Solvers included Bill Daly, Bernard McCabe, Dane Brooke, Stephen Kloder, and Jon Palin

Joseph DeVincentis -- The 12 cards are 0, 1, 2, 2, 3, 3, 4, 4,
5, 5, 6, 7. You were probably trying to make a set of cards so that the sums
when you draw 3 approach the distribution of rolling 3 6-sided dice. The result
is remarkably similar but the extremes are a little more likely.

Andy Juell -- While I'm not positive that this solution is unique, a random selection of any three of these 12 integers without replacement gives the same distribution of sums as the one you listed. 0, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 7 Thanks for the great site!