In a spacious room, Namita is standing facing a large mirror that covers the entire wall in front of her. She is wearing a hat with a unique symbol on the front. Karim enters the room and stands to Namita's right side, close enough to see Namita's reflection in the mirror but not directly facing the mirror himself; instead, he is looking towards Namita. There is a painting on the wall behind Namita and Karim, and both of them can easily see this wall if they turn their heads. Soon after, Layla enters the room from a door behind Namita and Karim, having a clear line of sight to the back of both Namita and Karim, as well as a full view of the painting on the wall. Given this situation, what can each person see directly and by reflection, and what can Layla infer about the symbol on Namita's hat? Provide a brief response.
Namita is facing the mirror and can see her own reflection, which means she can directly see the symbol on her hat's front via the mirror. Since she is facing the mirror, she cannot see Karim or Layla directly, who are positioned to her side and behind her, respectively. However, since mirrors reverse left and right but not up and down, in her peripheral vision or by slightly turning her head to the right, she may see Karim's reflection within the same mirror if he is within the field of view of the mirror.

Karim, standing to Namita's right, is not facing the mirror directly but towards Namita. He cannot see Namita's hat symbol directly because he is standing beside her; moreover, he cannot see the symbol's reflection because he's looking at Namita and not her reflection in the mirror. Depending upon how close he is standing to Namita and the angles involved, he might be able to catch a glimpse of the painting's reflection if he shifts his gaze to the mirror without moving completely to face it.

Layla, entering the room, can see the back of both Namita and Karim. She can see the painting directly because her view is not obstructed. She cannot see the symbol on Namita's hat directly because it is on the front of the hat, but she can see the unique symbol's reflection in the mirror since the mirror is in front of Namita. From Layla's perspective, she can deduce what the symbol on the front of Namita's hat looks like by looking at the mirror in front of Namita, which displays the reflection showing the symbol.

Therefore, Namita can see her own hat's symbol via reflection, Karim cannot see the symbol at all because he is neither facing the hat directly nor the reflection in the mirror, and Layla can deduce the appearance of the symbol by viewing the reflection of Namita in the mirror.