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 5 of 7: Set up the

x=4

level as a power tower

Previous step Next step

f(4,0)+3 = 2^{2^{2}}=16,\quad f(4,y+1)+3 = 2^{\,f(4,y)+3}

Detailed analysis

By rule (2) and Step 4, $f(4,0)=f(3,1)=2^{1+3}-3=13$ , so $f(4,0)+3=16=2^{2^{2}}$ , a power tower of three $2$ s (height $0+3=3$ ). By rule (3) and Step 4, $f(4,y+1)=f(3,f(4,y))=2^{f(4,y)+3}-3$ , so $f(4,y+1)+3=2^{f(4,y)+3}$ : each step in $y$ adds one more $2$ to the tower.