Number of Steps to Reduce a Number to Zero

Number of Steps to Reduce a Number to Zero

What is this problem about?

The Number of Steps to Reduce a Number to Zero problem asks: starting from a positive integer n, repeatedly perform: if n is even, divide by 2; if n is odd, subtract 1. Count the total number of steps until n becomes 0. This easy coding problem directly maps to counting set bits and total bits in the binary representation of n.

Why is this asked in interviews?

Microsoft, Amazon, Google, Bloomberg, and Hudson River Trading ask this easy problem because it has an elegant bit manipulation observation: each bit position requires one step (either a shift for '0' or a subtract for '1', plus an extra shift for '1' bits except the leading one). The math and bit manipulation interview pattern is the core.

Algorithmic pattern used

Bit counting formula. The number of steps = (number of bits in n's binary representation - 1) + (number of set bits in n's binary representation). Because: each '0' bit costs 1 step (shift), each '1' bit costs 2 steps (subtract then shift) except the leading '1' which costs 1 step (the final subtract to 0).

Simulation alternative: Just simulate the process: while n > 0, if even: n >>= 1; else: n -= 1; count++. O(log n) time.

Example explanation

n=14 (binary: 1110). Bits = 4, set bits = 3. Steps = (4-1) + 3 = 6. Verify: 14→7(odd,sub→13... wait 14 even→7, 7→6(odd-1→6), 6→3, 3→2, 2→1, 1→0. That's 6 steps ✓.

n=8 (binary: 1000). Bits=4, set bits=1. Steps = (4-1)+1=4. Verify: 8→4→2→1→0. ✓

Common mistakes candidates make

• Simulating without recognizing the O(log n) pattern.
• Using the formula incorrectly (forgetting the "-1" for the leading bit).
• Off-by-one: the leading '1' bit costs 1 step not 2 (final step to 0).
• Not handling n=0 (0 steps needed).

Interview preparation tip

This problem perfectly illustrates the "simulation vs formula" trade-off. The simulation is simple and correct for reasonable n. The formula (bit_length - 1) + popcount(n) requires deriving from first principles — write it out for n=5,6,7 to verify. Both approaches should be in your toolkit. For interviews, know the simulation (easier to explain), and mention the formula as an optimization — it demonstrates mathematical thinking that impresses interviewers.