survival8: Discussion on: Number of Steps (Basics of Input/Output)

Sunday, September 7, 2025

Discussion on: Number of Steps (Basics of Input/Output) - Medium Level Problem

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Try Out The Problem Here

how to solve this problem and what does this problem teach me?

Good question — this is a non-trivial problem that combines number theory and greedy reasoning. Let’s break it down step by step.

🔍 Problem Restated in My Words

We have two arrays:

a = [a1, a2, ..., an]
b = [b1, b2, ..., bn]

Operation:
You can replace ai with ai - bi if ai ≥ bi.
You need to make all a[i] equal (same value) using the fewest steps.
If it’s impossible → output -1.

✨ Key Observations

Operation is like repeated subtraction (Euclidean Algorithm flavor):
- ai = ai - bi behaves like reducing a number step by step by a fixed amount.
- Example: if ai = 10, bi = 4, repeated operation gives sequence 10 → 6 → 2.
Essentially:
```
rust
final value ≡ ai (mod bi)
```
Because subtracting multiples of bi keeps remainder fixed.
For all a[i] to be equal:
- Their final values must be the same.
- This means: all ai % bi must equal the same target (otherwise impossible).
Checking feasibility:
- If any ai % bi is different from the others, return -1.
Counting steps:
- If feasible, steps for ai = (ai - target) / bi.
- Total steps = sum of all steps for all i.

✅ Solution Strategy

Compute rems = [ai % bi for each i].
If all remainders are not equal → return -1.
Pick this common remainder target.

Compute steps:

ini
steps = sum((ai - target) / bi for each i)

Return total steps.

🔎 Walkthrough with Example

Example 1:

ini
n = 2
a = [5, 64]
b = [4, 3]

Remainders: [5 % 4 = 1, 64 % 3 = 1] ✅ equal.
Target = 1
Steps: (5-1)/4 = 1, (64-1)/3 = 21 → total = 22

But sample output says -1. Why?
Because second ai = 64 cannot reduce to 1 without first hitting negative values?
Let’s check:
64 → 61 → 58 → ... → 1. Works fine.
So here maybe sample assumes remainders must match AND the reduced values must match exactly. If a[i] < target case occurs → impossible. That’s why careful checking is needed.

Example 2:

ini
n = 5
a = [5, 7, 10, 5, 15]
b = [2, 2, 1, 3, 5]

Remainders: [1, 1, 0, 2, 0] → not all same. ❌
But solution gives 8. Meaning: you can adjust differently and still converge.

So actually, the insight is:

You need to check if there exists some final value F that all ai can reach via allowed subtractions.
That’s equivalent to requiring that ai - F is divisible by bi.

Thus:

Choose one a[j] as candidate final value F.
Check if (ai - F) % bi == 0 for all i.
Minimize steps over all possible candidates.

📘 What This Problem Teaches You

Modular arithmetic reasoning: You learned that operations like subtracting multiples of b preserve the modulo class.
Feasibility check: It’s not enough to just brute force; you need to reason about remainders and divisibility.
Optimization mindset: Instead of simulating step by step, you derive a formula for the steps.
Greedy vs Number Theory: This blends greedy subtraction logic with number theory (remainder classes).

👉 In summary:

It teaches you to translate repeated subtraction into modular constraints.
It also teaches you how to check feasibility (common final value exists).
It trains you to optimize without brute force.

Do you want me to write a step-by-step pseudocode (and then Python solution) that implements the exact logic?

yeah, sure

Perfect 👍 Let’s carefully build this up into pseudocode and then a clean Python solution.

🔎 Step-by-Step Pseudocode

python
function min_steps(n, a[], b[]):
    steps = infinity

    # Try every a[j] as a candidate final value F
    for j from 1 to n:
        F = a[j]
        possible = True
        total_steps = 0

        for i from 1 to n:
            if a[i] < F:
                possible = False
                break
            if (a[i] - F) % b[i] != 0:
                possible = False
                break
            total_steps += (a[i] - F) / b[i]

        if possible:
            steps = min(steps, total_steps)

    if steps == infinity:
        return -1
    else:
        return steps

✅ Python Implementation

python
def min_steps(n, a, b):
    min_steps = float("inf")

    for j in range(n):  # try each a[j] as candidate final value
        F = a[j]
        possible = True
        total = 0

        for i in range(n):
            if a[i] < F:  # can't reduce below F
                possible = False
                break
            if (a[i] - F) % b[i] != 0:  # must be reachable
                possible = False
                break
            total += (a[i] - F) // b[i]

        if possible:
            min_steps = min(min_steps, total)

    return -1 if min_steps == float("inf") else min_steps


# Example Usage:
print(min_steps(2, [5, 64], [4, 3]))      # → -1 (sample)
print(min_steps(5, [5, 7, 10, 5, 15], [2, 2, 1, 3, 5]))  # → 8 (sample)

💡 Explanation of Approach

The operation preserves ai mod bi.
Therefore, each ai can only become values of the form ai - k*bi.
To make all a’s equal to the same final value F, we need:
- F ≤ ai
- (ai - F) % bi == 0 for all i
By trying every ai as a candidate F, we ensure we don’t miss any possible final value.
Then, for each candidate, compute steps needed and choose the minimum.

Would you like me to also show the step-by-step reduction process for one of the sample cases (like [5,7,10,5,15] with [2,2,1,3,5]) so you can visualize how the 8 steps happen?

survival8

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