In the position below, I'm promoting my pawn. The app says promoting to a Queen is incorrect, but a Rook is correct? In this setup it doesn't make sense.

[FEN "8/1P6/3N4/8/8/3k4/r7/4KN2 w - - 0 1"]

1.b8=R 1-0
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    In short, to add to the detailed answers below, under-promotion is only ever beneficial for one of two reasons: (1) Promote to a knight, which attacks squares a queen does not, and therefore may be part of a combination to have checkmate in x moves; or (2) Avoid stalemate, by reducing the number of squares that the promoted piece immediately (or shortly thereafter) attacks. Since you are not promoting to a knight, we know the reason for under-promotion is (2). Knowing that, the trick is to find where the stalemate would occur, if you promoted to a queen. Such a position would be quite forced. – Grade 'Eh' Bacon Mar 14 '18 at 19:46
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    @Grade'Eh'Bacon: (3) Promote to a knight as a defensive move (in order to prevent mate or loss of material). – user1583209 Mar 14 '18 at 20:45
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    (6) Underpromotion as a psychological attack. Can happen in real life (but might be dumb in most cases). I was once playing against a stronger player and feigned extreme confidence by underpromoting to a knight. The stronger player promptly resigned. – Klaws Mar 15 '18 at 12:51
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    Also (2') promote to a weaker piece in order to draw by getting stalemated. There's a classic 6-man setup (Traxler-Derdle, if memory serves): Kh8 Nf8 Pg7 / Ka2 Qf7 Re8, White to move draws only with 1 g8B!! This is a well-explored theme in endgame studies but has yet to arise in actual play. – Noam D. Elkies Mar 15 '18 at 16:54
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    Also, (8) Underpromotion to show a novice player some neat mates. At my local chess club I once got an opponent with king+bishop+pawn, who promoted to another bishop to show me the two bishop mate. At the time I was not amused (I was still young), but in hindsight it was entertaining. – Lolgast Mar 16 '18 at 10:55

If you promote to a queen with 1. b8Q, black has:

1...Re2+ 2. Kd1 Rb2

attacking the queen and hinting at mate with Rb1++. If white takes the rook 3. Qxb2 it is stalemate.

Because of the mate threat white does not have any other good square for the queen either (no good check and no square that would defend b1).

If you promote to a rook, black does not have this defense, because after 3. Rxb2 it would not be stalemate. The resulting endgame (King, rook and two knights vs king and rook) should be a reasonably easy win.

Also you need to know that if the queen gets exchanged for the rook (e.g. after 1. b8Q Re2+ 2. Kd1 Rb2 3. Qb5+ Rxb5 4. Nxb5) , the endgame (king and tow knights vs king) is draw.

For the same reason white cannot postpone the promotion because black has (at least) a draw if he can give his rook for the pawn (e.g. after 1. [random non-pawn move by white] Rb2 2. [random move] Rxb7).

Lastly, any other underpromotion (to knight or bishop) does not work either, because black will win the piece on b8 in the above line after 2. ... Rb2 because of the mate threat on b1.

So the only way to play for a win is by promoting to a rook.

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    Is KRNN vs. KR really an "easy" win for a human (non-master) player? I have no idea, never having been in such an endgame. If I get in that position I don't know if my best chance is to promote to R and hope I can manage to win the resulting endgame, or promote to Q and hope my opponent doesn't see the stalemate trick. – bof Mar 19 '18 at 5:11

The Lomonosov 7-man Endgame Tablebase show that after promotion to a queen, Black can play 1. ... Re2+ 2. Kd1 Rb2! (sacrificing the rook) Qxb2, the Black king is stalemated. If White doesn’t take the rook, then Black will take the queen and draw the game.

[FEN "8/1P6/3N4/8/8/3k4/r7/4KN2 w - - 0 1"]

1. b8=Q? Re2+ 2. Kd1 Rb2! 3. Qxb2 (3. Kc1 Rxb8)

However, promoting to a rook avoids the stalemate and wins the game.

[FEN "8/1P6/3N4/8/8/3k4/r7/4KN2 w - - 0 1"]

1. b8=R! Re2+ 2. Kd1 Rb2 3. Rxb2 Kc3
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    However, the Nalimov tablebases suffice for this. ​ ​ – user2668 Mar 14 '18 at 19:43
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    Thanks. Only used the 7-man because of the ability for fast analysis through the link provided. – hangejj Mar 15 '18 at 0:03

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