2 Knight Mate

If you have a king and two knights again a king, you can't checkmate if your opponent doesn't make a big mistake.

To nitpick, you can from a very small number of positions e.g.

n9531I gives a nice example of this by Otto Blathy in Checkmate in 50 moves (99 plies).

There is an extensive analysis of mating with king and two knights against king and pawns in the last 60 pages of Collection of Chess Studies, A.A.Troitzky ISBN 0-923891-10-2.

But unless black captured something on e8, Kf7-e8 was a big mistake.

It is impossible to force checkmate with 2 knights and a king

Exactly. Silmilarly for Ke7-e8 and Kf8-e8. Similar comments apply to all mating positions with king and two knights aganst lone king which is why they're only one ply deep from positions with the same material. 

You said, "if your opponent doesn't make a big mistake" not "if your opponent hasn't make a big mistake". (I did say I was nitpicking.)

Each possible preceding black king move would be a mistake even if capturing (but a big mistake only if it were capturing a knight).

On the other hand the mate could also be achieved (somewhat suboptimally) from this position

without Black making any mistakes -though you might say he's made a mistake somewhere along the line given that he's a queen and knight down.

I've actually finished up with mates similar to the following against the Nalimov databases with optimal play on both sides.

You can't say that without qualification. See the previous post and the Otto Blathy study posted in Checkmate in 50 moves (99 plies).

What I precisely should have written was:

f you have a king and two knights again a king, You can either checkmate in 1 move or you can't checkmate if your opponent doesn't make a big mistake. 

If something is captured in checkmating move, it is not a situation with king and 2 knights against a king. But it is a two knight mate.

What was being asked in original post is not a 100 procent clear. I tried to answer what I thought was ment to be asked.

AndersElborg wrote:

What I precisely should have written was:

f you have a king and two knights again a king, You can either checkmate in 1 move or you can't checkmate if your opponent doesn't make a big mistake. 

Fair summary.

How many distinct checkmate positions, i. e. not reflections of each other, exist for king and two knights? I get 11 - what´s your count?

I count 15, 8 along the side, 7 in the corner.

I also counted 7 in a corner, but just 4 along a side.

The first of the 4:

The king can be on a1 and a2 and so it is for the other three.

Keep in mind that, when taking moves back, white could have captured material. That's the reason that all 15 positions may be valid ends to problems, studies and endgames.

By chance I just posted a puzzle in the puzzle forum "Take back your sorry move - for once" where taking back knight moves is a substantial ingredient.

Would not happen so it is not possible 

If you read my last post, you'd know that a mate with just two knights is very well possible even if both sides do their best! Example:

Looking at the logic of the thing:

All of the 15 checkmate positions inherently have to have the two knights on unlike colour.

The last move of the attacker must have been a knight move giving check with the knight.

Therefore, all positions half move from checkmate, and all positions one move from checkmate, must have the two knights on the same colour.

Wow very nice!