🚀 go-pugleaf

Consider an ending of K+P vs. K+P.  White pawn on a7, black on h2,
white king on a2, black on h7, white to move.

If white has the higher score, then perfect play by both sides will lead
to a win by white, with white still having a higher score.

If white has a score 10 or more lower than black, he can make a king move,
which gives black a winning position.  Since white only has 9 possible
moves, he can't make a move worse than 9, so his score will stay lower
than blacks, and thus black will win with a higher score.

If white has a score 9 or less lower than black, then let
white play these moves: a8Q, Qh1, Qh2, and let black respond with
three perfect moves.

The situation is now K+Q vs. K, with whites score no lower than
nine lower than blacks score.

Now let white, while keeping his king at a2, play a series of checks
by placing his unprotected queen next to the black king, making sure
that the black king has a move available other than taking the queen.

Since the white queen always has at least fourteen moves, and at most
three moves can satisfy the above conditions, there are at least 11
moves better than the ones white plays above.

Let black respond to these checks by not taking the queen.  There are
at most four moves available to black, so he can make no move worse
than fourth best, thus the score of white goes up at least 7 for each
check in this series.

Let white include at least two checks in the above series.  Then white
will have a higher score than black, but still have a won game.  Let both
sides then switch to perfect play, and white will win with a higher score.

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Tim Smith       sdcrdcf!ism780c!tim || ima!ism780!tim || ihnp4!cithep!tim