What is the sum of all the integers $n$ such that $\left|n-1\right|<\pi$?

[i]2016 CCA Math Bonanza Lightning #1.1[/i]
1. We start with the inequality given in the problem: 
   \[
   |n-1| < \pi
   \]
   This absolute value inequality can be split into two separate inequalities:
   \[
   -\pi < n-1 < \pi
   \]

2. We solve each part of the inequality separately:
   \[
   -\pi < n-1 \implies n > 1 - \pi
   \]
   \[
   n-1 < \pi \implies n < 1 + \pi
   \]

3. Combining these two inequalities, we get:
   \[
   1 - \pi < n < 1 + \pi
   \]

4. We know that \(\pi \approx 3.14159\). Therefore:
   \[
   1 - 3.14159 < n < 1 + 3.14159
   \]
   Simplifying this, we get:
   \[
   -2.14159 < n < 4.14159
   \]

5. Since \(n\) must be an integer, we consider the integers within this range:
   \[
   n = -2, -1, 0, 1, 2, 3, 4
   \]

6. We now sum these integers:
   \[
   -2 + (-1) + 0 + 1 + 2 + 3 + 4
   \]

7. Grouping and summing the integers:
   \[
   (-2 + 2) + (-1 + 1) + 0 + 3 + 4 = 0 + 0 + 0 + 3 + 4 = 7
   \]

Therefore, the sum of all the integers \(n\) that satisfy \(\left|n-1\right| < \pi\) is:
\[
\boxed{7}
\]