Statements: • All dolls are toys. • Some toys are gems. • Some gems are boxes. • All boxes are sticks. Conclusions: I. Some sticks are gems. II. Some gems are dolls. III. Some sticks are dolls. IV. Some toys are dolls. Choose the option that must follow.

Introduction / Context:We have two universal inclusions and two existentials. The key is spotting what must pass through the universal links, versus overlaps that are merely optional.

Given Data / Assumptions:

• Dolls ⊆ Toys.
• Some Toys are Gems.
• Some Gems are Boxes.
• All Boxes are Sticks.

Concept / Approach:Because all Boxes are Sticks, any particular Gem that is a Box is also a Stick. Hence “Some Sticks are Gems” is forced. Claims involving Dolls require existential import for Dolls or a specified overlap that does not exist in the premises.

Step-by-Step Solution:1) From “Some Gems are Boxes,” pick g1 ∈ Gems ∩ Boxes.2) From “All Boxes are Sticks,” g1 ∈ Sticks, so Sticks ∩ Gems ≠ ∅ ⇒ Conclusion I holds.3) Conclusion II needs Toys ∩ Gems to specifically include Dolls; not guaranteed.4) Conclusion III would require Dolls to reach Boxes; not given.5) Conclusion IV (“Some Toys are Dolls”) needs existence of Dolls. “All Dolls are Toys” does not assert that any Doll actually exists.

Verification / Alternative check:Model Dolls = ∅; Toys contain some non-doll elements overlapping Gems; Gems include a Box. Then I is true and II–IV fail.

Why Other Options Are Wrong:They rely on existential import for Dolls or unstated overlaps.

Common Pitfalls:Assuming “All A are B” implies “Some A exist.” It does not.

Final Answer:Only I follows.