user25767: Most of the winnable endgames require cornering the opposing king to checkmate, often using zugzwang. However, if you have an infinite board (infinite in every direction), you won't be able to corner the opposing king as easily, as they will be able to retreat unless there is a threat behind them. For example, queen vs king will be a draw in this scenario, as the king and queen will never cover all escape routes. I also believe that two rooks will be unable to deliver mate, as you cannot force opposition. However, queen and rook will be able to checkmate, by using the queen to create a "corner" and checkmating using rook and king. In general, if a combination mates in a regular board, that combination plus a queen will be able to mate here; but I'm looking for more interesting combinations.
So, which piece combinations will be able to mate in this scenario?

user12629: An infinite number of knights would be insufficient to force checkmate, assuming the enemy king starts some distance away and not surrounded. The knights are too slow and not enough of them can chase an enemy king that simply runs away.
A rook and two bishops, on the other hand, could force checkmate. It's rather easy to force a position like this:
[FEN "3BB3/8/4k3/R7/8/8/8/4K3 w - - 0 1"]

The Black king can only shuffle between two squares while the White king gets into position on d4 or e4 to set up Ra6#.
Four bishops could also force a checkmate:
[FEN "3BB3/8/4k3/8/3BB3/3K4/8/8 w - - 0 1"]  

1. Bc5 Ke5 2. Bd7 Kf4 3. Bd6#

I'm guessing that three bishops and a knight, or two bishops and two knights, could also force checkmate. The bishop pair is very effective at corralling the enemy king.

user17823: Here are some mates I found:

Queen and rook (limit the black king until the white arrives to help)
[Title "KQR vs k mate"]
[FEN "1R6/5k2/8/8/8/8/6Q1/K7 w - - 0 1"]

1. Qg8+ Ke7 2. Rd8 Kf6 3. Re8 Kf5 4. Kb2 Kf4
5. Qg2 Kf5 6. Kc3 Kf6 7. Kd4 Kf7 8. Qg8+ Kf6
9. Ke3 Kf5 10. Kf3 Kf6 11. Re6+ Kf5 12. Qg6#


Queen and pawn (not sure if the position can be reached with optimal play)
[FEN "8/8/4Q3/3Pk3/8/4K3/8/8 w - - 0 1"]


Queen and bishop (again, limit the king with the queen and the bishop, so the attacking king finds the time to approach)
[Title "KQB vs k mate"]
[FEN "Q7/2k5/8/8/B7/8/8/7K w - - 0 1"]

1. Kg2 Kb6 2. Kf3 Kc5 3. Qb7 Kc4 4. Ke3 Kc5
5. Ke4 Kd6 6. Qd7+ Kc5 7. Qd4#


Queen and knight (limit the black king with the queen and the knight as much as possible)
[Title "KQN vs k mate"]
[FEN "Q7/2k5/8/8/8/1N6/8/7K w - - 0 1"]

1. Kg2 Kd7 2. Kf3 Ke7 3. Ke4 Kf6 4. Qg8 Ke7
5. Ke5 Kd7 6. Kd5 Ke7 7. Nc5 Kf6 8. Ne4+ Ke7
9. Ke5 Kd7 10. Nd6 Kc6 11. Qb8 Kc5 12. Qb5#


Two rooks (Noam's answer mentions how the mate can be forced)
[FEN "8/8/R7/1R2k3/8/4K3/8/8 w - - 0 1"]




user8860: In fact K+R+R can force checkmate against K on an open board.
First move both Rooks away from the King and to different rows and columns.
Then check the King with one Rook, say on a row.
Whichever side the King goes, move the other Rook to limit it to a single row.
Now the Rooks, moving only horizontally,
can limit the defending King to just four squares on that row,
with enough spare moves for the attacking King to approach:
[Title "RR keep K boxed in"]
[FEN "R7/5k2/2R5/8/8/8/8/K7 w - - 0 0"]

1. Rh8 Kg7 2. Rch6 Kf7 3. Kb2 Ke7 4. Rc8 Kd7 5. Rhc6 Ke7 6. Kc3 Kf7 7. Rh8

Once the King has joined the fray, mate can soon follow.  For example:
[Title "KRR mate K on infinite board"]
[FEN "7R/5k2/2R5/8/3K4/8/8/8 w - - 0 0"]

