Farmer Michael has only to weigh the apples twice in order to find out whether the bags were labeled correctly.
1. Weighing the first time
On one side of the balance scales you put the bag with the description "6 kg," and on the other side the three bags displaying the information "1 kg," "2 kg" and "3 kg".
Now there are two options:
- The scale is not balanced. In this case, the bags were not labeled correctly. If they were, the scales would have to be in balance.
- The scale is in balance. In this case you can conclude the following:
1) The bag with the description "6 kg" also contains six kilograms and was therefore properly labeled.
2) The three bags on the left-sided pan are in total six kilograms. There is no other way to have on one scale pan three sacks and on the other scale pan only one bag, and have it to be in balance without the total sum of six kilograms. "6 kg" is the only sum you can achieve with three sacks.
2. Weighing the second time
The second weighing you put the two bags with the descriptions "6 kg" and "1 kg" on the left scale pan and the two bags with the details "5 kg" and "3 kg" on the right scale pan. Only for this combination, the right scale pan is heavier than the left one. In this case, the bags "6 kg" and "1 kg" as well as "5 kg"and "3 kg" were labeled correctly. If the right scale pan is lighter than the left scale, it proves that there must be a mistake in the labels on the bags. What cases are conceivable? See the table below.