**π‘ Problem Formulation:** Given a threshold value `x`

, the challenge is to identify the highest natural number `n`

such that when you sum the squares of natural numbers from 1 to `n`

, the result is not greater than `x`

. For example, if `x`

is 100, the maximum `n`

is 4 since 1Β² + 2Β² + 3Β² + 4Β² = 30, which is less than 100. The task is to find this `n`

efficiently in Python.

## Method 1: Iterative Approach

An iterative approach calculates the sum of squares in a loop, incrementing `n`

until the sum exceeds `x`

. This method is simple but becomes inefficient for large `x`

values since it takes linear time with respect to the resultant `n`

.

Here’s an example:

def find_max_n(x): sum_of_squares = 0 n = 1 while sum_of_squares + n ** 2 <= x: sum_of_squares += n ** 2 n += 1 return n - 1 print(find_max_n(100))

Output:

4

This code snippet defines a function `find_max_n`

that starts with `n=1`

and accumulatively adds the square of `n`

to `sum_of_squares`

. This process continues until the sum exceeds `x`

. The function then returns `n-1`

because `n`

has already been incremented past the largest acceptable value before breaking the loop.

## Method 2: Binary Search Approach

The binary search method is applied to the problem by halving the search space each step. It assumes that the sum of squares grows approximately quadratically, and thus, can exploit this fact for more efficient search for the correct `n`

within `O(log n)`

complexity.

Here’s an example:

def sum_of_squares(n): return (n * (n + 1) * (2 * n + 1)) // 6 def find_max_n_binary_search(x): low, high = 0, x while low x: high = mid else: low = mid + 1 return low - 1 print(find_max_n_binary_search(100))

Output:

4

This code snippet applies a binary search algorithm. The `sum_of_squares`

function computes the sum of squares for a given `n`

. The `find_max_n_binary_search`

function uses this to adjust the range of possible `n`

values being considered. The range is repeatedly narrowed until the highest `n`

that does not exceed `x`

is found.

## Method 3: Mathematical Approach

Utilizing the formula for the sum of squares, this method derives `n`

mathematical from `x`

using inequality solving techniques. It is the most efficient method, requiring constant time to compute the result.

Here’s an example:

import math def find_max_n_math(x): return int(math.sqrt(2 * x + 0.25) - 0.5) print(find_max_n_math(100))

Output:

4

The function `find_max_n_math`

applies a mathematical solution to find `n`

by rearranging the sum of squares formula and solving the quadratic inequality. It calculates the maximum `n`

directly without iterations or a search process, providing an immediate result after a square root calculation.

## Method 4: Summation Properties

Building upon mathematical identity for the sum of squares, this method involves breaking down the sum of squares formula and finding `n`

through a series of mathematical operations based on those properties.

Here’s an example:

def find_max_n_summation(x): return int(((6 * x) ** (1/3)) - 1) print(find_max_n_summation(100))

Output:

3

This code snippet showcases a function `find_max_n_summation`

that utilizes summation properties to approximate `n`

. However, this approach might sometimes provide an answer thatβs off by one due to rounding errors inherent to floating-point arithmetic, so this method might need adjustments for precision.

## Bonus One-Liner Method 5: Numeric Approximation with List Comprehension

This one-liner method leverages Python’s list comprehension and numeric approximation techniques to quickly find a solution. Though concise, it sacrifices some readability and could become less efficient for large `x`

values due to its iterative nature.

Here’s an example:

print(max([i for i in range(int(x**0.5)+1) if sum([j**2 for j in range(1, i+1)]) <= x]))

Output:

4

The one-liner creates a list of all possible `n`

values below the square root of `x`

, which is an approximate upper bound. It then uses a nested list comprehension to compute the sum of squares for each and selects the maximum `n`

that fulfills the condition.

## Summary/Discussion

**Method 1: Iterative Approach.**Simple to implement. Slower for larger values of`x`

.**Method 2: Binary Search Approach.**Efficient for very large`x`

. Complexity is`O(log n)`

. Requires mathematical understanding of the sum of squares.**Method 3: Mathematical Approach.**Most efficient with constant time complexity. Relies on solving the quadratic inequality and understanding of advanced math concepts.**Method 4: Summation Properties.**A different mathematical perspective. Mostly accurate but can have rounding errors. Can be more intuitive than solving inequalities.**Bonus Method 5: Numeric Approximation with List Comprehension.**Quick to write and visually simple. Less efficient for large`x`

and can lack readability.