Play continued:

2 Kh2 d2! snares the rook -- a useful tactic. Now the expected 2 ... g1/Q? would run into 3 Rd7+ Ke8 4 Rd8+! with perpetual check or stalemate. Black cannot wriggle out with 3 ... Kg8 4 Rd8+ Kh7 (5 Rd7+? Qg7!) because 5 Rh8+! draws all the same, and 2 ... g1/R 3 Rxe3 or 2 ... g1/N+ 3 Kh2 get Black nowhere, so...

And Black won after

With this knight on its ideal square the new bishop can't be stopped from gobbling white's weak pawns.

Hastens the end, but the immediate 5 Kg3 Be3 6 Rxh5 Bxf4+ wouldn't take long either.

Oops! Now the rook is trapped.

Now it's not hard to construct a position where a bishop promotion is required either offensively to avoid stalemate (as in Chan-Depasquale above) or defensively to force it, as in this 1909 study by K. Traxler and F. Dedrle (#1204 in Sutherland and Lommer's

Together with the advanced pawn at g7, White's two pieces would ordinarily balance the Black queen, but here they are both threatened and White's rook is loose too in such lines as 1 Kh7 fxg6 2 Rh1+ Kb2 3 g8/Q Re7+ 4 Kh6 Qf4+ 5 Kxg6 Qg3+ 6 Kf6 Qxg8 7 Kxe7 Qg7+ 8 Ke8 Qe5+ and 9 ... Qd5+. Hence the drawing combination:

And now the startling...

... forces

Multiple promotions are naturally much more challenging. Here's a modern setting in which the theme of Diagram 2 is multiplied, White having to promote two bishops to extract a draw:

This study earned Y. Afek a Commendation in the 1981 Guanabara Jubilee Tourney. The natural stalemate try 1 c8/Q?! Rxc8 2 dxc8/Q Rxc8 3 Rb6+ fails to 3 ... Kc4!, but contains the germ of the solution:

Forced here since after 1 ... Kc4?? 2 Rxb8 Black even loses; but with the White rook now gone the stalemate tries 2 c8/Q or 2 d8/Q fail to 2 ... B(x)c7! Thus:

White has his way after 2 ... Bxc7 or 2 ... Rxd8 3 cxd8/Q+ Rxd8, with stalemate in either case, as well as the tricky 2 ... Kb5!? 3 c8/Q Rxd8 (or B-any) 4 Qa6+!. Now 3 c8/Q+? Rxd8 wins, so of course White continues

Again!

And despite all of Black's squirming, it's stalemate after all!

We turn back to offensive underpromotions in the next example, showing all three underpromotions by the same pawn in three variations:

This is Herbstmann's second-prize study in

Not 1 Rg1+? Bg6! =. But now 1 ... Kxh7 loses to 2 exf8/N+! Kg7 3 Nxd7 (not 3 Rxd7? Kxf8 4 h6 Kg8 5 h7+ Kh8! =). Black has the stalemate tries

And finally a comical orgy of knight promotions by Korolkov (1937):

Black threatens to win with his own knighting: 1 ... c1/N+! 2 Kc1 Ndb3# or 2 ... Bd4#. Thus White must start checking.

Here and subsequently, 1 ... Kg5(h4)? allows 2 d8/Q+; that's why White did not check on f6.

Not 3 ... Kh8 4 Nxg6#!

Now the eighth rank is off limits.

A "model mate" (each of the king's escape squares is guarded just once) administered by

This concludes our exposition of underpromotions; the next column will feature more slapstick along the lines of Diagram 5.

*Noam Elkies is now Professor of Mathematics at Harvard University and is the author of "Chess Art in the Computer Age," published in ACJ 2 (1993). This article originally appeared in *Chess Horizons.

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