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The queen and king checkmate is a really simple checkmate. The key is to copy your opponent's moves!
First, move your queen a knight's distance away.
If the queen was a knight, that move would be a check. BUT DON'T GIVE ANY CHECKS!
This cuts the king off and forces him to the edge of the board, where you can finally deliver a checkmate.
Every time the king moves, move your queen the same way, like Simon Says.
When the opponent's king moves to the corner, don't follow him! That will be a stalemate!
Instead, keep your queen where she is and bring the king closer.
When your king is two squares away from the enemy king on your move your queen in to checkmate!
Like a queen sandwich!
Note: Moving to any of the green squares would also be checkmate.
Here's the whole checkmate if you wanted to see it:
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