2001 AMC 10 Problem 5

Check each shape for a mirror line, allowing horizontal, vertical, and diagonal axes.

Shapes with a line of symmetry:
- the straight bar (I): mirror along its length and across its middle;
- the T shape: mirror down its stem;
- the U shape: mirror between its two prongs;
- the L-shaped corner with equal arms of length $3$ (V): mirror along the diagonal through the corner;
- the staircase (W): mirror along the diagonal through the middle square;
- the plus sign (X): four mirror lines.

That is $6$ shapes.

Shapes without one: the F, the L (bar of four with one cell on the end), the N (offset $3$ and $2$), the P (a $2 \times 2$ block with a tail), the Y (bar of four with one cell off the second square), and the Z. The Z looks symmetric but only returns to itself under a half-turn, not a reflection.

The answer is $\boxed{\textbf{(D)}\ 6}$.

For a small, finite collection, a careful checklist beats cleverness. The one conceptual point is that reflectional symmetry is stricter than rotational symmetry: a figure like Z or N can look "balanced" while having no mirror line. Also remember to test diagonal axes, which catch V and W.

• Counting the Z (or S) pentomino, which has only 180-degree rotational symmetry, or missing the diagonal mirror lines of V and W.
• Losing track of which shapes have already been checked and double counting or skipping one of the twelve.