Name: Rob

Who is asking: Student
Level: All

Question:
Let f be a function whose domain is a set of all positive integers and whose range is a subset of the set of all positive integers with these conditions:

find f(955)

This seems to be a start:

1. We cannot have f(1) = 1, because then we would have f(f(1)) = f(1) = 1, but we know that f(f(1)) = 3 by rule b. Therefore f(1) > 1, and you can then use rule a and induction to prove that f(n) > n for all n.
   
   
2. We have f(f(1)) = 3, and by applying the rule f(n) > n to n = f(1), we get 3 = f(f(1)) > f(1) > 1. Therefore f(1) = 2.
   
   
3. Thus, f(1) = 2, f(2) = f(f(1)) = 3, and f(3) = f(f(2)) = 6. From this we get f(6) = f(f(3)) = 9; but since the sequence f(3) = 6, f(4), f(5), f(6) = 9 is an increasing sequence, we must have f(4) = 7, f(5) = 8.
   
   
4. Now we have the wheel going: We can deduce f(7) = f(f(4)) = 12, f(8) = f(f(15)) = 15 and f(9) = f(f(6)) = 18. We don't have f(10) right away, but we know that f(12) = f(f(7)) = 21, and this will force a value on f(10).

... From here, you are on your on; you must go up to f(955) in that way,

or find shortcuts.