Certainly! Let's break down the test case generator for the given problem statement "B: Mapping".

**Problem Restatement:**
1. We have two sets $S=\{a_1, ..., a_N\}$ and $T=\{b_1, ..., b_N\}$ where $a_i$ maps to $b_i$.
2. We need to determine if for any element $x$ in $S$, all mappings $g, h: S \rightarrow S$ that satisfy $g(f(x))=h(f(x))$ also satisfy $g(x)=h(x)$. If not, we provide a counterexample.

**Constraints:**
- $1 \le N \le 10^5$
- $1 \le a_i, b_i \le N$
- $a_i \neq a_j \ (i \neq j)$

Here is a test case generator for the problem:

```python
import random

def tcgen():
    N = random.randint(1, 10**5)  # Randomly choose N within the given constraints

    # Generate a permutation of 1 to N for a
    a = list(range(1, N + 1))
    random.shuffle(a)

    # Generate a permutation of 1 to N for b
    b = list(range(1, N + 1))
    random.shuffle(b)

    return N, a, b

# Example usage
N, a, b = tcgen()
print(N)
print(" ".join(map(str, a)))
print(" ".join(map(str, b)))
```

**Explanation:**
1. We first generate a random number $N$ within the range [1, 100000].
2. We create a permutation of the list [1, 2, ..., $N$] for $a$ to ensure $a_i \neq a_j$ for all $i \neq j$.
3. Similarly, we create a permutation of the list [1, 2, ..., $N$] for $b$ to ensure valid mappings.

**Note:**
- This generator ensures that constraints are satisfied by construction.
- The output provides two permutations of numbers from 1 to $N$, ensuring that the conditions $1 \le a_i, b_i \le N$ and $a_i \neq a_j$ are met.

**Example Output:**
```
5
3 1 4 2 5
5 3 1 4 2
```

This generator will produce inputs that you can use to test the problem statement "B: Mapping". Each run of `tcgen()` will produce a different test case within the problem constraints.