jmc

Alex wants to minimize the number of coins he gives to his friends without giving any two of them the same number of coins. The minimum number of coins he can give to a friend is 1. He then gives 2 coins to another friend, then 3 to another, then 4, and so on, until the last friend receives 12. The total number of coins Alex has given away is $1+2+3+\cdots +12=\frac{12\cdot 13}{2}=78$. Thus, Alex needs $78-63=\boxed{15}$ more coins.