Data Arrangement & Constraint-Based Puzzles — XAT Decision Making
Every other chapter in Decision Making tests calibrated judgment with no single provably correct answer. This one is the exception, and it's worth knowing that explicitly: a constraint puzzle has exactly one arrangement that satisfies every stated condition, and finding it is a matter of method, not judgment. Treat it like a QA problem that happens to live inside the DM section, not like a caselet.
1. What XAT actually asks
Data Arrangement & Constraint-Based Puzzles form a smaller but reliably scoring slice of Decision Making's 21 questions (see docs/exam-briefs/xat-2026-brief.md). A typical item gives you a small set of entities — people, projects, time slots, resources — and a list of conditions linking them (direct assignments, exclusions, relative positions, conditional rules), then asks one or more questions about the resulting arrangement.
Unlike the caselets covered in the previous two chapters, these puzzles are self-contained logic problems. There is no ambiguity in what counts as a correct answer, and no judgment call about proportionality or ethics — only whether you have correctly applied every stated constraint.
2. Why this chapter is different from the rest of DM
The previous two chapters were explicit that XAT DM has no algorithm and tests calibration under ambiguity. Constraint puzzles are the deliberate exception, and mistaking one for the other costs time in both directions: treating a puzzle like a caselet (looking for "the most reasonable" arrangement) wastes time on a problem that has an exact answer, while treating a caselet like a puzzle (hunting for a hidden formula) wastes time on a problem that doesn't have one.
The tell: if a question gives you discrete entities and explicit linking conditions ("X is not adjacent to Y," "if A is assigned Monday, B must be assigned Wednesday"), it's a puzzle — solve it with a grid. If it gives you a scenario, a decision-maker, and options describing different courses of action, it's a caselet — apply the previous two chapters' methods instead.
3. The grid method
- List every entity and every attribute being assigned (people, days, roles, whatever the puzzle links) in a simple grid or table before touching the conditions.
- Enter every direct, unconditional constraint first — "X is assigned to Monday" goes straight into the grid, no reasoning required.
- Enter every exclusion next — "Y is not assigned to Tuesday" marks a cell as impossible, narrowing future placements without yet fixing anything.
- Work through relative and conditional constraints last, in the order that lets each one eliminate the most possibilities. A constraint like "if A is first, then B is last" is only useful once you know something about A's position — sequence your reasoning to reach the conditions with the most leverage first.
- Re-scan all constraints against your near-final grid before answering. A single overlooked condition, applied only at the end, is the most common source of an entirely correct-seeming but wrong solution.
4. The four recurring puzzle shapes
| Shape | What it looks like | Key technique |
|---|---|---|
| Linear/sequential arrangement | Entities placed in an ordered sequence (days of a week, positions in a queue, ranks) | Track fixed positions first, then work outward from them using relative constraints |
| Circular/relative arrangement | Entities placed around a table or in a cycle, defined by relative position, not absolute rank | Fix one entity's position arbitrarily as a reference point, since circular arrangements have no true "first" position |
| Grouping/assignment | Entities distributed into categories or paired with resources (people to teams, tasks to days) | Build a grid with entities on one axis and categories on the other; mark constraints as confirmed, excluded, or undetermined |
| Conditional/if-then | Rules that only activate under a specific condition ("if X does A, then Y cannot do B") | Solve unconditional constraints first to establish as much of the grid as possible before testing which conditional rules actually activate |
Worked example
Puzzle: Five employees — P, Q, R, S and T — are each assigned to exactly one of five weekdays (Monday through Friday) for a mandatory training session, one employee per day. The following conditions apply:
- P is scheduled two days before S.
- Q is not scheduled on Monday or Friday.
- R is scheduled the day immediately after T.
- S is not scheduled on Friday.
Q1. On which day is T scheduled?
Pick an option to check your answer.
Show explanation
Solution. Start with the strongest constraint: "P is two days before S" fixes a pair with a gap of exactly one day between them. Since S is not on Friday, and P must be two days before S, the possible (P, S) pairs are (Monday, Wednesday), (Tuesday, Thursday) only — (Wednesday, Friday) is excluded since S can't be Friday.
