I drew a 12 x 12 table, columns labelled 1 to 12, rows 1 to 12. In each cell I put the total of header + row where this is a Fibonacci number. We see straight away that in our solution 12 can be flanked only by 1 and 9, and then 11 only by 2 and 10. Looking at remaining numbers in succession we build up and join series of numbers.
I'd start with ---the numbers 1 to 12 add up to 78 so we need 6 pair's totaling 13 each.
I would then use the list of fibonachi numbers from Google or from some posts here