insufficient material
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Is it a rule that a game would be considered insufficient material when one even with queen on board fails to checkmate an opponent's king when it's the only piece left, after 6 moves?
No. Insufficient material refers to a draw where your opponent has run out of time, but you lack material to checkmate him so it's a draw. USCF and chess.com don't consider "helpmates" valid but FIDE apparently does. Of course if the guy with a Q runs out of time and all his opponent has left is his K, then it's insufficient material and a draw...Hope that made sense. If not, you could try a forum search for "insufficient material"....like this...
https://www.chess.com/forum/search?keyword=insufficient+material