9.3 Constraint Deduction Drills: Ordering, Arrangement & Assignment

9.3 Constraint Deduction Drills: Ordering, Arrangement & Assignment

Scope note — read this first. Bocconi publishes exactly two item types for the critical-thinking area of the standard Online Bocconi Test: data propositions (choose which statements are true or false depending only on the data provided) and passage-plus-assertion items (decide whether an assertion is true, false, or cannot be deduced). Constraint puzzles are not on that list, so do not budget test-day time for seating plans or floor assignments. Sequenced-condition deduction items are described for the separate Online Bocconi Test – Law, and the discipline they build — exhausting every forced consequence of the given conditions before concluding anything — is precisely what the data-proposition items in Section 9.4 reward. Treat this section as training, not as a forecast.

Constraint puzzles do not test general knowledge or economic vocabulary; they evaluate algorithmic deduction under constraints. You are given a finite set of elements (candidates, presentations, office floors) and a set of explicit rules. Your objective is to translate verbal clues into spatial or tabular models, eliminate impossible permutations, and extract uniquely determined assignments.

---

1. The Four Primary Puzzle Genres

Almost every constraint puzzle maps onto one of four standard geometric or relational frameworks:

Genre 1: Linear Sequencing and Ordering

• Scenario: Arranging entities across a discrete sequence with a defined direction (e.g., race finishing order 1st through 6th; speaking schedule Monday through Saturday; project priority queues).
• Key Spatial Relations:
  • Immediately adjacent: [A B] or [B A].
  • Fixed gap offset: A is two slots ahead of B: [A _ B].
  • Relative ordering: A finishes before B, but after C: C < A < B.

Genre 2: Circular Seating Arrangements

• Scenario: Seating N individuals around a circular conference table.
• Crucial Orientation Rule:
  • Facing Center: An individual's Left corresponds to the Clockwise direction; their Right corresponds to the Counter-Clockwise direction.
  • Facing Outward: An individual's Left is Counter-Clockwise; their Right is Clockwise (everything inverts!).
  • Opposite Positions: For an even number of individuals N, the person sitting directly opposite position k sits at position k + N/2. At a 6-person table, position 1 is opposite position 4; position 2 is opposite 5; position 3 is opposite 6.

Genre 3: Spatial and Multi-Floor Grids

• Scenario: Assigning individuals or corporate departments to floors in a multi-story headquarters (e.g., Floors 1 to 6, where Floor 1 is the ground floor and Floor 6 is the top floor).
• Key Relational Clues: "Between two entities", "on an even-numbered floor", "at least two floors above".

Genre 4: Multi-Variable Matching and Matrix Grids

• Scenario: Matching entities across multiple distinct categories simultaneously (e.g., 5 Analysts matched to 5 European Cities and 5 Industrial Sectors).
• Diagramming Tool: A 2D cross-tabular matrix where rows represent entities and columns represent attributes, populated with affirmative checkmarks (✓) and negative crosses (✗).

---

2. The Anchor-First Diagramming Protocol

Never start testing hypothetical permutations in your head. Execute this structured 5-step diagramming protocol:

                  THE ANCHOR-FIRST DIAGRAMMING PROTOCOL
                  
  +-------------------------------------------------------------+
  | STEP 1: Establish the Fixed Framework                       |
  | Identify the naturally ordered set (e.g., Ranks 1 to 6,    |
  | Days Mon-Sat, Floors 1-6) and draw the empty slots.         |
  +-------------------------------------------------------------+
                                 |
                                 v
  +-------------------------------------------------------------+
  | STEP 2: Place Definite "Anchor" Clues                       |
  | Fill in clues that establish absolute positions             |
  | (e.g., "Elena presents in slot 4", "Floor 1 is Zurich").     |
  +-------------------------------------------------------------+
                                 |
                                 v
  +-------------------------------------------------------------+
  | STEP 3: Symbolize Blocks and Negative Rules                 |
  | - Fixed blocks: [B E]                                       |
  | - Negative constraints: write slashed initials below slots  |
  |   (e.g., ~A below slot 1 means Alba ≠ Slot 1).              |
  +-------------------------------------------------------------+
                                 |
                                 v
  +-------------------------------------------------------------+
  | STEP 4: Convert Conditional Rules into Contrapositives      |
  | Rule: If P is in Slot 2 -> Q is in Slot 5                   |
  | Contrapositive: If Q is NOT in Slot 5 -> P is NOT in Slot 2.|
  +-------------------------------------------------------------+
                                 |
                                 v
  +-------------------------------------------------------------+
  | STEP 5: Scenario Splitting (Bifurcation)                    |
  | When a key entity has exactly two valid positions, branch   |
  | into Scenario A and Scenario B; eliminate the contradiction.|
  +-------------------------------------------------------------+

---

3. Handling Conditional Rules and Block Packing

Block Packing Technique

When given a multi-entity block—for instance, "Giulia presents exactly two time slots after Marco"—treat this relationship as a single indivisible composite unit: [M _ G] This block has an effective width of 3 slots. In a 6-slot schedule, [M _ G] can only occupy four possible positions: (1, 3), (2, 4), (3, 5), or (4, 6). Intersecting this block with existing anchor clues or negative constraints immediately narrows the puzzle down to 1 or 2 viable states.

