Only kings and horses

Question

This is the fourth entry in my decentralised or "Swiss" chess mini series. See bottom for a recap of the modified rules of Swiss chess.

Enough warm up! It is time for a proper Swiss chess problem:

Task:

Consider the Swiss KNN-K endgame: All pieces except for the two kings and two white knights have been traded and a neutral (no immediate threats on either side) position has been reached. In regular chess this is a theoretical draw: Assuming best play from both sides white despite being two pieces up cannot force check mate. But what about Swiss chess? Is the Swiss king + 2 Swiss knights vs Swiss king endgame winnable or a draw? Please back up your answer by summarising the successful side's strategy.

By "summarising" I mean: Give just one or two example variations demonstrating each major step the strategy may consist of. I'm not asking for an exhaustive search tree

Note: This may look rather daunting at first sight but Swiss rules actually make it much easier to analyse than the standard chess KNN vs K

Swiss chess primer:

In Swiss chess, the advantage in terms of mobility of a centrally placed piece over a piece in the corner is compensated for by opening up additional squares a marginalised piece has access to. This applies to all pieces except rooks and pawns. To keep rules simple, the queen like in standard chess is a bishop+rook even though that does mean it is a bit weaker on edge-of-the-board squares.

Credit to @AxiomaticSystem for the following streamlined description of Swiss moves:

0. Movements ending at the starting square are implicitly forbidden

1. The King may either take a diagonal step, or make an orthogonal jump. This jump can skip at most as many squares as the number of edges the King is on.

2. A bonk is a unit move (knight leap for the Knight, diagonal step for the bishop), except if this move would take them to a square off the board, the piece moves to the closest on-board square instead.

3. A Knight can always bonk. In addition, if it's in the corner, it can also take a diagonal step .

4. A Bishop can always move as normal. If the number of possible normal moves (assuming otherwise empty board) is smaller than the maximum, 13, then the bishop can also bonk off the nearest edge. (If the Bishop is on one of the long diagonals there are two nearest edges and both are allowed.) This extension is, however, capped at 13 possible moves total, meaning that if necessary the longest bonk moves are pruned.

5. The Queen moves like a Rook or Bishop.

Here are some animated gifs, showing the full set of legal moves from a representative set of squares for the Swiss knight, the Swiss bishop, the Swiss queen and the Swiss king. If you need more info, please comment or find some at the other Swiss chess posts.

Hint:

Note that due to the "asymmetric" way the Swiss king moves it is sometimes possible for a Swiss king to give check to the other Swiss king. Not very royal behaviour, admittedly, but useful to know.

Answer 1

General strategy:

Using the denser footprint of Swiss knights on the edge we can build this simple prison

which confines the defending king to a 12-square "L"-shaped area.
Please note how perfectly the shapes of the king's and knights' sets of allowed moves match on the edge where the knight has just enough to prevent the king from utilising his two-square jump.

How difficult is setting up such a prison?

Shouldn't be too difficult. We just need to get the white king to the centre. Depending on where the other pieces are we can choose and switch the most convenient corner for the prison by a single move of the king. For example, if the black king were in the top right corner then it would take just three moves Ke5, Nb8, Nh2 to relocate the prison to that corner.

Where from here?

Once the black king is cornered we can start closing in the walls by advancing the knights to a5 and e1. I assume there will be a way of forcing checkmate from this position.

A few checkmates:

Checkmate 1:

Checkmating a cornered king:

    1.   ...    Ka1-b2
    2. Kd4-c4   Kb2-a2
    3. Kc4-b4   Ka2-a1
    4. Na5-a4   Ka1-b1
    5. Kb4-c3   Kb1-a1
    6. Na4-b2+  Ka1-a2
    7. Ne1-c1#
 

Checkmate 2

Checkmating a careless king:

