Consider you are given \(4\) distinct numbers and want the sum of any two of them to be a square. Euler gave the solution \(\{722, 432242, 2814962, 3246482\}\). In fact, there is another solution \(\{2, 359, 482, 3362\}\) that has a smaller largest value. Furthermore, this solution has the smallest largest value and we call such a solution minimal. Suppose instead of \(4\) numbers we have \(n\) numbers. OEIS lists minimal solutions up to \(n=5\). In this problem we will consider finding a minimal solution when \(n=6\).
Please submit a CSV file that contains \(6\) numbers. Call this set of \(6\) numbers \(\mathcal{A}\). In order to be scored the sum of any two of these numbers must be a square. Your goal is to minimize $$\max\{\mathcal{A}\}$$
The following is a Python implementation of the score function to help you get started.
from math import isqrt
from itertools import combinations
def score(nums: [int]):
N = 6
# Not the correct size.
if len(nums) != N:
return None
# Not distinct
if len(nums) != len(set(nums)):
return None
for x, y in combinations(nums, 2):
# A pair did not sum to a square
if isqrt(x + y) ** 2 != x + y:
return None
return max(nums)