1. Ke5 Kg7 (1... Ke7 2. Rc7#) 2. Rch6 Kf7 3. Kd6 Kg7 4. Kd5 Kf7 
5. Ke5 Kg7 (5... Ke7 6. R6h7#) 6. Kf5 Kf7 7. R6h7#


25767: <p>Most of the winnable endgames require cornering the opposing king to checkmate, often using zugzwang. However, if you have an infinite board (infinite in every direction), you won't be able to corner the opposing king as easily, as they will be able to retreat unless there is a threat behind them. For example, queen vs king will be a draw in this scenario, as the king and queen will never cover all escape routes. I also believe that two rooks will be unable to deliver mate, as you cannot force opposition. However, queen and rook will be able to checkmate, by using the queen to create a &quot;corner&quot; and checkmating using rook and king. In general, if a combination mates in a regular board, that combination plus a queen will be able to mate here; but I'm looking for more interesting combinations.</p>
<p>So, which piece combinations will be able to mate in this scenario?</p>
 12629: <p>An <em>infinite</em> number of knights would be insufficient to force checkmate, assuming the enemy king starts some distance away and not surrounded. The knights are too slow and not enough of them can chase an enemy king that simply runs away.</p>
<p>A rook and two bishops, on the other hand, could force checkmate. It's rather easy to force a position like this:</p>
<pre><code>[FEN &quot;3BB3/8/4k3/R7/8/8/8/4K3 w - - 0 1&quot;]
</code></pre>
<p>The Black king can only shuffle between two squares while the White king gets into position on d4 or e4 to set up Ra6#.</p>
<p>Four bishops could also force a checkmate:</p>
<pre><code>[FEN &quot;3BB3/8/4k3/8/3BB3/3K4/8/8 w - - 0 1&quot;]  

1. Bc5 Ke5 2. Bd7 Kf4 3. Bd6#
</code></pre>
<p>I'm guessing that three bishops and a knight, or two bishops and two knights, could also force checkmate. The bishop pair is very effective at corralling the enemy king.</p>
 17823: <p>Here are some mates I found:</p>
<ol>
<li><p>Queen and rook (limit the black king until the white arrives to help)</p>
<pre><code>[Title &quot;KQR vs k mate&quot;]
[FEN &quot;1R6/5k2/8/8/8/8/6Q1/K7 w - - 0 1&quot;]

1. Qg8+ Ke7 2. Rd8 Kf6 3. Re8 Kf5 4. Kb2 Kf4
5. Qg2 Kf5 6. Kc3 Kf6 7. Kd4 Kf7 8. Qg8+ Kf6
9. Ke3 Kf5 10. Kf3 Kf6 11. Re6+ Kf5 12. Qg6#
</code></pre>
</li>
<li><p>Queen and pawn (not sure if the position can be reached with optimal play)</p>
<pre><code>[FEN &quot;8/8/4Q3/3Pk3/8/4K3/8/8 w - - 0 1&quot;]
</code></pre>
</li>
<li><p>Queen and bishop (again, limit the king with the queen and the bishop, so the attacking king finds the time to approach)</p>
<pre><code>[Title &quot;KQB vs k mate&quot;]
[FEN &quot;Q7/2k5/8/8/B7/8/8/7K w - - 0 1&quot;]

1. Kg2 Kb6 2. Kf3 Kc5 3. Qb7 Kc4 4. Ke3 Kc5
5. Ke4 Kd6 6. Qd7+ Kc5 7. Qd4#
</code></pre>
</li>
<li><p>Queen and knight (limit the black king with the queen and the knight as much as possible)</p>
<pre><code>[Title &quot;KQN vs k mate&quot;]
[FEN &quot;Q7/2k5/8/8/8/1N6/8/7K w - - 0 1&quot;]

1. Kg2 Kd7 2. Kf3 Ke7 3. Ke4 Kf6 4. Qg8 Ke7
5. Ke5 Kd7 6. Kd5 Ke7 7. Nc5 Kf6 8. Ne4+ Ke7
9. Ke5 Kd7 10. Nd6 Kc6 11. Qb8 Kc5 12. Qb5#
</code></pre>
</li>
<li><p>Two rooks (<a href="https://chess.stackexchange.com/a/33385">Noam's answer</a> mentions how the mate can be forced)</p>
<pre><code>[FEN &quot;8/8/R7/1R2k3/8/4K3/8/8 w - - 0 1&quot;]
</code></pre>
</li>
</ol>
 8860: <p>In fact K+R+R can force checkmate against K on an open board.</p>
<p>First move both Rooks away from the King and to different rows and columns.</p>
<p>Then check the King with one Rook, say on a row.
Whichever side the King goes, move the other Rook to limit it to a single row.</p>
<p>Now the Rooks, moving only horizontally,
can limit the defending King to just four squares on that row,
with enough spare moves for the attacking King to approach:</p>
<pre><code>[Title &quot;RR keep K boxed in&quot;]
[FEN &quot;R7/5k2/2R5/8/8/8/8/K7 w - - 0 0&quot;]

1. Rh8 Kg7 2. Rch6 Kf7 3. Kb2 Ke7 4. Rc8 Kd7 5. Rhc6 Ke7 6. Kc3 Kf7 7. Rh8
</code></pre>
<p>Once the King has joined the fray, mate can soon follow.  For example:</p>
<pre><code>[Title &quot;KRR mate K on infinite board&quot;]
[FEN &quot;7R/5k2/2R5/8/3K4/8/8/8 w - - 0 0&quot;]

1. Ke5 Kg7 (1... Ke7 2. Rc7#) 2. Rch6 Kf7 3. Kd6 Kg7 4. Kd5 Kf7 
5. Ke5 Kg7 (5... Ke7 6. R6h7#) 6. Kf5 Kf7 7. R6h7#
</code></pre>
