the attacking King can't
get in:

[Event "Fortresses"]
[Site "?"]
[Date "????.??.??"]
[Round "?"]
[White "Q"]
[Black "NB"]
[Result "1/2-1/2"]
[FEN "k7/1b6/8/3n4/2KQ4/8/8/8 w - - 0 1"]
[Setup "1"]

1/2-1/2

[Event "Fortresses"]
[Site "?"]
[Date "????.??.??"]
[Round "?"]
[White "Q"]
[Black "RP"]
[Result "1/2-1/2"]
[SetUp "1"]
[FEN "4k3/4p3/3r4/5Q2/2K5/8/8/8 w - - 0 1"]
[PlyCount "0"]

1/2-1/2

We played around with variations on that last fortress, and then looked
it up. It's sort of interesting: if the pawn is advanced, the Queen can
get behind your defences and you lose, but if the pawn is advanced and
on the edge, you are able to hold again.

[Event "Fortresses"]
[Site "?"]
[Date "????.??.??"]
[Round "?"]
[White "Q"]
[Black "RP"]
[Result "1/2-1/2"]
[FEN "4k3/4p3/3r4/5Q2/2K5/8/8/8 w - - 0 1"]
[Setup "1"]

1/2-1/2

[Event "Fortresses"]
[Site "nearly"]
[Date "????.??.??"]
[Round "?"]
[White "Q"]
[Black "RP"]
[Result "1-0"]
[FEN "8/4k3/4p3/3r4/2K2Q2/8/8/8 w - - 0 1"]
[Setup "1"]

1-0

[Event "Fortresses"]
[Site "nearly"]
[Date "????.??.??"]
[Round "?"]
[White "Q"]
[Black "RP"]
[Result "1-0"]
[FEN "6k1/7p/6r1/2K2Q2/8/8/8/8 w - - 0 1"]
[Setup "1"]

1-0

[Event "Fortresses"]
[Site "?"]
[Date "????.??.??"]
[Round "?"]
[White "Q"]
[Black "RP"]
[Result "1/2-1/2"]
[FEN "8/7k/6p1/5r2/2K1Q3/8/8/8 w - - 0 1"]
[Setup "1"]

1/2-1/2

[Event "Fortresses"]
[Site "?"]
[Date "????.??.??"]
[Round "?"]
[White "Q"]
[Black "RP"]
[Result "1/2-1/2"]
[FEN "8/7k/5Q1p/6r1/4K3/8/8/8 w - - 0 1"]
[Setup "1"]

1/2-1/2

To break down the defence, you have to use zugzwang to make the Rook
abandon the King.

The same ideas occur in the 'simple' Q vs R endgame, which six-times US
champion GM Walter Browne failed to win against a computer in the 1970s,
before looking up the right technique... Try it against your own
computer!

[Event "Fortresses"]
[Site "nearly"]
[Date "????.??.??"]
[Round "?"]
[White "Q"]
[Black "R"]
[Result "1-0"]
[FEN "7k/6r1/8/4QK2/8/8/8/8 w - - 1 1"]
[Setup "1"]

1. Qe8+ Kh7 2. Kf6 Ra7 ( 2... Kh6 3. Qf8 ) ( 2... Rg1 3. Qe4+ Kh8 4. Qa8+ Kh7 5. Qa7+ ) 3. Qe4+ Kg8 4. Qd5+ Kh8 5. Qh1+ Kg8 6. Qg1+ Kh7 7. Qxa7+ 1-0

Fortresses in practice

Here's one from a game I had a year or two ago: I got through, but I
think I shouldn't have been allowed to.

[Event "East Devon Open"]
[Site "Exeter"]
[Date "2013.03.02"]
[Round "3"]
[White "Regis, David"]
[Black "Bartlett, Simon"]
[Result "1-0"]
[ECO "A00"]
[Annotator "Regis,Dr.Dave"]
[SetUp "1"]
[FEN "8/p5k1/1p6/1Pp1p1Pb/2P1B3/P2P2K1/8/8 w - - 0 46"]
[PlyCount "9"]
[EventDate "2013.03.??"]

