user29886: In the attached position, why can't white mate?
Is it not possible to go Rxc6 and depending on blacks response go Rxc6.
[FEN "4r1k1/1p3ppp/2b5/5n2/pp6/8/P4PPP/2R1R1K1 w - - 0 1"]

Source: Bobby Fischer Teaches Chess (1966) - Bobby Fischer & Stuart Margulies

user24511: Based of your question it appears that you have correctly assessed and identified the critical piece holding blacks position together. The Bishop on c6.
Rxc6 - Removing the Defender.
[FEN "4r1k1/1p3ppp/2b5/5n2/pp6/8/P4PPP/2R1R1K1 w - - 0 1"]

1. Rxc6 Rxe1#

Results in a completely lost position which allows black to mate in one.
Rxe8 - "Correct" Variation.
[FEN "4r1k1/1p3ppp/2b5/5n2/pp6/8/P4PPP/2R1R1K1 w - - 0 1"]

1. Rxe8 Bxe8 Rb1 b5 Rxb4

Doesn't significantly change much. Black has a promotion threat on the a/b file. The rook is unable to defend because any attempts to dislodge the bishop will result in either loss of material or promotion. In addition to this the knight has the ability to apply pressure on whites rook.
Engine Evaluations would indicator that white is down approximately 5 pawns (or a rook) of material from the starting position.

user15223: If White goes for 1. Rxc6??, Black has 1...Rxe1#

29886: <p>In the attached position, why can't white mate?
Is it not possible to go Rxc6 and depending on blacks response go Rxc6.</p>
<pre><code>[FEN &quot;4r1k1/1p3ppp/2b5/5n2/pp6/8/P4PPP/2R1R1K1 w - - 0 1&quot;]
</code></pre>
<p>Source: Bobby Fischer Teaches Chess (1966) - Bobby Fischer &amp; Stuart Margulies</p>
 24511: <p>Based of your question it appears that you have correctly assessed and identified the critical piece holding blacks position together. The Bishop on c6.</p>
<p><strong>Rxc6 - Removing the Defender.</strong></p>
<pre><code>[FEN &quot;4r1k1/1p3ppp/2b5/5n2/pp6/8/P4PPP/2R1R1K1 w - - 0 1&quot;]

1. Rxc6 Rxe1#
</code></pre>
<p>Results in a completely lost position which allows black to mate in one.</p>
<p><strong>Rxe8 - &quot;Correct&quot; Variation.</strong></p>
<pre><code>[FEN &quot;4r1k1/1p3ppp/2b5/5n2/pp6/8/P4PPP/2R1R1K1 w - - 0 1&quot;]

1. Rxe8 Bxe8 Rb1 b5 Rxb4
</code></pre>
<p>Doesn't significantly change much. Black has a promotion threat on the a/b file. The rook is unable to defend because any attempts to dislodge the bishop will result in either loss of material or promotion. In addition to this the knight has the ability to apply pressure on whites rook.</p>
<p>Engine Evaluations would indicator that white is down approximately 5 pawns (or a rook) of material from the starting position.</p>
 15223: <p>If White goes for 1. Rxc6??, Black has 1...Rxe1#</p>
