Santa Claus tells the three super-smart elves Alpha, Beta, and Gamma, “Each of you has received a card that carries a number from 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. Your three numbers are pairwise distinct, and the largest one equals the sum of the two smaller ones. Now, please have a look at your cards with your number, but don't show it to the two others!”

Alpha, Beta and Gamma stare at their cards for some time.

- After some time
**Alpha**says, “According to my knowledge, there are at most eight possible candidates for Beta's number.” - Then,
**Beta**says, “According to my current knowledge, there are exactly three possible candidates for Gamma's number.” **Gamma**shouts, “I see! Now, I know Alpha's number.”**Alpha**thinks about it and says, “I still don't know Beta's number.”**Beta**exclaims, “I see! Now, I also do know Alpha's number.”

Of course, we would like to know: What is Alpha's number?

Calculations are required!