------
## 最终结论
```python
def max_volume(s):
    if s < 3:
        return 0
    # The maximum volume of a cuboid with a given sum of sides is achieved when the cuboid is as close to a cube as possible.
    # Let the dimensions be x, y, z. Then we have:
    # x + y + z = s / 2 (since each edge is counted twice in the sum of sides)
    # To maximize the volume V = x * y * z, we can use the fact that for a fixed sum, the product is maximized when the numbers are as equal as possible.
    
    # The optimal way to split s/2 into three parts is to take floor(s/6) and distribute the remainder.
    half_s = s / 2
    x = half_s // 3
    y = half_s // 3
    z = half_s // 3
    
    remainder = half_s % 3
    
    if remainder == 1:
        x += 1
    elif remainder == 2:
        y += 1
    
    return int(x * y * z)

# Test cases
assert max_volume(8) == 18
assert max_volume(4) == 2
assert max_volume(1) == 0
```