Solve the following problems independently of each other
If a plane is divided into 22 distinct regions, find the minimum number of coplanar lines which divide this plane in such a way that neither two lines are parallel nor any three lines are concurrent.

Options

A. 3
B. 4
C. 8
D. 6

Answer

To solve the problem of finding the minimum number of coplanar lines that divide a plane into 22 distinct regions, where no two lines are parallel and no three lines are concurrent, we can follow these steps:

### Step 1: Understand the relationship between lines and regions
The maximum number of regions \( R(n) \) that can be formed by \( n \) lines in a plane, under the given conditions, is given by the formula:
\[ R(n) = \frac{n(n + 1)}{2} + 1 \]
This formula accounts for the fact that each new line can potentially intersect all previous lines, thereby increasing the number of regions.

### Step 2: Set up the equation
We need to find the smallest integer \( n \) such that:
\[ R(n) = 22 \]
Substituting the formula into the equation gives:
\[ \frac{n(n + 1)}{2} + 1 = 22 \]

### Step 3: Simplify the equation
Subtract 1 from both sides:
\[ \frac{n(n + 1)}{2} = 21 \]
Now, multiply both sides by 2 to eliminate the fraction:
\[ n(n + 1) = 42 \]

### Step 4: Solve for \( n \)
Now we need to find two consecutive integers \( n \) and \( n + 1 \) that multiply to 42. We can factor 42:
\[ 42 = 6 \times 7 \]
Thus, we can set:
\[ n = 6 \quad \text{and} \quad n + 1 = 7 \]

### Step 5: Conclusion
The minimum number of coplanar lines required to divide the plane into 22 distinct regions is:
\[ \boxed{6} \]