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Two questions from a junior tournament today
Interesting. On the day, the arbiter's take on question 1 was that picking up the clock and "making the flag fall" was equivalent to playing an illegal move when your opponent has <5 minutes left and restarting your opponent's clock. White was awarded an extra 2 minutes, then checkmated black easily.

For question 2, your suggestion was exactly what happened. One of the other board stewards / managers was busy telling white "did you already agree it was checkmate - if you did the game is over" but the managers of the two players involved went with just going back to the point where the illegal move was made. 3 moves later, black won with a genuine checkmate.

For question 2, would the situation be different if white had stopped the clocks to show that they agreed that it was checkmate?

Which leads to hypothetical question 3, which I have seen happen in the past. It is stalemate but the players both agree that it is checkmate. My understanding is that if it's not stalemate, then agreeing that it's checkmate when it's not is equivalent to the loser resigning. However I would think that stalemate must end the game, making it impossible for a player to resign after the stalemate has happened? At what point is a controller no longer able to say that the result is actually a draw? I am imagining a scenario where the players may go off to hand in a results slip before resetting the pieces, but might also reset the pieces before filling in the results slip.
"Heather's clever book" - as plugged by the Rampant Chess team.
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