Tom and Ray said that there was no algebraic solution to this puzzler. But there is. Here is the condensed version:
Variable Description Domain
F1 Fahrenheit Temp: 10s digit integer
F2 Fahrenheit Temp: 1s digit integer (1,2)
F3 Fahrenheit Temp: Decimal [0,1)
C1 Celcius Temp: 10s digit integer (1,2)
C2 Celcius Temp: 1s digit integer
Identities
F = 10F1 + F2 + F3
C = 10C1 + C2
F1 = C2
F2 = C1
F = 1.8C + 32
solve for F1
F1 = (17F2 - F3 + 32) / 8.2
F2 = 1
F1 = 5.97560975609756 --> 6
F2 = 2
F1 = 8.04878048780488 --> 8
check
F C
61 16
82 27
Sorry - wrong forum