user11071: The puzzle (this may have a known answer) is to construct a position such that:

If Black makes move A, it's mate in 1 for white.
If Black makes move B, it's mate in 2 for white.
...
If Black makes move X, it's mate in N for white.

For instance, in this position
6k1/7p/5K1P/8/8/4Q3/8/8 b - - 0 1


If 1..Kh8 2. Qe8#
If 1..Kf8 2. Kh8 Qe8#

So this position has N=2.
What's the highest possible N? (All of the 1-N moves must exist)
Extra details:

It doesn't count as having mate-in-X if white also has a shorter forced mate.
It is acceptable for white to have several different mate-in-X for a given black response.
It is acceptable for black to have additional moves available that don't lead to any particular mate-in-X.
It is acceptable for black to have two different moves that both forced mate in the same number of moves.

(It might be considered "nicer" to have solutions that don't rely on the last three points though).

user11144: A N=4 demo (board from Black view), not necessarily the stuff that you wanted, depending on your demands.
[FEN "8/6B1/8/8/B2Rp3/8/3p1n1n/K1k5 b - - 0 1"]

If Black plays...
~: 1.Rc4#
d1B: 1.Bh6+ e3 2.Bxe3#
d1Q: 1.Bh6+ e3 2.Bxe3+ Qd2 3.Bxd2#
d1N: 1.Bf8 (threat Ba3) Nd3 2.Bh6+ e3 3.Rxd3 Nd1~/~ 4.Bxe3/Rxd1#
Observe this isn't perfect from my view: ~ is not exact, since Black could play Nd1 with a #2. (I'd prefer ~ is a single move with a #1.) In the #3 line Black might shorten it with Qd2. Also, even worse, Black can play d1R, leading to a dual Bxd2/Rc4 in the mating move. In the #4 variant, the line splits again. It shouldn't since it obscures the theme. But at least all White moves are unique.
Thus, again, please clarify the exact demands. My own maximum demands on the task would be
X%: 1.A#!
Y%: 1.B! Y2! 2.B2#!
Z%: 1.C! Z2! 2.C2! Z3! 3.C3#!
and so on, where "%" means "all of the moves, but choice possible" and "!" only black move/only White move that mates in the required number of moves. In contrast, my minimum demand would be:
X: 1.A#!
Y: 1.B! Y2 2.B2#!
Z: 1.C! Z2 2.C2! Z3 3.C3#!
i.e., only that some forced unique lines with n=1,2,...,N exist.
EDIT:
[FEN "8/k1K5/p1R5/8/8/P7/8/8 b - - 0 1"]

"Superexact" N=2 example. All moves are forced (Black's as they are the only legal ones, White's as they are the only fulfilling the stipulation, also no #2 exists in the #1 line!): 0...Ka8 1.Rxa6#, 0...a5 1.a4 Ka8 2.Ra6#.

11071: <p>The puzzle (this may have a known answer) is to construct a position such that:</p>
<ul>
<li>If Black makes move A, it's mate in 1 for white.</li>
<li>If Black makes move B, it's mate in 2 for white.</li>
<li>...</li>
<li>If Black makes move X, it's mate in N for white.</li>
</ul>
<p>For instance, in this position</p>
<pre><code>6k1/7p/5K1P/8/8/4Q3/8/8 b - - 0 1
</code></pre>
<ul>
<li>If 1..Kh8 2. Qe8#</li>
<li>If 1..Kf8 2. Kh8 Qe8#</li>
</ul>
<p>So this position has N=2.</p>
<p>What's the highest possible N? (All of the 1-N moves must exist)</p>
<h2>Extra details:</h2>
<ul>
<li>It doesn't count as having mate-in-X if white also has a shorter forced mate.</li>
<li>It is acceptable for white to have several different mate-in-X for a given black response.</li>
<li>It is acceptable for black to have additional moves available that don't lead to any particular mate-in-X.</li>
<li>It is acceptable for black to have two different moves that both forced mate in the same number of moves.</li>
</ul>
<p>(It might be considered &quot;nicer&quot; to have solutions that don't rely on the last three points though).</p>
 11144: <p>A N=4 demo (board from Black view), not necessarily the stuff that you wanted, depending on your demands.</p>
<pre><code>[FEN &quot;8/6B1/8/8/B2Rp3/8/3p1n1n/K1k5 b - - 0 1&quot;]
</code></pre>
<p>If Black plays...</p>
<p>~: 1.Rc4#</p>
<p>d1B: 1.Bh6+ e3 2.Bxe3#</p>
<p>d1Q: 1.Bh6+ e3 2.Bxe3+ Qd2 3.Bxd2#</p>
<p>d1N: 1.Bf8 (threat Ba3) Nd3 2.Bh6+ e3 3.Rxd3 Nd1~/~ 4.Bxe3/Rxd1#</p>
<p>Observe this isn't perfect from my view: ~ is not exact, since Black could play Nd1 with a #2. (I'd prefer ~ is a single move with a #1.) In the #3 line Black might shorten it with Qd2. Also, even worse, Black can play d1R, leading to a dual Bxd2/Rc4 in the mating move. In the #4 variant, the line splits again. It shouldn't since it obscures the theme. But at least all White moves are unique.</p>
<p>Thus, again, please clarify the exact demands. My own maximum demands on the task would be</p>
<p>X%: 1.A#!</p>
<p>Y%: 1.B! Y2! 2.B2#!</p>
<p>Z%: 1.C! Z2! 2.C2! Z3! 3.C3#!</p>
<p>and so on, where &quot;%&quot; means &quot;all of the moves, but choice possible&quot; and &quot;!&quot; only black move/only White move that mates in the required number of moves. In contrast, my minimum demand would be:</p>
<p>X: 1.A#!</p>
<p>Y: 1.B! Y2 2.B2#!</p>
<p>Z: 1.C! Z2 2.C2! Z3 3.C3#!</p>
<p>i.e., only that some forced unique lines with n=1,2,...,N exist.</p>
<p>EDIT:</p>
<pre><code>[FEN &quot;8/k1K5/p1R5/8/8/P7/8/8 b - - 0 1&quot;]
</code></pre>
<p>&quot;Superexact&quot; N=2 example. All moves are forced (Black's as they are the only legal ones, White's as they are the only fulfilling the stipulation, also no #2 exists in the #1 line!): 0...Ka8 1.Rxa6#, 0...a5 1.a4 Ka8 2.Ra6#.</p>
