That is clever! Before, I did F1F1 = 241 x 257 in my head, but I couldn't do 3 digit by 3 digit multiplication in my head without a high risk of mistakes. I really should have thought of the 8^2 thing. Thanks for sharing your solution!
yeah, kind of did the same as 81929128's last thought: F*(1010) + 101->15*(4096+16) + 257 ... though confess to having to write down a couple of intermediate stages in the answer box as I was going!
F1F1 = F01 x 101 (both in base 16) = 241 x 257
Then I used a bunch of mental calculation tricks:
241 x 257 = 249^2 - 8^2
249^2 = 248x250 + 1 = 62x1000 + 1 = 62001
so F1F1 = 62001 - 64 = 61937
Took me about 2 minutes because I subtracted one instead of adding one the first try :P
Thanks for the puzzle, haha