To solve this problem, we need to find the minimal number of operations to reduce a given number (N) to 1. The allowed operations are:

1. If (N) is even, divide it by 2.
2. If (N) is odd, either add 1 or subtract 1.

Approach

The key insight to minimize operations lies in how we handle odd numbers:

• For even numbers: Directly divide by 2 (right shift in binary, which reduces the number quickly).
• For odd numbers:
  • If (N = 3), subtract 1 (since (3 \rightarrow 2 \rightarrow 1) is optimal).
  • If the second least significant bit (bit 1, 0-indexed) is set (i.e., (N & 2 != 0)), add 1. This flips trailing 1s to 0 and carries over, leading to more trailing zeros (allowing more divisions).
  • Otherwise, subtract 1 (this makes the number even with fewer trailing 1s).

Solution Code

def min_operations(N):
    steps = 0
    while N > 1:
        if N % 2 == 0:
            N = N // 2
        else:
            if N == 3:
                steps += 2
                break
            if (N & 2):
                N += 1
            else:
                N -= 1
        steps += 1
    return steps

Explanation

• Even Numbers: Dividing by 2 is the most efficient way to reduce the number, so we do this directly.
• Odd Numbers:
  • Special Case (N=3): Subtract 1 to get to 2, then divide by 2 to reach 1 (2 steps total).
  • Second Bit Set: Adding 1 turns trailing 1s into 0s (e.g., (15 \rightarrow 16)), which allows multiple divisions (16 → 8 →4 →2 →1, 4 steps after adding 1).
  • Second Bit Not Set: Subtract 1 to get an even number with fewer trailing 1s (e.g., (9 \rightarrow8), then 8→4→2→1, 3 steps after subtracting 1).

This approach ensures we take the minimal steps by leveraging binary properties, leading to an efficient (O(\log N)) time complexity (since each step reduces the number by at least half).

Example Test Cases:

• (N=7): (7→8→4→2→1) (4 steps).
• (N=9): (9→8→4→2→1) (4 steps).
• (N=15): (15→16→8→4→2→1) (5 steps).
• (N=3): (3→2→1) (2 steps).

This solution efficiently handles all cases, including large values of (N) (like (10^9) or (2^{63}-1)).)

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