Question

For all non-negative integers \( x \) and \( y \), \( f(x, y) \) is defined as below:
\( f(0, y) = y + 1 \)
\( f(x + 1, 0) = f(x, 1) \)
\( f(x + 1, y + 1) = f(x, f(x + 1, y)) \)
Then, what is the value of \( f(1, 2) \)?

• A. Two
• B. Four
• C. Three
• D. Cannot be determined

Hint:

Apply the three rules repeatedly, working from the innermost bracket outwards; the rule for \( f(0,y) \) ends the recursion.

Answer 1

The Correct Option is B

Step 1: Reduce \( f(1,2) \) using the third rule.
\( f(1,2) = f(0,\, f(1,1)) \).

Step 2: Find \( f(1,1) \).
\( f(1,1) = f(0,\, f(1,0)) \).

Step 3: Find \( f(1,0) \) using the second rule.
\( f(1,0) = f(0,1) = 1 + 1 = 2 \).

Step 4: Work back up.
\( f(1,1) = f(0,2) = 2 + 1 = 3 \).
\( f(1,2) = f(0,3) = 3 + 1 = 4 \).

The value is a definite number, so "Cannot be determined" is wrong.

Final Answer:
The correct option is (2): Four