Logical Puzzles: Matching and Allocation

Matching and allocation puzzles require you to assign items from one category to items in another category based on a set of constraints. These puzzles are common in the UCAT Decision Making subtest and are distinct from ordering problems because they involve pairing rather than sequencing. This lesson teaches you the grid method — the most reliable technique for solving these puzzles under time pressure.

---

What Are Matching and Allocation Puzzles?

These puzzles typically involve:

• Assigning people to roles, tasks, or locations (e.g., three doctors to three wards)
• Pairing items from two categories (e.g., matching five students to five projects)
• Allocating resources (e.g., distributing equipment across departments)

The constraints typically include:

• Direct assignments ("A is assigned to Ward 1")
• Exclusions ("B is NOT assigned to Ward 2")
• Conditional rules ("If C is in Ward 3, then D is in Ward 1")
• Capacity constraints ("Each ward has exactly one doctor")

---

The Grid Method

The grid method is the most efficient approach for matching puzzles. Here is how it works:

Step 1: Create a Grid

Draw a table with one category along the rows and the other along the columns.

	Ward 1	Ward 2	Ward 3
Dr A
Dr B
Dr C

Step 2: Mark Eliminations

For each constraint that excludes a pairing, place an ✗ in the corresponding cell.

Step 3: Mark Confirmations

For each constraint that confirms a pairing, place a ✓ in the corresponding cell. When you confirm a pairing, you can also eliminate all other cells in that row AND that column (because each person is assigned to exactly one ward, and each ward has exactly one doctor).

Step 4: Chain Deductions

When a row or column has only one empty cell remaining, that cell must be the assignment. Mark it with ✓ and eliminate other cells accordingly. Repeat until the grid is complete.

---

Worked Example 1: Three-Way Assignment

Three nurses — Kenji, Lara, and Mei — are each assigned to one of three shifts: Morning, Afternoon, and Night. Each nurse works exactly one shift, and each shift has exactly one nurse.

• Kenji does not work the Night shift
• Lara does not work the Morning shift
• Mei does not work the Afternoon shift

Question: Which shift does each nurse work?

Step 1: Set up the grid.

	Morning	Afternoon	Night
Kenji			✗
Lara	✗
Mei		✗

Step 2: Look for forced assignments.

Examine each row and column:

• Kenji's row: Morning or Afternoon (2 options)
• Lara's row: Afternoon or Night (2 options)
• Mei's row: Morning or Night (2 options)
• Morning column: Kenji or Mei (2 options — Lara is eliminated)
• Afternoon column: Kenji or Lara (2 options — Mei is eliminated)
• Night column: Lara or Mei (2 options — Kenji is eliminated)

No single cell is forced yet. We need to try cases.

Step 3: Try Kenji = Morning.

	Morning	Afternoon	Night
Kenji	✓	✗	✗
Lara	✗
Mei	✗	✗

Now Mei's row: only Night remains → Mei = Night. Lara's row: only Afternoon remains → Lara = Afternoon.

	Morning	Afternoon	Night
Kenji	✓	✗	✗
Lara	✗	✓	✗
Mei	✗	✗	✓

Check all constraints: Kenji ≠ Night ✓, Lara ≠ Morning ✓, Mei ≠ Afternoon ✓. Valid.

Step 4: Try Kenji = Afternoon.

	Morning	Afternoon	Night
Kenji	✗	✓	✗
Lara	✗	✗
Mei		✗

Lara's row: only Night remains → Lara = Night. Mei's row: only Morning remains → Mei = Morning.

	Morning	Afternoon	Night
Kenji	✗	✓	✗
Lara	✗	✗	✓
Mei	✓	✗	✗

Check: Kenji ≠ Night ✓, Lara ≠ Morning ✓, Mei ≠ Afternoon ✓. Also valid.

Two valid solutions exist. A UCAT question would add an additional constraint or ask "Which of the following must be true?" The answer would be something true in both solutions — for example, "Lara does not work the Morning shift" (true in both, since it is a given constraint) or ask "Which of the following could be true?"

---

Worked Example 2: Four-Way Matching with Conditional

Four students — Amy, Bilal, Celia, and Dan — each choose one of four elective modules: Ethics, Finance, History, and Law. Each student chooses exactly one module, and each module is chosen by exactly one student.

• Amy chooses either Ethics or History
• Bilal does not choose Finance
• If Celia chooses Law, then Dan chooses Ethics
• Dan does not choose History

Question: Who chooses Finance?

Step 1: Set up the grid.

	Ethics	Finance	History	Law
Amy		✗		✗
Bilal		✗
Celia
Dan			✗

Amy chooses Ethics or History → eliminate Amy from Finance and Law. Bilal ≠ Finance. Dan ≠ History.