Just six?
Yeah, I couldn't resist. The plot is the card version of the "triple prediction" using the principle that Rev mentioned. Here are some sources of similar routines:
1. Magician's Willpower,
Art of Magic, p. 176 (1909) (Magician names cards and spectator correctly selects -- Attributed to David Devant's
Magic Made Easy (1903))
2. Cards of Chance,
Encyclopedia of Card Tricks, p. 21 (1937) (Written predictions match three named cards that were then selected from the deck).
3. Hummer Detection,
Encyclopedia of Card Tricks, p. 159 (1937) (Magician finds three selected cards including the first one that is inserted to a deck and shuffled by the spectator).
4. Wizard's Will,
Encyclopedia of Card Tricks, p. 224 (1937) (Magician calls cards that the spectator selects them using full deck stack).
5. Marlo's Think a Card,
Early Marlo, p. 26 (1941) (Thought of cards predicted on billets).
6. Calling the Cards,
Scarne on Card Tricks , p. 9 (1950) (Magician calls cards as spectator points to them).
7. Best Card Trick,
Secret Ways of Al Baker, p.683 (2003 publication but Baker's original work is much older) (Using stack and billets)
8. Wonderful Four Cards and Pellet Mystery,
Secret Ways of Al Baker, p.760 (2003 publication but Baker's original work is much older) (Using stack, Svengali and billets crumpled into pellets) (also in
Lost Notebooks of John Norhern Hilliard, p. 53.
9. The Clairvoyant Spectator,
Secret Ways of Al Baker, p. 205 (2003) (naming cards to be pulled out by the spectator and using a marked deck)
10. Any Cards Called For,
Mnemonica, p. 211 (2004) (Spectator selects 3 cards, magician selects 1 - all cards named on open predictions and shown in order using unique Mexican Turnover within a packet)
11. Menage a Trois,
Hidden in Plain Sight p.80 (2005) (Magician finds two cards thought by spectators after looking at deck and mate for selected card put into card box).
I didn't find anything in writing that was exactly what
@Jack H1 said -- where the cards are used as the prediction placed down ahead of the spectator calling the card. However, it is close to Devant's / Down's Magician Willpower, Scarne's Calling the Cards and Al Baker's Clairvoyant Spectator except the roles are reversed in that the spectator selects the cards and the magician calls them out in those effects My sense is that the effect Jack is describing would have the magician look at the faces of the cards between each spectator naming the cards. That to me makes the effect potentially able to be reverse-engineered, especially in light of the need for a final phase. I'd prefer to do it using a stack or a marked deck.
My concern with those effects is with the final selection. There isn'at a good way to make it seem natural. I like the
Menage a Trois and
Hummer Detection effects because the first selection is done with the magician's hands off of the deck which can be accomplished using a stack, marked deck or a one-way deck. Heck, you could even use a force.
I'd be interested in a link to the AGT performance of the effect if you can find it.