Here's why:

* Surface Area: A full moon presents the entire illuminated side of the moon to us, whereas a half moon only shows half.

* Direct Sunlight: The illuminated side of the moon reflects direct sunlight. A full moon has the largest area reflecting sunlight directly.

Why it might seem like a half moon is brighter:

* Contrast: A half moon's sharp crescent shape can appear more intense against the dark background of the sky. Our brains sometimes perceive this contrast as brightness.

* Atmospheric Conditions: The brightness of the moon can be affected by things like clouds, dust, and moisture in the atmosphere. A half moon might appear brighter under certain atmospheric conditions, even though it's reflecting less light overall.

In summary: While a half moon might sometimes appear brighter due to contrast and atmospheric conditions, a full moon is actually objectively brighter because it reflects more direct sunlight from a larger illuminated area.