What pieces does White need to beat Black's veto?

Question

Consider the following chess variant: before each of White's moves, Black chooses one move which White is not allowed to make.

(The rules for which positions constitute checkmate are unchanged - Black can't nullify a checkmate by "vetoing" the move where White takes Black's king.)

Suppose Black has a lone king. What is the minimum amount of material White needs to guarantee a win? Assume that the white pieces start on the first rank, and the black king starts on e5.

Let's say that quantity of material is defined by adding up the classic point values of White's pieces: Q=9, R=5, B=3, N=3. White isn't allowed any pawns. And of course White always has a king.

Full disclosure: I don't know the answer to this puzzle.

Answer 1

Having tried various options to get below 9 points, I've been very successful in forcing a checkmate with

8 points: Rook + Knight

The end goal is to get black into a position where:

White has 2 moves that both give a checkmate. Therefore black cannot veto both and the game is over.

Initially, I expect that black will veto:

Moves where the white rook makes his available movable area the smallest. Therefore, when black vetos a white rook move, my strategy is to pick the next best move. Doing this, I was successful in forcing black to either row 8 or column a/h, any of which is required for the checkmate.

My plan is to get to the following position (or a rotation/reflection of it):

What I was worried about being the largest issue here - black vetoing the king or knight moves, stops being an issue once I get black into a position where every other move has a rook checkmate.

Given this position, there are 2 possibilities, depending on who's move it is. White to move:

1. Nc6/Nd7+ Ka8
2. Ra7/Rc8++

Black to move:

1. .. Ka8
2. Rd7 Kb8 (Rc8++ blocked)
3. Nc6+ Kc8 (Rd8++ blocked)
4. Rc7/Rd8++
Note: if 3. .. Ka8, then 4. Ra7/Rd8++

Answer 2

I believe the minimum is

Either 8 or 9 pts, most likely 9.

White cannot win with

just a rook (5 pts). Even if white manages to drive the black king to the edge of the board, the mating move with the rook is always an "only move" checkmate when there's no additional material on board. All black has to do to draw is to veto the mating move whenever one is available.

White also cannot win with

two minor pieces (6 pts). KNN v K is a draw even without the veto rule, and KBB v K suffers from the same problem as a lone rook: if there's ever a checkmate available, it's the only winning move and can be vetoed by black. I believe there are ways to mate with KBN v K where the knight is the one delivering mate, in which case there could be a situation where there are two mating moves available, but the immediately preceding move would have to be a bishop check and black can just veto that.

I think white can win with

a queen (9 pts). Intuitively this should be a win for white with the usual strategy of restricting the black king along a rank (or, equivalently, a file) using the queen and then inching the king over to aid in pushing the black king towards the edge of the board. Both the king moves and the queen moves by white have some degree of freedom since the king can approach in multiple different ways and also the queen can change rank in more than one way. However, I can't conclusively prove this without going through tons and tons of variations. Each move by white has to have two winning options, so the number of winning lines grows exponentially compared to simple find-a-mate problems.

The only way to improve this would be

either rook+bishop or rook+knight for 8 pts. These look very difficult and I strongly suspect both are draws, but I don't see any immediate reason that has to be the case. (Also, since OP mentioned they don't know the answer, there might not even be a neat way to show this even if it is a draw.)