9.2 Analytical Problem Solving

Turning Clues into a Grid

Analytical problem solving on the CSE means constraint puzzles: you are told the members of a set and a handful of rules (the clues), and you must deduce a single valid arrangement - who is assigned where, what order things fall in, or which items group together. The reliable method is elimination on a grid: list every possibility, translate each clue into a mark, and let confirmed facts erase the rest. You never guess; you cross out until one arrangement survives. Because the answer is forced by the clues, a careful examinee can reach 100% certainty on these items.

The elimination method, step by step

1. List the categories and their members.
2. Draw a grid with one category on the rows and another on the columns.
3. Translate each clue into a mark: X for 'impossible', O for 'confirmed'.
4. Propagate: an O eliminates the rest of that row and column, since each member matches exactly one.
5. Read the surviving cells for the answer.

Worked Example 1 - Assignment grid

Three clerks - Ana, Ben, Carlos - are each assigned to exactly one division: Records, Budget, or Human Resources (HR). Clues: (1) Ana is not in Budget; (2) Carlos is not in Records and not in Budget; (3) Ben is not in HR. Mark the 'not' clues as X:

Carlos's row has X on Records and Budget, so Carlos = HR. HR is now taken, so cross out Ana-HR. Ana's row then has X on Budget and HR, so Ana = Records. Records and HR are gone, leaving Ben = Budget. Final: Ana-Records, Ben-Budget, Carlos-HR. The whole puzzle was solved from 'not' statements - negative clues are as powerful as positive ones.

Worked Example 2 - Ordering / ranking

Four applicants - W, X, Y, Z - each earned a different score. Clues: (1) X scored higher than Y; (2) Z scored lower than Y; (3) W scored higher than X. Chain the inequalities: from clues 1 and 3, W > X > Y; from clue 2, Y > Z. Combine into W > X > Y > Z. So the highest scorer is W and the lowest is Z, with X second. Ranking puzzles go fastest when you write each clue as a '>' chain and then link the chains at their shared member - here, Y.

Worked Example 3 - Grouping under a rule

A supervisor must send exactly two of four staff - Rey, Sol, Tina, Ugo - to a seminar, following two rules: if Rey goes, Sol must go; and Tina and Ugo cannot both go. Test 'Rey goes': then Sol goes too (that is already two), so Tina and Ugo both stay and the second rule holds - valid group {Rey, Sol}. Test 'Rey stays': pick two from Sol, Tina, Ugo, but Tina and Ugo cannot both go, so the only valid pairs are {Sol, Tina} or {Sol, Ugo}. Thus {Tina, Ugo} can never attend together, and if Rey attends, Sol must.

Worked Example 4 - Day-and-person schedule

Three inspectors - Lito, Mina, Noel - each conduct one inspection on a different day: Monday, Tuesday, or Wednesday. Clues: (1) Mina's inspection is before Noel's; (2) Lito does not inspect on Monday. From clue 2, Monday must be Mina or Noel. Clue 1 says Mina is before Noel, so Noel cannot be Monday (nothing precedes Monday) - therefore Mina = Monday. Lito is then Tuesday or Wednesday; testing Lito = Tuesday leaves Noel = Wednesday, and Mina (Mon) is still before Noel (Wed). The only schedule is Mina-Mon, Lito-Tue, Noel-Wed. An ordering clue plus one 'not' clue pinned the entire timetable.

The marks at a glance

When the clues do not fully solve it

Sometimes the clues leave two arrangements standing. When that happens, use a trial branch: assume one open possibility is true, follow it to the end, and see whether it breaks a clue. If it does, the other branch is the answer; if both survive, the correct exam choice is usually 'cannot be determined'. In Worked Example 3 the grouping had more than one valid pair, so the question could only ask what must or cannot happen - never a single forced group. Reading the stem for the words must, cannot, or could tells you whether one unique answer even exists before you invest time hunting for it.

Common traps

• Reading order from the clue order. Clue 1 is a comparison, not 'first place'.
• Ignoring negative clues. 'Carlos is not in Records' removed a cell and forced the solution.
• Adding assumptions. Use only what is written; never invent a preference that no clue states.
• Stopping too early. Keep propagating until every member has exactly one confirmed cell.

Work every constraint puzzle the same disciplined way - grid, translate, propagate, read - and even the 'hard'-tagged items on the Professional exam reduce to a few clean strokes of the pen.