> I then say clearly that I will remove 6 of the other cups. And, to retain the mystery, I will not reveal the prize.

It wasn't a criticism.

I just wanted to draw attention to the point that in the random game (I.e. player marks two, you reveal 6 random unmarked, if no prize is revealed, player decides to switch to one or to keep two) it is the better strategy to never switch.

Do you think it would be an interesting variant to play the game like that?

You say you don’t reveal the prize, but one can remove six cups at random without revealing their contents.

“Removing cups” is not the same as “removing empty cups.”

If the rules of the game permit you to remove the cup with the prize, it is significant. I think the math and optimal strategy changes.

One can, but one cannot guarantee it.

Actually, one can. One can choose 6 at random from the ones that are neither selected, nor hide the prize.

My point is that my phrasing is, I believe, sufficient to allow the listener to determine that I will turn six cups without revealing the prize. In particular I say that I will remove 6 of the other cups and ... will not reveal the prize.

I honestly don't see how that can be interpreted in any way other than to say that I will deliberately not turn a cup that reveals the prize.