Question

A historical research archive is curating three new collections of ancient manuscripts. Each collection will contain four manuscripts: two from classical antiquity and two from the medieval period. Moreover, no single manuscript may appear in all three collections, and no two collections may share more than one of the same manuscript. It is further known that at least one collection must include both a manuscript by Homer and a manuscript by Bede. So far, three manuscripts in each collection have been selected:

Collection 1	Collection 2	Collection 3
Homer (antiquity)	Bede (medieval)	Augustine (medieval)
Virgil (antiquity)	Anselm (medieval)	Hildegard of Bingen (medieval)
Hildegard of Bingen (medieval)	Plato (antiquity)	Homer (antiquity)

Select a manuscript that must be added to one of the collections, and a manuscript that cannot be added to any collection. Make only two selections, one for each column.

Correct Selections:

• Must be added to one of the collections: Bede (medieval)
• Cannot be added to any collection: Homer (antiquity)

Hint:

For set theory questions without overlap data:

The \textbf{maximum} possible intersection (AND) is the minimum of the individual set sizes.
The \textbf{minimum} possible union (OR) is the maximum of the individual set sizes.

Answer 1

Step 1: Analyze the rules and the current state

• Rule 1 (Composition): Each collection must have 2 antiquity + 2 medieval manuscripts.
• Rule 2 (Uniqueness): No manuscript may appear in all three collections.
• Rule 3 (Sharing): Any two collections may share at most one manuscript.
• Rule 4 (Required Pair): At least one collection must contain both Homer and Bede.

Current needs:

• Collection 1: Homer (A), Virgil (A), Hildegard (M) → needs 1 medieval.
• Collection 2: Bede (M), Anselm (M), Plato (A) → needs 1 antiquity.
• Collection 3: Augustine (M), Hildegard (M), Homer (A) → needs 1 antiquity.

Step 2: Which manuscript cannot be added?

Test Homer (antiquity):

• Collection 1: Already has 2 antiquity → cannot add.
• Collection 3: Already contains Homer → duplicate.
• Collection 2: Needs antiquity, but then Homer would be in all three collections → violates Rule 2.

Therefore, Homer cannot be added to any collection.

Step 3: Which manuscript must be added?

To satisfy Rule 4 (Homer + Bede together):

• Option A: Add Bede to Collection 1 → valid, since it needs a medieval and already has Homer.
• Option B: Add Homer to Collection 2 → would satisfy the pair, but violates Rule 2 (forbidden).

Thus, Bede must be added to Collection 1.

✅ Must be added: Bede (medieval)
❌ Cannot be added: Homer (antiquity)