In 1916, Lawrence S. Crane or Frank Hopkins (Pritchard and Keller give different information here) proposed the following variant: the first player that gives check wins the game. However, one can show (as was first done by Frank Marshall and Walter Penney, and later by Martin Gardner) that white has a forced win in four moves in this variant.
Thus, variants of this basic form of single check chess were proposed.
Presto Chess (possibly also invented by Lawrence Crane in the beginning of the 20th century) is probably the most played `single check chess'-variant:
The first player that gives check with a piece that cannot be taken wins the game.
Origins of this variant are unknown. In this variant, the first player that moves his king loses the game.
Frank Hopkins, when confronted with the forced win of white in the original single check, proposed to play single check chess, but with all pawns originally on the third and sixth rows. Pawns do no longer have an initial double step.
Nick Pelling suggests a playing this variant without Queens. He provides the following sample game:
Nick Pelling vs. Graham Waddingham, correspondence, April 2005.
1. Nc3 e5
2. Nd5 Kd8
3. Nf3 f6
4. c3 c6
5. Nxe5!! fxe5
6. d4 Be7
7. Nxe7 Nxe7
8. Bg5 Ke8
9. Bxe7 Kxe7 (9. ... d5! was probably Black's only saving move)
10. dxe5 Rf8
11. O-O-O Ke8
12. Rd6 Resigns
FIDE Master Dave Gertler has developed a page devoted to this variant, including analysis of a forced win for White.