Okay, we are tasked with adding 3393 and 3187. Let's begin!

Step 1: We'll start by adding the digits 3 & 7 in column 1 and get 10.
We'll write down the last digit 0 and carry the 1 to the next column.

Step 2: We'll start by adding the digits 9 & 8 in column 2 and get 18.
We'll write down the last digit 8 and carry the 1 to the next column.

Step 3: We'll start by adding the digits 3 & 1 in column 3 and get 5.

Step 4: We'll start by adding the digits 3 & 3 in column 4 and get 6.

So, 3393 + 3187 = 6580.

