I hadn't thought about this forever but did so today. I couldn't quite remember how my dad said it so I went to the internet for answers. This discussion has set me right. Thank you internet! lol
Venture Rinktums No returns (we would say the entire phrase) was said by the newly haircutted person to avoid the knuckle up the back of the head.Rinktums was said to allow the person saying it to inflict the knuckle. The game wasn't about pain. It was mostly a demonstration of "I'm more aware of the situation than you. I win."
My dad was born in 1916 in Dallas and graduated from Texas Tech in 1939.
He might have picked it up there in West Texas just as several here have previously mentioned their own West Texas roots.
This seems to be a very specific regional thing. I'm guessing, based on Texas' many German immigrants, that the origin somehow has it's roots in the German language. We are of German descent.