The Flying Knight

White gets all of his pieces in the beginning and gets one move per turn. Black gets a king knight and a king and 2 moves per turn. (spoilers black will probably win).

It's impossible to checkmate with King and knight

Black gets two moves each turn.
He can either move his king + knight, or twice either of them. The king has twice the 'radius' a normal king does and the knight controls twice as much squares wherever he is. It is very possible, though kind of hard to pull off an actual mate. White usually resigns earlier as it is possible to capture a piece and hop back with the knight, or capture two pieces in one turn. The king also is able to move out most checks more easily, and capture nearby pieces so white is almost lost after losing his bishops and rooks (well most of the time).

Even if black captures ALL the pieces he won't be able to checkmate!

Smothered mates are just as possible as in normal chess though harder to pull off. Truth be told, I have only played it a few times and my opponents resigned after losing their rooks (just imagine how easy it is to take them still in the opening if you're careful about your movements), but I don't see why would it be impossible to checkmate.
Yes, it's harder and most games end either with white resigning or black jumping around with his king until a draw is reached, but it's still possible to deliver a mate.

Because you have no reason to castle and sure not to put your king in the corner - the only place where it can be theoretically checkmated (ok, putting pieces on every free square around the king might work to, which is even more stupid).

Sure it's very hard for white to win - no sane person will exchange his only piece, and no tactics works because of black having 2 moves each time - enough to go out pin or fork. But it's even harder for white to lose, so it's dead draw.

Then try it and see for yourself. Black wins more times however crazy it seems.

Can you post couple of games won by black with checkmate? Just want to see it.

You have no null move in pgn but you can post it as text, not as diagram - like

1. Nf3 Nf6 2. -- Nh5 3. Nc3 ... etc.

As I've said I have never pulled off one, my opponents resigned, and I didn't even record them in any way.

It may be good calculating practice for both sides, but I won't believe checkmate by any side is possible without heavy blunders

I suppose black king can't enter a square under white's control during his double-move, can he?

The black King alone can checkmate white under these rules, right? It doesn't even need the Knight. E.g. white Ke1, black Kd3 would be checkmate, as wherever the white King moves, he would be captured on the next turn by the black King.

It depends if entering a square opponent controls is legal. In classic chess, the king must not go through a square under opponent's control in his castling double-move. I cannot google exact rules and I suppose it is not legal, but I understand that arbitrarily created small-potato chess variant can have different rules.

If white plays 1. e4, 2. Bc4, 3. Qf3, then black could never win, because it is impossible for black cover d1, e1, e2 and f1 with only 2 kings and a knight.

Assuming that:

- Checkmate means checkmate, aka when white king is in check and white to move, and there is no legal move to get him out of check. Thus capturing the white king is NOT checkmate.

- When it is white to move, black's kings may not be in check. (You could say that if there are no legal moves for black to get both kings out of check in his 2 turns, he is checkmated. But this is irrelevant for the fact about "black will probably win").

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So unless there are some additional rules, I fail to see how black will probably win :)

Well, unless it is explicitly forbidden (as in the case of castling in orthodox Chess), I would expect it to be allowed. Being 'in check' is defined as exposing your King to capture, and when it is not your opponent's turn to move, you are obviously not exposed to anything.

Castling is a special case, which should not automatically be generalized to this variant: you are also not allowed to castle out of check, and it was sort of implied by the OP that black could use double King moves here to escape mates while in check. And the rule that double moves are only available to pieces that have not moved yet obviously does not apply here either.

In any case a double-move Knight would not suffer any such restrictions, and should have little problems checkmating the opponent with the aid of its own King, even when this was just a normal King. (E.g. white Ka1, black Kb3, Na3) Double-move pieces are exceedingly strong, as we know from Chu Shogi, where the Lion, which can do two King moves per turn, is worth about 1.6 times as much as a Queen (which itself is enhanced in value because of the larger board).

By that logic delivering checkmate with K + B + N is also impossible once the opponent King moves to e2, as it is impossible to cover d1, e1, f1, d2, e2, f2, d3, e3, f3 all at once with just a King, Bishop and Knight...

Not sure why you mention a second King anyway. Black has only two pieces here, right? And in fact the double-moving King alone already seems able to cover the squares you mention. Or an ordinary King (on d3, covering e2) and a double moving Knight (on e3, covering d1, f1 and e1). So not much of what you wrote seems to make any sense.

I noticed that the OP mentioned "king knight and king", though, and I am not sure if this was a simple typo, or whether 'king knight' means a compound piece.

I read it as:

"Black gets a king, knight and a king. Also black gets 2 moves per turn."

But then again, regardless of what he actually meant, and assume your interpetation of a "king which also has to ability to move as a knight", abiding my assumption, it is impossible for black to win:

"Checkmate means checkmate, aka when white king is in check and white to move, and there is no legal move to get him out of check. Thus capturing the white king is NOT checkmate."

But if capturing the white king implies checkmates too, and black allowed to be in check between his moves, Black should always win, assuming he makes no enormous blunders, as it seems impossible for white to cover the enormous range that black king has.

If black king may not be in check between his moves, white has to cover in the worst case 17 squares. And I mean cover all these 17 squares, because if there is one not covered, he gets additional squares to run to (or stalemate if the only one not covered is the one the black king is standing on).

If black may be in check between his moves, white has to cover in the worst case 61 (!) squares, simply impossible, unless black allowed white to have like 8 queens.

I don't follow your arithmetic. If black is not allowed to 'pass through check', white would have to cover 9 squares in the worst case, the one on which the black King is, and those next to it over which he could escape. And on an edge that would only be 6, and in a corner only 4. Just like in normal Chess.

And with K+R you cannot cover the 9 squares, but we all know that K+R can still force a mate, because you will not be checkmating the opponent in the worst place, but drive him to the best place first. And you only need to cover the single square on which he is to make sure he cannot stay there. So 'worst case' is totally irrelevant.

If black can pass through check, covering 25 squares would be enough to ceckmate him in the middle of the board. But in a corner 9 would suffice, and a King + 2 Rooks could do that. So '8 Queens' seems a gross exaggeration.

'King knight' simply means king-side knight. So black only has two pieces: A king, and a knight on the king-side.

OK, clear. And how about passing through check? If white's King is on e1, and black plays his King to e3, is white now in check (if d2, e2 or f2 is empty)?