Hi Dr. Foote,

The alien children have seven fingers.
This is assuming that the number of fingers (7) is the base of their numbering system just as 10 is the base of our numbering system. Changing the base to number of fingers (f) would yield
(f+3)+(f+5)=(3f+1) which can be solved to find f=7
(7+2)+(7+5)=(21+1)
22=22

6 x 3 = (2f+4) Check by plugging in the f=7 from the first equation
18 = (14+4)
18=18

f*f=f^2 holds true for any numbering system. 

I hope that's right.
Matt Hodges