Let $ n $ be a natural number. How many numbers of the form $ \pm 1\pm 2\pm 3\pm\cdots\pm n $ are there?
1. First, we need to understand the sum of the first \( n \) natural numbers. This sum is given by:
   \[
   S_n = 1 + 2 + 3 + \cdots + n = \frac{n(n+1)}{2}
   \]

2. We are interested in the number of distinct sums that can be formed by taking each number from 1 to \( n \) and either adding or subtracting it. Let's denote this sum as \( T_n \):
   \[
   T_n = \pm 1 \pm 2 \pm 3 \pm \cdots \pm n
   \]

3. The range of possible values for \( T_n \) is from \( -S_n \) to \( S_n \). This is because the smallest possible sum is when all terms are negative, and the largest possible sum is when all terms are positive:
   \[
   -S_n \leq T_n \leq S_n
   \]

4. Additionally, \( T_n \) must have the same parity (even or odd) as \( S_n \). This is because the sum of an even number of terms will always be even, and the sum of an odd number of terms will always be odd. Since \( S_n = \frac{n(n+1)}{2} \), we need to consider its parity:
   - If \( n \) is even, \( n(n+1) \) is even, so \( S_n \) is an integer.
   - If \( n \) is odd, \( n(n+1) \) is even, so \( S_n \) is an integer.

5. Therefore, \( T_n \) can take on all integer values from \( -S_n \) to \( S_n \) that have the same parity as \( S_n \). The number of such values is:
   \[
   \text{Number of values} = \frac{S_n - (-S_n)}{2} + 1 = \frac{2S_n}{2} + 1 = S_n + 1
   \]

6. Substituting \( S_n = \frac{n(n+1)}{2} \) into the equation, we get:
   \[
   \text{Number of values} = \frac{n(n+1)}{2} + 1
   \]

The final answer is \(\boxed{\frac{n(n+1)}{2} + 1}\).