Test (Tuesday, Thursday) for P and S: that leaves Monday, Wednesday, Friday for Q, R, T. Q cannot be Monday or Friday, so Q must be Wednesday, leaving R and T to split Monday and Friday. But "R immediately after T" needs consecutive days, and Monday/Friday aren't consecutive — this branch fails entirely.
Test (Monday, Wednesday) for P and S: that leaves Tuesday, Thursday, Friday for Q, R, T. Q cannot be Friday, so Q is Tuesday or Thursday. "R immediately after T" needs a consecutive pair from {Thursday, Friday} (Tuesday isn't consecutive with either remaining day once Q takes one of them) — so T is Thursday and R is Friday, leaving Q as Tuesday. Full arrangement: P-Monday, Q-Tuesday, S-Wednesday, T-Thursday, R-Friday. Answer: (d) Thursday.
Q2. If R were instead scheduled on Wednesday, which employee would be scheduled on Thursday?
Pick an option to check your answer.
Show explanation
Solution. The only valid base arrangement found in Q1 has R fixed on Friday — every other constraint (P-Monday, S-Wednesday's slot, Q's Monday/Friday exclusion, T-immediately-before-R) was shown to force that specific arrangement uniquely, with no alternative branch surviving.
Since S already occupies Wednesday in the only valid arrangement, R cannot be moved there without violating either the P-S gap constraint or creating a conflict with S's fixed position. No valid arrangement satisfies the original four conditions with R on Wednesday. Answer: (e).
6. Common traps
- Solving conditions in the order they're listed, rather than the order that eliminates the most possibilities. Starting with the loosest, most flexible constraint wastes time exploring branches a stronger constraint would have closed immediately.
- Forgetting to re-check all constraints against the final grid. A puzzle can look solved after satisfying the last constraint you worked with, while silently violating one you fixed earlier and never revisited.
- Missing that a constraint only applies conditionally. "If A is first, B is last" says nothing when A isn't first — treating it as an unconditional rule forces false eliminations.
- Assuming a unique solution exists when the puzzle asks about a hypothetical variation. As Q2 shows, a follow-up "what if" condition can have no valid arrangement at all — "no valid arrangement exists" is sometimes the correct answer, not a sign of a mistake.
- Confusing this chapter's puzzles with the caselets elsewhere in Decision Making, and either searching a puzzle for "the most reasonable" answer or hunting a caselet for a hidden exact rule — recognise the tell from Section 2 immediately.
- Not using a grid at all, and trying to track five or six linked conditions mentally — this is the single most common cause of running out of time on an otherwise solvable puzzle.
7. When to guess, and why
Within your first 8 skips across all of Part 1, a constraint puzzle you haven't solved with a grid in a reasonable time is a good candidate to skip — unlike a caselet, more time spent almost always does eventually resolve a puzzle, but that time may not be available.
Beyond your 8th skip, a blind 1-in-5 guess has an expected value of exactly 0, while a blank costs -0.10 — but here, unlike ambiguous caselets, even a partially completed grid often eliminates two or three options outright, making a late guess meaningfully better than blind chance.
Summary
- Constraint puzzles are the one part of Decision Making with a single, determinate correct answer — treat them like a QA problem, not a caselet.
- The tell: discrete entities plus explicit linking conditions means a puzzle (use a grid); a scenario plus a decision-maker plus courses of action means a caselet (use the previous two chapters' methods).
- Grid method: list entities and attributes, enter direct constraints first, then exclusions, then relative/conditional constraints in order of leverage, then re-scan everything against the final grid.
- Four recurring shapes: linear/sequential, circular/relative, grouping/assignment, conditional/if-then — each has a specific starting technique.
- Solve the strongest, most restrictive constraints first — they eliminate the most branches fastest.
- "No valid arrangement exists" is sometimes the genuinely correct answer to a hypothetical variation, not evidence of an error.
- Past your 8th free skip in Part 1, a partially completed grid usually eliminates enough options to make a late guess meaningfully better than blind chance.
