Some cases immediately come to mind why you would not want to promote to queen. 1) Promotion to queen turns a winning position into a stalemate. In this case, a rook will usually do the trick.
(For example White: Pf7, Kh5, Black: Kh7. 1. F8=Q?
Draw. On the other hand, 1. F8=R is winning.) 2) Because of its particular movement, the knight has properties other pieces don't.
In certain particular positions, the knight is preferable. (For example White: Kd7, pe7 pa6, Black: Kg7, Qf6. 1.
E8=Q? Qxa6 ,again, merely draws. On the other hand: 1.
E8=N forks Queen and king 1... Kg6 2. NxQ KxN 3. A7, with the a7 pawn queening on the next move.) 3.
Related to "2" above; what you need is a tempo and the Queen is just not giving it to you. (White: Kd6, Nb6, Pc2, Pe7; Black Kg7, Qa4.) In this position 1. E8=Q probably wins, but you're still going to have to battle it out.
On the other hand... 1. E8=N+! Kf8 2.
NxQ KxN easily wins. 4. You might also be in need of perpetual check in an otherwise lost position.
White: Ke7 Pf7 Black: Kg7, Qb3, Bg8, Pg7, Ph6. 1. F8=Q loses to 1... Qa6+ 2.
Kd7 QxQ. But 1. F8=N+!
Saves the day with 1... Kh8 being the only move followed by 2. Ng6+ Kh7 3. Nf8+ etc... draw.
I can also think of a fifth case, which might be of interest. Suppose you are losing badly and a stalemate draw would actually be your only hope. White: Kg7 Pa7, Black: Kb5, Qe1, Rd6, Bb6, Bd7, Nc6.
1. A8=B! And should black play 1... Qe8?
, figuring mate is one move away, you've got stalemate! Of course, depending on a blunder is never good, but then again, to paraphrase Tennessee Williams: "I've always depended on my opponents' mistakes". Chess is a game of the imagination.
It looks mechanical until you reach that point where you can actually let your mind go.
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