Consider the recursive functions represented by the following code segment:…
2026
Consider the recursive functions represented by the following code segment:
The smallest positive integer n for which foo(n) returns 5 is ______. (answer in integer) Note: Ignore syntax errors (if any) in the function.
Attempted by 37 students.
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Correct answer: 65536
First, analyze the function bar(n). It returns the number of times n can be divided by 2 before reaching 1. Mathematically, bar(n) = floor(log2(n)). For example, bar(1)=0, bar(2)=1, bar(4)=2, bar(8)=3.
Next, analyze foo(n). It adds 1 for each recursive step: foo(n) = 1 + foo(bar(n)). We want foo(n) = 5.
This implies foo(bar(n)) = 4, foo(bar(bar(n))) = 3, foo(bar(bar(bar(n)))) = 2, and foo(bar(bar(bar(bar(n))))) = 1.
Since foo(1) = 1, we need bar(bar(bar(bar(n)))) = 1.
Working backwards: bar(x) = 1 for x in [2, 3]. bar(y) = 2 for y in [4, 7]. bar(z) = 3 for z in [8, 15].
Continuing, bar(w) = 4 for w in [16, 31] and bar(w) = 15 for w in [32768, 65535]. So bar(n) must be in [16, 65535].
Finally, bar(n) = 16 for n in [65536, 131071]. The smallest positive integer n is 65536.