46. Bf5 {Black had a long think here, and allowed me to draw a number of
unhappy conclusions. White would like to play the King to e4, but how?
Crawling via f2 allows ...Bg6!} Kf8 $4 {Removing the threat of Bg6} (46... Kf7
$1 {is tough! I couldn't find a way in then or now...}) (46... Bd1 $4 47. Kf2
$1 Bh5 48. Ke3 Bg6 49. Ke4) 47. Kf2 Ke7 48. Ke3 Bf7 49. Ke4 Kd6 50. g6 $18 (50.
g6 Be6 (50... Bg8 51. Bc8 Kc7 52. Bh3 Kd6 53. Bg4 $22 {is Zugzwang}) 51. a4 Bg8
52. Bc8 Be6 (52... Kc7 53. Bh3 Kd6 54. Bg4 $22) 53. g7 $1 Bg8 54. Bb7 Bh7+ 55.
Ke3 Ke7 56. Bd5 $18) 1-0

A couple of well-known ones at GM level: a great swindle by Zsuzsa
Polgar:

[Event "Biel"]
[Site "Biel"]
[Date "1987.??.??"]
[Round "?"]
[White "Chandler, Murray G"]
[Black "Polgar, Zsuzsa"]
[Result "1/2-1/2"]
[ECO "B33"]
[WhiteElo "2575"]
[BlackElo "2485"]
[SetUp "1"]
[FEN "6n1/6k1/8/5KP1/4B3/8/7P/8 w - - 0 53"]
[PlyCount "8"]
[EventDate "1987.07.??"]
[EventRounds "14"]
[EventCountry "SUI"]
[EventCategory "13"]
[Source "ChessBase"]
[SourceDate "1988.04.01"]

53. Ke6 Nh6 $3 {Surely a blunder in time trouble?} 54. gxh6+ Kh8 $3 (54... Kxh6
55. Kf7 {and White wins!}) 55. Bd5 Kh7 56. Kf7 Kh8 1/2-1/2

Another sad tale of the wrong Rook's Pawn!

And a spectacular save:

[Event "Schilde op U18 6th"]
[Site "Schilde"]
[Date "1970.07.25"]
[Round "9"]
[White "Petrosian, Arshak B"]
[Black "Hazai, Laszlo"]
[Result "1/2-1/2"]
[ECO "E80"]
[PlyCount "110"]
[EventDate "1970.07.??"]
[EventType "swiss"]
[EventRounds "9"]
[EventCountry "BEL"]
[Source "ChessBase"]
[SourceDate "2005.11.24"]

1. c4 g6 2. Nc3 Bg7 3. d4 Nf6 4. e4 d6 5. f3 e5 6. d5 Na6 7. Be3 Nh5 8. Qd2
Qh4+ 9. Bf2 Qe7 10. O-O-O O-O 11. Be3 f5 12. Bd3 f4 13. Bf2 Bf6 14. Nge2 Bh4
15. Bg1 Bd7 16. Kb1 b6 17. Nc1 Nc5 18. Bc2 a5 19. Bxc5 bxc5 20. Ba4 Bxa4 21.
Nxa4 Rfb8 22. Qc2 Kh8 23. Rd3 Rb4 24. Ra3 Qe8 25. Nd3 g5 26. h3 Nf6 27. Rc1 Kg7
28. b3 Nd7 29. Qd2 Rb7 30. Ndb2 Nb6 31. Ka1 h5 32. Nd1 Nxa4 33. Rxa4 Rb4 34.
Nc3 Qc8 35. Nb5 Bg3 36. Qe2 Kf6 37. Rb1 Ke7 38. a3 Rxa4 39. bxa4 Kd8 40. Nc3
Rb8 41. Rb5 Rxb5 42. axb5 Qa8 43. Na4 Qa7 44. Ka2 Kc8 45. Qb2 {[%tqu "Amazing
move coming up","","",Qb6,"",10]} Qb6 $1 46. Nxb6+ cxb6 47. h4 gxh4 48. Qd2 h3
49. gxh3 h4 50. Kb3 Kb7 51. Ka4 Ka7 52. Qg2 Kb7 53. Qb2 Ka7 54. Qc2 Kb7 55. Qc3
Ka7 1/2-1/2

P.S. Foxes are said to eat hedgehogs by rolling them into water, when the hedgehog will unroll to swim, and loses its life.

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Chess Quotes

"Chess is played with the mind and not with the hands!"
— -- RENAUD and KAHN