Step-by-step Solution:

Quick Solution

This is a classic puzzle that uses the "worst-case scenario" logic. To be absolutely sure of getting a pair, we have to think about what would happen if we were as unlucky as possible in our picks.

1. Simulating the Worst-Case Scenario

The goal is to get one pair of socks of the same colour. The worst possible luck would be to avoid making a pair for as long as possible by picking one of each available colour.

• First Sock Picked: It could be either Black or Brown. Let's say you pick a Black sock.
  (You now have 1 sock, but no pair yet).
• Second Sock Picked: To continue being unlucky, the second sock you pick must be a different colour. So, you pick a Brown sock.
  (You now have 2 socks, one of each colour, but still no matching pair).
• Third Sock Picked: You have already picked one of each possible colour. The third sock you pick has no choice but to be either Black or Brown.
  
  • If it's Black, it matches the first Black sock you picked.
  • If it's Brown, it matches the second Brown sock you picked.
  In either case, this third sock guarantees that you have a matching pair.

2. Conclusion

The number of socks of each color in the drawer (10 of each) is extra information designed to be a distractor. The logic only depends on the number of different colours available (which is two).

This problem is an example of the Pigeonhole Principle: if you have N categories (pigeonholes), you need to pick N+1 items (pigeons) to guarantee at least one category has two items. Here, N=2 colours, so you need 2+1=3 picks.

Final Answer: The smallest number of socks to be taken to be sure of a pair is 3.