Reasoning

Let the universe be “all people.” Define predicates:
: “ is a Fields medalist ”
: “ has won an International Mathematics Olympiad medal ”
: “ is a poet ”
: “ has dropped out of college ”
Let denote June Huh. From the passage we have the facts:
1) (June Huh is a Fields medalist.)
2) (He did not win any IMO medals.)
3) (He dropped out of college.)
4) (He was also a poet.)
Now evaluate each option using these facts alone.
Option (A): “Every Fields medalist has won an IMO medal.”
Formal form: .
A single counterexample falsifies a universal statement. We know and , hence is a counterexample.
Therefore (A) is False.
Option (B): “Everyone who has dropped out of college has won the Fields medal.”
Formal form: .
From the passage we only know and for one person. This does not justify a universal rule about all dropouts.
Construct a model consistent with the passage but making (B) false: add a person with and . None of the given facts mention , so the passage remains true while (B) fails.
Hence (B) cannot be inferred; it is Not entailed.
Option (C): “All Fields medalists are part-time poets.”
Formal form: .
Again, one instance is insufficient to conclude a universal rule. We can conceive a person with and without contradicting any given fact.
Therefore (C) is Not entailed.
Option (D): “Some Fields medalists have dropped out of college.”
Formal form: .
This is an existential statement and is directly witnessed by since and are both true. No additional assumptions are needed.
Therefore (D) is Certainly True.
Key logical lesson:
- Universal claims ( ) require evidence about all members; a single counterexample suffices to refute them.
- Existential claims ( ) require evidence about at least one member; a single confirmed instance proves them.
- The passage gives one concrete instance (June Huh) with properties , , , . Thus only the existential statement in (D) is logically forced by the data.

Label the grid cells by coordinates with from left to right and from bottom to top.
From the figure, the initially black squares are (top row, third column), (second row, first column), and (bottom row, third column).
Symmetry conditions:
- Reflection about the vertical midline maps .
- Reflection about the diagonal (the line ) maps .
Close the set under both reflections (and hence under their composition).
Start with .
Apply : , , , so add .
Apply : and ; the images of are already in . Add .
Applying the reflections again yields no new cells (closure reached).
Final symmetric set:
— a total of squares.
Since were already black, the minimum additional squares needed .