Problem 6

The function

f(x,y)

satisfies

f(0,y) = y+1,

f(x+1,0) = f(x,1),

f(x+1,y+1) = f\big(x, f(x+1,y)\big)

for all non-negative integers

x,y

. Determine

f(4,1981)

Step 6 of 7: Iterate the tower up to

y=1981

Previous step Next step

f(4,1981)+3 = \underbrace{2^{2^{\cdot^{\cdot^{2}}}}}_{1984 \text{ twos}}

Detailed analysis

Starting from $f(4,0)+3$ , a tower of height $3$ , and applying Step 5's rule $1981$ more times each adds one more $2$ to the tower; tracking the height, $f(4,1981)+3$ is a power tower of exactly $3+1981=1984$ twos.