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To solve this problem, let's break down the given conditions and analyze them systematically:

1. There are 6 towers, labeled A, B, C, D, E, and F.
2. Each student, after a period of free movement, finds that they can only see towers A, B, C, and D, and cannot see towers E and F.

**Analyzing the conditions:**

1. **Condition 1**: The positions of the students and the towers they can see are all on the same plane, and no two positions overlap.
2. **Condition 2**: A, B, C, D, E, and F are 3 non-collinear points.
3. **Condition 3**: The students are positioned such that the only way to be unable to see a tower is if their line of sight is obstructed by other towers. For example, if a student's position is on the line connecting A and B, then they cannot see the tower at B.

From these conditions, it is clear that each student’s position is determined by the visibility of towers A, B, C, and D, and the invisibility of towers E and F. 

Since the problem states the students see only A, B, C, and D but not E and F, we need to find positions such that E and F are blocked by the line of sight of the other four towers.

Considering the given positions and conditions:

- Each student can be at a position where two towers block the view of E and F.
- The number of unique ways this can occur can be deduced from the combinatorial arrangements of 3 points among A, B, C, and D.

Using combinatorial mathematics, the number of ways to choose 3 points out of 6 is given by the combination formula:
\[ \binom{6}{3} = \frac{6!}{3!(6-3)!} = 20 \]

However, since the problem specifically asks about the possible number of students that could form such a group given the conditions, we need to look at the constraints and symmetry provided by the visibility of A, B, C, and D, and the obstructions E and F.

Each unique triplet of towers blocking another two will be the configurations we need.

**Solution:**
Upon reviewing the problem constraints and the necessary geometric configurations, the possible number of students that satisfy the conditions and can form such a group is:
\[ \boxed{6} \]

Do you need further details or explanations? Here are five questions you might find interesting:

1. Can we determine specific coordinates for each student's position?
2. How would the solution change if more towers were added?
3. What is the significance of non-collinear points in this context?
4. How would the solution be affected if some towers were aligned?
5. Can we generalize this solution to three-dimensional space?

**Tip:** When solving problems involving visibility and obstruction, visualizing the setup with sketches can greatly aid in understanding and finding a solution.

Explore a mathematical problem involving tower visibility and obstruction, where students must determine configurations based on given conditions. This problem utilizes combinatorial mathematics and geometric configurations to find optimal solutions.

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