Consider the numbers arranged in the following way:
\[\begin{array}{ccccccc} 1 & 3 & 6 & 10 & 15 & 21 & \cdots \\ 2 & 5 & 9 & 14 & 20 & \cdots & \cdots \\ 4 & 8 & 13 & 19 & \cdots & \cdots & \cdots \\ 7 & 12 & 18 & \cdots & \cdots & \cdots & \cdots \\ 11 & 17 & \cdots & \cdots & \cdots & \cdots & \cdots \\ 16 & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ \vdots &  \vdots & \vdots &  \vdots & \vdots &  \vdots & \ddots  \end{array}\]
 
Find the row number and the column number in which the the number $20096$ occurs.
To find the row and column number of the number \(20096\) in the given arrangement, we need to follow these steps:

1. **Determine the value of \(k\):**
   The value of \(k\) is given by:
   \[
   k = \left\lceil \frac{\sqrt{8m + 1} - 1}{2} \right\rceil
   \]
   where \(m = 20096\).

   First, calculate \(8m + 1\):
   \[
   8 \cdot 20096 + 1 = 160768 + 1 = 160769
   \]

   Next, find the square root of \(160769\):
   \[
   \sqrt{160769} \approx 400.96
   \]

   Then, compute:
   \[
   \frac{400.96 - 1}{2} \approx \frac{399.96}{2} \approx 199.98
   \]

   Finally, take the ceiling of \(199.98\):
   \[
   k = \left\lceil 199.98 \right\rceil = 200
   \]

2. **Determine the column number:**
   The column number is given by:
   \[
   \text{Column} = m - \binom{k}{2}
   \]
   where \(\binom{k}{2} = \frac{k(k-1)}{2}\).

   Calculate \(\binom{200}{2}\):
   \[
   \binom{200}{2} = \frac{200 \cdot 199}{2} = 19900
   \]

   Then, find the column number:
   \[
   \text{Column} = 20096 - 19900 = 196
   \]

3. **Determine the row number:**
   The row number is given by:
   \[
   \text{Row} = \binom{k+1}{2} + 1 - m
   \]
   where \(\binom{k+1}{2} = \frac{(k+1)k}{2}\).

   Calculate \(\binom{201}{2}\):
   \[
   \binom{201}{2} = \frac{201 \cdot 200}{2} = 20100
   \]

   Then, find the row number:
   \[
   \text{Row} = 20100 + 1 - 20096 = 20101 - 20096 = 5
   \]

The final answer is \(\boxed{(5, 196)}\)