user15543: What is the total number of KNN vs. K checkmate positions? Assume that White is checkmating.
Firstly the Black king must be mated on a corner. It is impossible to do it on the edge. The calculations on one corner can be multiplied by four for the final results.
Addendum: It actually can be done as @Glorfindel’s answer shows. 
[FEN "k6k/8/8/8/8/8/8/k6k w - - 0 1"]

Let’s choose this corner for now.
[FEN "8/8/8/8/8/8/8/k7 w - - 0 1"]

The white king needs to be on one of these four squares.
[FEN "k6k/8/8/8/8/8/8/k6k w - - 0 1"]

The checkmating knight needs to be on one of these two squares.
[FEN "8/8/8/8/8/1N6/2N5/8 w - - 0 1"]

Finally, the knight used to block off the black king’s escape route has to be on one of these five squares.
[FEN "8/8/8/8/1N6/N1N5/3N4/2N5 w - - 0 1"]

Can anyone calculate the total number of possiblities?

user7297: TL;DR: 120
You're on the right track, but combinatorics can be hard sometimes.
If the black king must be in the corner, just pick a1. If the white king is on c1, one knight (the white one below) must give check on c2 or b3, and the other one (the black one below) must cover a2, so be on c3 or b4. That's four positions (we'll cover symmetry later).
[FEN "8/8/8/8/1n6/1Nn5/2N5/k1K5 w - - 0 1"]

If the white king is on c2, the knight giving (the white one below) check must be on b3 and the other one (the black one below) must cover a2, so be on c1, c3 or b4. That's three positions.
[FEN "8/8/8/8/1n6/1Nn5/2K5/k1n5 w - - 0 1"]

Now all seven positions can be flipped along the long diagonal; that's 14 (there are no duplicates; the white king isn't on the diagonal). And finally all positions can happen on all four corners, for a total of 14 times 4 = 56 positions ... in the corner. @Remellion found another mating pattern:
[FEN "8/8/8/8/8/1N2N3/K7/K1k5 w - - 0 1"]

(the white king can be at two places) and this position can be shifted one square to the right four times. (I missed it because I only looked for mates with the black king on b1 and assumed I didn't need to consider the other edge squares. I was wrong ...) We again have an eight-fold rotation symmetry (four edges times two, for the mirrored version), so this pattern yield 4 times 2 times 8 = 64 positions.
Together, there are 120 mating positions.

15543: <p>What is the total number of KNN vs. K checkmate positions? Assume that White is checkmating.</p>

<p>Firstly the Black king must be mated on a corner. <strong>It is impossible to do it on the edge.</strong> The calculations on one corner can be multiplied by four for the final results.</p>

<p><strong>Addendum</strong>: It actually can be done as @Glorfindel’s answer shows. </p>

<pre><code>[FEN "k6k/8/8/8/8/8/8/k6k w - - 0 1"]
</code></pre>

<p>Let’s choose this corner for now.</p>

<pre><code>[FEN "8/8/8/8/8/8/8/k7 w - - 0 1"]
</code></pre>

<p>The white king needs to be on one of these four squares.</p>

<pre><code>[FEN "k6k/8/8/8/8/8/8/k6k w - - 0 1"]
</code></pre>

<p>The checkmating knight needs to be on one of these two squares.</p>

<pre><code>[FEN "8/8/8/8/8/1N6/2N5/8 w - - 0 1"]
</code></pre>

<p>Finally, the knight used to block off the black king’s escape route has to be on one of these five squares.</p>

<pre><code>[FEN "8/8/8/8/1N6/N1N5/3N4/2N5 w - - 0 1"]
</code></pre>

<p>Can anyone calculate the total number of possiblities?</p>
 7297: <h3>TL;DR: 120</h3>
<p>You're on the right track, but combinatorics can be hard sometimes.</p>
<p>If the black king must be in the corner, just pick a1. If the white king is on c1, one knight (the white one below) must give check on c2 or b3, and the other one (the black one below) must cover a2, so be on c3 or b4. That's four positions (we'll cover symmetry later).</p>
<pre><code>[FEN &quot;8/8/8/8/1n6/1Nn5/2N5/k1K5 w - - 0 1&quot;]
</code></pre>
<p>If the white king is on c2, the knight giving (the white one below) check must be on b3 and the other one (the black one below) must cover a2, so be on c1, c3 or b4. That's three positions.</p>
<pre><code>[FEN &quot;8/8/8/8/1n6/1Nn5/2K5/k1n5 w - - 0 1&quot;]
</code></pre>
<p>Now all seven positions can be flipped along the long diagonal; that's 14 (there are no duplicates; the white king isn't on the diagonal). And finally all positions can happen on all four corners, for a total of 14 times 4 = <strong>56</strong> positions ... <strong>in the corner</strong>. @Remellion found another mating pattern:</p>
<pre><code>[FEN &quot;8/8/8/8/8/1N2N3/K7/K1k5 w - - 0 1&quot;]
</code></pre>
<p>(the white king can be at two places) and this position can be shifted one square to the right four times. (I missed it because I only looked for mates with the black king on b1 and assumed I didn't need to consider the other edge squares. I was wrong ...) We again have an eight-fold rotation symmetry (four edges times two, for the mirrored version), so this pattern yield 4 times 2 times 8 = <strong>64</strong> positions.</p>
<h3>Together, there are 120 mating positions.</h3>
