project_euler.problem_190.sol1

Project Euler Problem 190: https://projecteuler.net/problem=190

Maximising a Weighted Product

Let S_m = (x_1, x_2, …, x_m) be the m-tuple of positive real numbers with x_1 + x_2 + … + x_m = m for which P_m = x_1 * x_2^2 * … * x_m^m is maximised.

For example, it can be verified that |_ P_10 _| = 4112 (|_ _| is the integer part function).

Find Sum_{m=2}^15 = |_ P_m _|.

Solution: - Fix x_1 = m - x_2 - … - x_m. - Calculate partial derivatives of P_m wrt the x_2, …, x_m. This gives that

x_2 = 2 * x_1, x_3 = 3 * x_1, …, x_m = m * x_1.

  • Calculate partial second order derivatives of P_m wrt the x_2, …, x_m. By plugging in the values from the previous step, can verify that solution is maximum.

Functions

solution(→ int)

Calculate sum of |_ P_m _| for m from 2 to n.

Module Contents

project_euler.problem_190.sol1.solution(n: int = 15) → int

Calculate sum of |_ P_m _| for m from 2 to n.

>>> solution(2)
1
>>> solution(3)
2
>>> solution(4)
4
>>> solution(5)
10