     1. Nh1-f1+  Kd1-c2
     2. Ng1-e1+  Kc2-c3
     3. Nf1-d1+  Kc3-d2?
     4. Ne1-c1   Kd2-c2
     5. Nd1-b1   Kc2-b2
     6. Ke5-d4   Kb2-c2
     7. Kd4-c4   Kc2-b2
     8. Kc4-b4   Kb2-c2
     9. Kb4-a5   Kc2-b2
    10. Ka5-a4   Kb2-c2
    11. Ka4-a2#
 

What a typical endgame may look like:

Step 1:

Claiming the centre for the white king:

    1. Ka1-b2   Kd4-d3
    2. Kb2-b3   Kd3-d4
    3. Kb3-b4   Kd4-d5
    4. Kb4-c3   Kd5-c6
    5. Nb7-a5+  Kc6-d5
    6. Kc3-d3   Kd5-e5
    7. Ng2-e1   Ke5-d5
    8. Ne1-f3   Kd5-c5
    9. Kd3-d4     ...
 

Step 2:

The centred white king on his own leaves two wide and two narrow edges for the black king. We'll force the black king onto the narrow ones.

     9.  ...     Kc5-b5
    10. Na5-a3   Kb5-b4
    11. Na3-a1   Kb4-c3
    12. Nf3-g1   Kc3-c4
    13. Ng1-f1   Kc4-c3
    14. Nf1-d1+  Kc3-c4
    15. Nd1-b1   Kc4-c5
    16. Nb1-a2   Kc5-c4
    17. Na1-a3+  Kc4-c5
    18. Ke4-e5   Kc5-c6
    19. Na2-a4   Kc6-d7
    20. Na3-b5   Kd7-e7
    21. Nb5-d4   Ke7-f7
    22. Nd4-f3   Kf7-e7
    23. Nf3-h4   Ke7-d7
    24. Na4-a5   Kd7-c7
    25. Ke5-d5   Kc7-b6
    26. Na5-a4+  Kb6-c7
    27. Nh4-h6   Kc7-d7
    28. Nh6-g8
 

The prison is complete.

Step 3:

The writing is on the wall.

    28.   ...    Kd7-d8
    29. Na5-a7   Kd8-c7
    30. Ng8-e7    ...
 

and now

    30.   ...    Kc7-d7
    29. Na7-c8   Kd7-c7
    30. Ne7-d8    ...
 

followed by checkmate 2
or

    28.   ...    Kc7-b6
    29. Ne7-c6   Kb6-b7
    30. Nc6-b7   Kb7-c7
    31. Na7-a8#
 

Answer 2

This answer is incorrect. I'm just leaving it up here to show what my though process was.

My strategy was to start with checkmate and go backwards.
There are three positions of the black king we must consider:

1. Black king in the center (not touching walls)

Not physically possible to checkmate just yet...
I say this be cause in order to checkmate, 9 squares must be controlled: The black king and the squares immediately surrounding it. It is not physically possible to control 9 the most is 8. white The king can usually do 3 but 4 if its in the corner. The knights can do 2 each. Below is a picture capturing 8, resulting in stalemate:

2. Black king on the side (touching 1 wall)

It seems possible to checkmate the king. There are a checkmated position shown below: However, notice that in this situation, there are 2 pieces checking the black king. This means the black king must have moved itself into check, which is illegal.
There for it is not possible to checkmate the king just yet.

3. Black king in the corner (touching 2 walls)

Behold: the only checkmate positions: The circled knights are options for the second knight. They all work. The problem is that to get the king there, the black king must have previously been on b8. This makes the white king unable to get to B6, and therefore renders this impossible.

Therefore

checkmate is impossible

Answer 3

Not really an answer, but (in reaction to Ankid's post): I like the following endgame:

Black is almost cornered, and decides on Kh8, threatening h5.
After all even if white plays Ke7 black can escape to g6 after Kg8 Nh4h6.

Alas checkmate 'off the borders' in 3

with Nh4h6 Kf8(forced) Nh6h7 Kf7(forced) Nh7h8.
(Regrettably I did not see a way to force winning like this.)