Contrapositive Translation

Puzzles frequently introduce conditional constraints:

• Rule: "If Chiara is assigned to Milan, then Dario must be assigned to Paris" (Chiara_Milan → Dario_Paris).
• Immediate Deduction (Contrapositive): If Dario is not in Paris, Chiara cannot be in Milan (¬Dario_Paris → ¬Chiara_Milan).
• Fallacy Trap: Do not commit the inverse fallacy: if Chiara is not in Milan, Dario might still be in Paris!

---

4. Pacing Discipline When Conditions Pile Up

The transferable habit here is knowing when a chain of conditions has stopped paying for the time it consumes. Remember that the Online Bocconi Test is forward-only: there is no second pass, so a stalled deduction is not deferred, it is lost along with the other two questions sharing that screen.

Abandonment Checklist: When to Stop Deducing and Start Eliminating

Judge the scenario in the first 15 seconds. Switch to elimination and answer immediately if:

1. There are more than 6 entities across 3 or more variable dimensions.
2. There are zero definitive anchor clues, so every clue is relative or conditional and the solution requires triple-branch scenario splitting.
3. The question asks something like "which of the following must be false if any two adjacent entities swap places?", which forces you to rebuild the model several times.

With 5 to 6 entities and at least one anchor clue, the full deduction is normally faster than guessing — run it.

---

5. Worked Constraint Puzzle Walkthrough

The Scenario

Six Bocconi MSc students—Alba, Bruno, Chiara, Dario, Elena, and Federico—are scheduled to deliver capstone research presentations across six consecutive hourly time slots numbered 1 through 6.

The Constraints

1. Elena presents immediately after Bruno: [B E].
2. Dario presents in an even-numbered slot (D ∈ {2, 4, 6}).
3. Chiara presents at some time before Federico, but at some time after Elena (E < C < F).
4. Alba does not present in slot 1 or slot 6 (A ≠ 1, A ≠ 6).
5. Federico presents in slot 6 (F = 6).

Step-by-Step Complete Deduction Trace

1. Place Definitive Anchors:

   • By Constraint 5, slot 6 is locked: Slot 6 = Federico. _ _ _ _ _ F(6)

2. Analyze Relative Ordering Chain:

   • Combine Constraints 1 and 3: Bruno is immediately before Elena, and Elena is before Chiara, who is before Federico: B < E < C < F(6)
   • Because B and E are strictly adjacent ([B E]), and both must precede C, and C must precede F(6):
   • Slot 5 cannot be B or E (since C must follow E and precede F).
   • Therefore, the latest possible slot for C is Slot 5.

3. Locate Dario and Alba:

   • Dario must be in an even-numbered slot: D ∈ {2, 4, 6}. Since slot 6 is occupied by F, D ∈ {2, 4}.
   • Alba cannot be in slot 1 or slot 6: A ≠ 1, A ≠ 6.

4. Evaluate Placement of the Block [B E]:

   • Where can [B E] go? It needs two consecutive slots before C.
   • Can [B E] occupy Slots 3 and 4?
     • If B=3, E=4, then C must be in Slot 5 (since E < C < F).
     • The remaining slots are 1 and 2, which must be filled by A and D.
     • Since Dario must be even, D=2. That leaves A=1.
     • But Constraint 4 states A ≠ 1! Contradiction. Thus, [B E] cannot be in (3, 4).
   • Can [B E] occupy Slots 2 and 3?
     • If B=2, E=3, then Dario (who must be even) must be in Slot 4 (D=4).
     • Then C must be in Slot 5 (E < C < F).
     • That leaves Slot 1 for A, which again violates A ≠ 1! Contradiction.
   • Therefore, [B E] MUST occupy Slots 1 and 2:
     • Slot 1 = Bruno
     • Slot 2 = Elena

5. Resolve Remaining Slots (3, 4, 5):

   • Unassigned students: Alba, Chiara, Dario.
   • Unassigned slots: 3, 4, 5.
   • Dario must be in an even slot: since Slot 2 is Elena and Slot 6 is Federico, Dario must be in Slot 4 (D = 4).
   • Now unassigned students are Alba and Chiara in slots 3 and 5.
   • We know E < C < F, which is satisfied whether C is in 3 or 5.
   • Does Alba have restrictions? A ≠ 1, A ≠ 6. Both 3 and 5 are valid for Alba!
   • Final robust deduced framework: B(1), E(2), A/C, D(4), C/A, F(6).
   • Every single condition is completely fulfilled!