Brainteaser 1805: Find the deuce

From The Sunday Times, 20th April 1997 [link]

Adam, Bill, Charlie and Dave play a game using the ace (counting as 1) and 2 to 10 of spades. These cards are shuffled and placed face down on the table. Each player in turn draws one card and declares whether it is odd or even. Each then in turn draws a further card and must declare a number being either the sum or the product of the two cards held. The winner is the first to deduce which cards remain on the table. They are all perfect logicians, draw every possible inference, and rely on the fact that the others do too.

In a recent game the calls went as follows:

Adam: “odd”
Bill: “odd”
Charlie: “odd”
Dave: “odd”
Adam: “6”
Bill: “12”
Charlie: “15”

But as Dave reached for his second card, a player claimed the game.

Who was the winner and where was the 2?

This puzzle was included in the book Brainteasers (2002, edited by Victor Bryant). The puzzle text above is taken from the book.