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Is there any program or algorithm that determines conclusively whether or not a person at severe material disadvantage can claim a win on time?

This question got me thinking about precise parameters needed for flagging. I noticed that bishop vs queen can't flag for a win, and neither can bishop vs rook, or bishop vs. rook and queen. The friendly piece blocking the king can always interpose. Then I saw bishop vs two pawns and rook and queen had a case where it happened:

8/8/8/5R2/3k1KQ1/5PP1/3b4/8 w - - 0 1

However, this doesn't seem to work so well when the pawns aren't connected. Put one on the b-file and one on the g-file, and it's a draw. My rough proof was this:

  • Black King guards a maximum of 3 squares
  • Black bishop checks and guards a maximum of 2 squares--but if the King guards 3 squares, that overlaps with the bishop. The King guards two adjacent corners in the other King's possible moves, but the bishop guards two opposite ones.
  • White thus needs to have 4 pieces to block the way out

However, in this case, white can underpromote to a bishop then cooperate. So clearly it's not immediately trivial to tell off-hand if there is a win.

Plus if White has just one less pawn but it's a rook pawn, this disaster can always happen, if the queen and rook wander off.

5k1K/7P/5b2/8/8/8/Q7/1R6 w - - 0 1

Time forfeits could thus cause some arguments over whether a position with overwhelming material disadvantage is possible to win.

So is there an established and relatively quick set of rules to determine this? Or is there an actual program that calculates everything? I imagine it's doable with brute force and, so said brute force doesn't take forever, guesswork. Just wondering about the level of complexity.

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When can a player with just a bishop/knight vs overwhelming material claim a win on time?

A win on time can be claimed when it is possible to "checkmate the opponent's king by a series of legal moves". In other words when helpmate is possible. So basically we have to determine when helpmate is possibel and when it isn't.

In both cases King + minor piece is not enough to deliver helpmate against a bare king because the king plus minor piece cannot cover all the squares around the lone king and also give check. The lone king needs at least one extra piece to block the square the the king + minor piece can't cover and at the same time not be able to take the minor piece or block the check in the case of the bishop.

First let's look at all the possible combinations of king + minor piece vs king + one piece where the one piece can block the square next to the king not covered by the opposing king or minor piece. This will tell us which combinations work.

King + Bishop vs King + Rook or Queen

If the opponent has a rook or a queen they can always block the check from the bishop. Hence checkmate not possible.

[fen "KR6/8/k1b5/8/8/8/8/8 w - - 1 1"]

King + Bishop vs King + Knight

The following helpmate position is checkmate.

[fen "KN6/1b6/1k6/8/8/8/8/8 w - - 1 1"]

King + Bishop vs King + Opposite Coloured Bishop

The following helpmate position is checkmate.

[fen "KB6/1b6/1k6/8/8/8/8/8 w - - 1 1"]

King + Knight vs King + Queen

This doesn't work because the queen can always either take the knight or the position is illegal because both kings are in check. Hence checkmate not possible.

[fen "KQ6/2n5/k7/8/8/8/8/8 w - - 1 1"]

King + Knight vs King + Rook

The rook isn't quite as mobile as the queen so this does work.

[fen "KR6/2n5/k7/8/8/8/8/8 w - - 1 1"]

King + Knight vs King + Knight

This works.

[fen "KN6/2n5/1k6/8/8/8/8/8 w - - 1 1"]

King + Knight vs King + Bishop

This works.

[fen "KB6/8/kn6/8/8/8/8/8 w - - 1 1"]

So we have K+B wins against:

K+N
K+opposite coloured bishop

K+N wins against:

K+R
K+B
K+N

These are the minimum requirements for the superior side so if the superior side has at least this much material then helpmate is possible.

This also means that if the superior side has at least one pawn then helpmate is possible because the pawn can promote to a knight and in both cases helpmate is possible.

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    Only one addendum is needed to Brian's concise list: The position may be so hopelessly blocked that no helpmate is possible (independent of the actual material). But this is fairly easy to decide (case in point - I don't know a single chess problem with the stipulation "is helpmate possible?").
    – Hauke Reddmann
    Commented Aug 3, 2021 at 11:08
  • @HaukeReddmann In practice, though, is the result of a timed-out game decided by the feasibility of helpmate in complicated scenarios? I doubt whether online platforms (or arbiters) will try to solve a puzzle to prove helpmate is possible or even query a tablebase - isn't it likelier a simple rule is implemented, like the ones Brian wrote?
    – Mobeus Zoom
    Commented Aug 4, 2021 at 21:32
  • @MobeusZoom the arbiter may not be motivated but the losing player may be
    – Michael West
    Commented Aug 5, 2021 at 3:22
  • @MobeusZoom: The probability of a practical game ending by flag in a blocked position is already minuscule, and the probability of a position where "helpmate possible?" takes longer than a glance on the board (and applying Brian's rules) is infinitesimal. If this would ever happen in my arbiter life, I would rather be inclined to believe that two jokesters want to troll me :-)
    – Hauke Reddmann
    Commented Aug 5, 2021 at 9:08
  • @HaukeReddmann in otb classical sure! But online games? ... MichaelWest: interesting point, but on the other hand if the situation is hard to deduce, the loser's 'solution' can hardly be considered unbiased
    – Mobeus Zoom
    Commented Aug 5, 2021 at 21:10

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