Paired sums

What this combinatorial constraint problem tests

This is a medium-difficulty puzzle that combines strict ordering with algebraic constraints. It appears frequently in quant interview loops because it rewards clear logical reasoning and the ability to translate abstract conditions into a concrete, countable set.

The problem asks you to find all 6-tuples of positive integers that simultaneously satisfy a strict descending order and a set of pairwise sum equations. Success requires setting up the constraint system carefully, identifying the degrees of freedom, and then systematically enumerating valid solutions without over- or under-counting. The key is recognizing how the fixed sums interact with the ordering to restrict the parameter space.

• Systems of linear constraints
• Strict vs. non-strict inequalities
• Counting valid integer solutions in a bounded region
• Substitution and reduction of variables