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If you modify chess so that white gets to make one extra move within the first 10 moves, is it possible to show there is a non-losing strategy for white?

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Student T gives an explicit answer, but you can also view it from a game-theoretical perspective.

Suppose normal chess is a draw (or a win for black). Then, white can play 1. Ng1-f3 and Nf3-g1, effectively becoming black and the rest of the game is normal, so it is a draw (or a win for white).

Suppose normal chess is a win for white. This will almost certainly involve moving a pawn two steps from its initial position within the first 10 moves, or moving a bishop, queen, rook or king. You can use your extra move to perform this move in two steps, e.g. e2-e3-e4 or Bf1-d3-e2.

In any case, White will not lose the game.

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  • That is a very elegant answer!
    – GloriaVictis
    Apr 4 '16 at 8:12
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    This is a non-constructive proof. You must be a mathematician!
    – Dag Oskar Madsen
    Apr 4 '16 at 8:39
  • This is not a proof. The knight manoeuvre cost two tempi, not only one. Thus it does not prove that Black has no winning strategy.
    – jk - Reinstate Monica
    Apr 4 '16 at 10:06
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    @jknappen - the second tempo is 'paid' by the extra move White gets. And DagOskarMadsen, you are right.
    – Glorfindel ♦
    Apr 4 '16 at 10:38
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Yes. 1.e4, then move the white Queen out to f3|g4|h5. Prepare for Qxf7+ then Qxe8 or Qxd7+ then Qxe8. Now white wins because the black king is gone.

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    Say 1.e4 Nf6. Can you win Black's king by force now? Or are you just going to grab a piece? Can you prove that White can win or draw with an extra piece?
    – bof
    Apr 4 '16 at 9:03
  • 2. e5 and your knight can't run (otherwise 3. e6 wins) so white can at least grab your knight without using his extra move.
    – jf328
    Apr 4 '16 at 12:17
  • 2.Bc4 can give Black some trouble. 1.e4 Nf6 2.Bc4 (threatens 3.Qh5,Qxf7#) g6 3.e5 (threatens 4.e6,exf7# or 4.Qf3,Qxf7# if the knight moves anywhere but d5, where it is lost.) e6 4.exf6. Now if 4…Qxf6 5.Qf3,Qxf6 and if anything else then 5.Qg4 threatens 6.Qxe6,Qxf7# or even just 6.Qxe6,Qxe8#. 5…Be7 doesn't prevent mate on f7, and 5…Qe7 or 5…Qf6 loses the queen. Also 5…d5 just loses to 6.Bb5,Bxe8# Sacrificing the knight on d5 on move 2 or 3, followed by e6, seems best for Black, but probably still loses.
    – Jivan Scarano
    Jul 11 '16 at 9:42

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