Imagine that you have five blue socks and five brown socks in a drawer that are not paired up. How many socks must you take out from the drawer (without looking) to be sure of getting a pair of the same color?

To be sure of getting a pair of the same color, you need to apply the Pigeonhole Principle. This principle states that if you have more items to put into containers than there are containers, at least one container must contain more than one item.

In this case, you have two colors, blue and brown, and you want to find a pair of the same color. Since you have five socks of each color, you have a total of 10 socks.

To ensure you have a pair of the same color, you would need to draw out at least 6 socks. Here's why:

- The worst-case scenario would be that you draw out 5 blue socks and then draw out the first brown sock, leaving 1 blue and 1 brown sock remaining in the drawer. In this case, you wouldn't have a pair of the same color yet.
- However, on the next draw, no matter what color sock you get (either blue or brown), you will definitely have at least one pair of the same color since there will be three socks of that color remaining in the drawer.

Therefore, to be sure of getting a pair of the same color, you need to take out at least 6 socks from the drawer.