From All squares are rectangles. Some rectangles are blue. Can we conclude that some blue shapes are squares?

• A. Yes
• B. No
• C. Not necessarily
• D. Cannot be determined.

Answer: (C)

The important idea is how subsets and overlaps work here. All squares are rectangles, but not all rectangles are squares. So the fact that some rectangles are blue only tells us there exists a blue rectangle. It says nothing about whether any of the blue rectangles are squares. The blue shape could all be non-square rectangles, in which case no blue square exists. To guarantee a blue square, we’d need extra information linking the blue shapes specifically to the square set. For example, if we knew some blue rectangle was a square, or that every blue shape were a square, then we could conclude there are blue squares. Without that, the conclusion isn’t forced.