The Worst Case Scenario

Reasoning with extremes is a particularly useful strategy to solve some problems. It can also be seen as a “worst case scenario” strategy. The best way to appreciate this kind of thinking is through example. So let’s begin to appreciate some really nice reasoning strategies.

In a drawer, there are 8 blue socks, 6 green socks, and 12 black socks. What is the minimum number of socks you must take from the drawer, without looking, to be certain that you've 2 socks of the same color?

The phrase “……. certain ……. two socks of the same color” is the key to the problem. The problem does not specify which color, so any of the three would be correct. To solve this problem, have reason from a “worst case scenario.” You pick one blue sock, one green sock, and then one black sock. You now have one of each color, but no matching pair. (True, you might have picked a pair on your first two selections, but the problem calls for “certain.”) Notice that as soon as you pick the fourth sock, you must have a pair of the same color.

Consider a second problem:

In a drawer, there are 8 blue socks, 6 green socks, and 12 black socks. What is the minimum number of socks you must take from the drawer, without looking; to be certain that you've two black socks?

Although this problem appears to be similar to the previous one, there is one important difference. In this problem, a specific color has been specified. Thus, it is a pair of black socks that we must guarantee being selected. Again, let’s use deductive reasoning and construct the “worst case scenario.” Suppose you first pick all of the blue socks (8). Next you pick all of the green socks (6). Still not one black sock has been chosen. You now have 14 socks in all, but none of them is black. However, the next two socks you pick must be black, since that is the only color remaining.

To be certain of picking two black socks, you must select 8 + 6 + 2 = 16 socks in all.