Optimal Weight Measurement: Weighing 1 kg to 100 kg with Just 5 Weights

In the realm of weight measurement, an intriguing problem from a campus placement interview caught the attention of many. This article explores the concept of utilizing just five weights to accurately determine a material's weight ranging from 1 kg to 100 kg. The keys to solving this puzzle lie in the strategic use of weights and the principles of mathematical balance.

Introduction to the Weight Measurement Challenge

One of the questions posed in a campus placement interview required candidates to determine how to weigh from 1 kg to 100 kg using only five weights. This problem, while seemingly simple, requires a profound understanding of mathematical logic and the concept of balance, both in the physical and metaphorical sense.

Solving the Puzzle: Powers of 3 and the Seesaw Balance

The solution to this puzzle involves utilizing weights that are powers of 3. Specifically, the weights chosen are 1 kg, 3 kg, 9 kg, 27 kg, and 81 kg. Why these particular weights? Let's delve deeper into the reasoning behind this choice.

The Concept of Balance

Imagine that you are using a seesaw to measure weights. You can place the weights on either side of the seesaw, not just on one side. This unique feature transforms a simple weighing problem into a complex puzzle that requires a deep understanding of mathematical principles.

The Mathematical Foundation

The weights (1, 3, 9, 27, 81) are chosen because they are powers of 3. This choice is not arbitrary; it is based on the fact that any weight between 1 kg and 100 kg can be represented as a sum or difference of these weights. The key insight is that any integer within this range can be expressed in the form of a balanced ternary number system.

Example: Weighing 55 kg

To understand the practical application of these weights, let's consider an example. Suppose you need to measure 55 kg of a material. Here’s how you would use the 1 kg, 3 kg, 9 kg, 27 kg, and 81 kg weights to achieve this: 1. Place the 81 kg weight on one side of the balance. 2. Place the 27 kg and 9 kg weights on the same side as the 81 kg weight to make a total of 117 kg. 3. Place the 55 kg material on the other side of the balance. 4. Adjust the placement of the remaining weights (1 kg, 3 kg, and 27 kg) on the opposite side until the balance is even. For instance, by adding the 27 kg and 9 kg weights to the material, we get 81 27 - 9 - 27 - 27 55 kg. This process ensures that the material’s weight is accurately measured.

Generalization: Measuring Any Weight from 1 kg to 100 kg

The principle of using weights that are powers of 3 can be generalized to measure any weight from 1 kg to 100 kg. The weights (1, 3, 9, 27, 81) provide a flexible and efficient way to achieve this. Any weight within this range can be represented as a combination of these weights, either by adding or subtracting them.

Conclusion and Further Exploration

In conclusion, the use of five weights (1 kg, 3 kg, 9 kg, 27 kg, and 81 kg) offers a unique solution to the problem of weighing materials from 1 kg to 100 kg. This method not only showcases the elegance of mathematical principles but also highlights the importance of using strategic and optimal solutions in real-world scenarios. Whether you are a student, a professional, or just someone interested in puzzles, the concept of using weights that are powers of 3 is a fascinating area of exploration.

Further Reading:

Mathematical Puzzles and Logic Games Real-World Applications of Mathematics Tips for Campus Placement Interviews