A through I sum puzzle

This puzzle asks you to work with nine distinct numbers arranged in a line and tied together by several overlapping sum conditions. Each condition involves a consecutive block of variables, and every block must add up to the same target value. Because the variables are required to be a permutation of a fixed small set of integers, you are forced into a very tight combinatorial structure where local constraints interact strongly with the global one. The challenge is to narrow down which assignments are possible, and to see if any variables are uniquely determined or if multiple consistent configurations exist.

To tackle it, you need comfort with linear relationships and with subtracting equations to exploit the overlaps between sums. The problem leans on recognizing invariants, cancelling common terms, and spotting how certain positions in the chain are more constrained than others. An interviewer would be looking for systematic reduction of the system, clear reasoning about symmetry and uniqueness, and disciplined case checking rather than brute-force enumeration or guesswork.

What it tests

When a set of variables is constrained by overlapping linear sum equations, the key is to recognize how subtracting these equations isolates small groups of variables, often revealing fixed relationships or constants between them. This is because the overlap means each variable appears in multiple equations, so differences between equations cancel out shared terms and highlight the incremental changes. The structure is governed by the way the variables are 'chained' together through these overlaps, and the total sum of all variables (if known) can be used as a global constraint to further limit possibilities. The principle holds because linearity ensures that each subtraction reduces the dimensionality of the system, turning a complex web of constraints into a series of manageable, often pairwise, relationships. This approach works broadly for problems where variables are arranged in sequences or cycles with overlapping group sums, and the sum of all variables is fixed or can be deduced.