Weighing Tasks

This material is suitable for children of any school age. It helps develop auditory perception through solving mathematical tasks, trains auditory memory, and enhances logical and mathematical thinking. To improve memory capacity in children, special exercises are needed. The volume of remembered material increases if, instead of mechanical memorization based on simple associations, the child uses logical connections.

Weighing tasks are mathematical problems in which you need to establish a fact using a balance scale without weights or dials. In these tasks, through logic and mathematical actions, you can distinguish a counterfeit coin from real coins, determine which is heavier or lighter.

Task #1

Out of 27 coins, one is counterfeit and is lighter than the others. What is the minimum number of weighings on a balance scale without weights needed to identify the counterfeit coin?

Solution:

1. Divide all the coins into 3 groups of 9 coins each and, for the first weighing, compare 2 of the groups: if they are equal, the counterfeit coin is in the remaining group; if not, the counterfeit coin is in the lighter group.
2. Divide the 9 coins from the group with the counterfeit coin into 3 groups of 3 coins each and, for the second weighing, compare 2 of the groups of 3 coins. If they are equal, the counterfeit coin is in the remaining group; if not, the counterfeit coin is in the lighter group.
3. Take 2 coins from the group with the counterfeit coin and compare them in the third weighing. If they are equal, the counterfeit coin is the remaining one; if not, the counterfeit coin is the lighter one.

In this type of task, the goal is to determine the minimum number of weighings. The steps are described in detail.

Task #2

There is a balance scale without weights and 4 coins that are identical in appearance. One of the coins is counterfeit, and it is unknown whether it is lighter or heavier than the real coins. How many weighings are required to determine the counterfeit coin?

Answer: 2 weighings

Task #3

A housewife has a balance scale and a 100g weight. How can she weigh out 700g of grain in three weighings?

Solution:

1. Place the 100g weight on one side of the scale and pour 100g of grain on the other side. When the sides balance, the weight will be equal.
2. Move the weight to the side with the grain. Now it holds: 100 + 100 = 200 grams. Pour 200 grams of grain on the other side.
3. Transfer all the grain to the side with the weight. Now it holds: 200 + 200 = 400 grams (including the 100g weight). Pour 400 grams of grain on the other side.

In total, the two sides will hold: 400 + 300 = 700 grams of grain.

Through solving weighing tasks, children develop visual and imaginative thinking. The cognitive development of visually impaired children follows the same patterns as that of children with normal vision.