1981 IMO Problems/Problem 6

Problem

The function $\displaystyle f(x,y)$ satisfies

(1) $\displaystyle f(0,y)=y+1,$

(2) $\displaystyle f(x+1,0)=f(x,1),$

(3) $\displaystyle f(x+1,y+1)=f(x,f(x+1,y)),$

for all non-negative integers $\displaystyle x,y$ . Determine $\displaystyle f(4,1981)$ .

Solution

We observe that $\displaystyle f(1,0) = f(0,1) = 2$ and that $\displaystyle f(1, y+1) = f(1, f(1,y)) = f(1,y) + 1$ , so by induction, $\displaystyle f(1,y) = y+2$ . Similarly, $\displaystyle f(2,0) = f(1,1) = 3$ and $\displaystyle f(2, y+1) = f(2,y) + 2$ , yielding $\displaystyle f(2,y) = 2y + 3$ .

We continue with $\displaystyle f(3,0) + 3 = 8$ ; $\displaystyle f(3, y+1) + 3 = 2(f(3,y) + 3)$ ; $\displaystyle f(3,y) + 3 = 2^{y+3}$ ; and $\displaystyle f(4,0) + 3 = 2^{2^2}$ ; $\displaystyle f(4,y) + 3 = 2^{f(4,y) + 3}$ .