This problem has a very simple and elegant mathematical solution. But to go there, let's do some visual analysis first. After we get our zigzagged string, let's try to assign character on each line to some list, like on the picture below,

Question is how to know beforehand which character goes to which list. In order to do so, let's put character indices as values on ** X** axis and corresponding list indices on

**axis. For a given string (A**

*Y*_{1}... A

_{35}) plot will look like as follows

You can immediately notice that it's a periodic function. In general that means that if you take the minimal part of the plot as below

and shift right and left as infinitely many times you get all possible values of the function.

In our case, period is 10. The blue segment of the plot is a simple function

**y = |x|**. Now, how to find right red segment? According to the plot, we need to shift the red segment by 10, so it's

**y = |x - 10|**, the next right segment would be

**y = |x - 20|**and so forth. How to write it in a most general way? Obviously

**y = |x - 10k|**, but the value of

**k**depends on which segment you're on. You can easily guess that

**x**values on each segment are bound by

**10k - 5 ≤ x ≤ 10k + 5**, or if you convert it k = 10 ⌊(x + 5)/10⌋

You can also guess, that period 10 is actually connected to the number of rows as

**.**

*2(numRows - 1)*After you find that mapping, rest is the mere technicality of joining characters in each list and then joining again across all lists. To wrap it up, here's the python code

```
class Solution(object):
def convert(self, s, numRows):
"""
:type s: str
:type numRows: int
:rtype: str
"""
new_string = [[] for a in range(numRows)]
period = (numRows - 1)*2
for i, char in enumerate(s):
new_string[abs(i - period*((i+period/2)//period))].append(char)
return ''.join([''.join(a) for a in new_string])
```

PS: beware, the code above won't pass all the test cases upon submission, since there's no checks for edge cases, like zero period or period being longer than two length of the string, etc.