You have 5 coins, one of which is counterfeit and is slightly lighter than the other four, which are identical. With a balance scale, how can you determine which coin is counterfeit in just two weighings?

To determine which coin is counterfeit in just two weighings, you can follow these steps:

1. First, divide the 5 coins into three groups: two groups of 2 coins each, and one group of 1 remaining coin.

2. Perform the first weighing by comparing the two groups of 2 coins. There are three possible outcomes from this weighing:

a) If both sides of the scale are equal: This means that the counterfeit coin is in the group of 1 remaining coin, and since there is only one coin in this group, you can directly conclude that this coin is the counterfeit one.

b) If one side of the scale is lighter: This means that the counterfeit coin is in the lighter group of 2 coins.

3. If the result is case (b), perform the second weighing by comparing the two coins in the lighter group from the first weighing:

a) If one side of the scale is lighter: This means that the coin on the lighter side is the counterfeit one.

Here is a visual illustration of the process (C represents the counterfeit coin and R represents the real coins):

First Weighing:

(1,2)-(3,4)      (1,2)=(3,4)        (1,2)<(3,4)
1    2   3  4     3    4   1  2     1    2   3  4
|____|   |__|     |____|   |__|     |____|   |__|
   (A)     (B)       (C)     (A)       (D)     (C)

Second Weighing (if needed):

1-2            1=2           1<2
|_|           _| |_          |_|
 (A)           (B)            (C)

In this example, you can see that in case (A) or (D), you would need to perform both weighings. In any case, you will be able to determine the counterfeit coin in just two weighings.