If I got this right the "magic day" would be the day/month/year that his birthday falls on, in which his age matches up with the last 2 dates of the year he was born?
Well, here's one possible solution. The birth date fell on:
The birthday ("magic day") was:
If the above is his DOB then HIS magic day would be:
Otherwise, we'd need to know his age.
Obviously the day and month had to be the same in both dates. We should try 1/9 first, it's the most logical assumption as it's closest to us. Meaning it's September the 1st. Going through the trial and error method, we eventually find 1/9/39. We then check 1/9/78 because obviously the year has to be doubled for the person's age to match the last 2 digits. Low and behold, it's on a Friday. Then of course to find his magic day we go ahead of 1978 by, obviously, 78, to get 2056!
He is in fact still alive.
It took me a while to answer this, because it seemed to me as if the question wanted the DOB and the magic day, I didn't get the part about HIS magic day and the other person's magic day being his DOB.
If you go back farther than 1939 you can also find DOB: 1911 and Magic Day: 1922 making his birthday/magic day: 1944
But this won't fit the question as the question states his birthday on his magic day "will be" indicating that this Magic day is in future