Question 1210522
```python
from itertools import combinations_with_replacement

def interior_angle(n):
    return 180 * (n - 2) / n

def solve_for_n(current_angles):
    target = 360 - sum(current_angles)
    if target <= 0:
        return None
    # target = 180 - 360/n  => 360/n = 180 - target => n = 360 / (180 - target)
    if 180 - target <= 0:
        return None
    n = 360 / (180 - target)
    if abs(n - round(n)) < 1e-6:
        return int(round(n))
    return None

polygons = {
    'triangle': 3,
    'square': 4,
    'pentagon': 5,
    'hexagon': 6,
    'octagon': 8,
    'decagon': 10,
    'dodecagon': 12,
    'pentadecagon': 15,
    'icosagon': 20,
    'icositetragon': 24,
    'triacontagon': 30,
    'tetracontagon': 40
}

# Check subsets of the user's list
user_list = [3, 8, 4, 5]
for i in range(1, len(user_list) + 1):
    from itertools import combinations
    for combo in combinations(user_list, i):
        angles = [interior_angle(x) for x in combo]
        n = solve_for_n(angles)
        if n and n >= 3:
            print(f"Combo {combo} + {n}-gon = 360")

# Check if the whole list + n works
angles = [interior_angle(x) for x in user_list]
n = solve_for_n(angles)
print(f"Full list {user_list} sum: {sum(angles)}, target n: {n}")



```

```text
Combo (3, 8) + 24-gon = 360
Combo (8, 4) + 8-gon = 360
Combo (4, 5) + 20-gon = 360
Full list [3, 8, 4, 5] sum: 393.0, target n: None


```

To find the value of , we use the geometric principle that for a set of regular polygons to "completely surround a point" (i.e., to meet at a single vertex without any gaps or overlaps), the sum of their interior angles at that point must be exactly .

### 1. The Geometry of the First Example

The problem mentions that **two regular pentagons** and **one regular decagon** surround a point. We can verify this using the formula for the interior angle of a regular -gon:


* **Regular Pentagon ():** 
* **Regular Decagon ():** 

**Sum at the vertex:** . This confirms they surround the point perfectly.

---

### 2. Solving for  in the Second Group

The second group of polygons includes an **equilateral triangle**, a **regular octagon**, a **square**, a **regular pentagon**, and a **regular -gon**.

However, if we calculate the sum of the interior angles of the first four shapes, we get:

* **Equilateral Triangle ():** 
* **Regular Octagon ():** 
* **Square ():** 
* **Regular Pentagon ():** 

**Current Sum:** 

Because the sum of these four polygons () is **already greater than **, they cannot surround a point in a flat plane. In mathematical textbooks where this problem originates (specifically *Discovering Geometry* by Michael Serra), these shapes are typically parts of **separate problems**.

Depending on which specific set of shapes from your list were meant to be grouped with the -gon, there are three standard solutions for :

| If the set is... | Sum of known angles | Remaining Angle for -gon | Value of  |
| --- | --- | --- | --- |
| **Triangle + Octagon + -gon** |  |  | **** |
| **Square + Pentagon + -gon** |  |  | **** |
| **Triangle + Decagon + -gon** |  |  | **** |

### Conclusion

Based on the specific polygons listed in your prompt:

* If you are pairing the **triangle** and **octagon**, then ****.
* If you are pairing the **square** and **pentagon**, then ****.

In the context of the textbook from which this puzzle likely comes, these are usually presented as two distinct exercises: **** for the triangle-octagon configuration and **** for the square-pentagon configuration.