The Big Idea About the One and Two Pan Scale Puzzle

The principle of the two-pan scale is based on the idea of mass equality, whereby the weight on one side must exactly match the weight on the other for the scale to achieve balance. This system is particularly fascinating because it allows for a variety of weights to be represented through combinations of smaller weights, especially when utilizing powers of 3, such as 1 gram, 3 grams, 9 grams, and 27 grams, which measure any number from 1 to 40.

The two-pan scale operates using a balanced ternary system, which incorporates three distinct digits: -1, 0, and +1. In this context, 0 signifies that a weight is not used, -1 indicates that the weight is placed on the same side as the item being weighed, and +1 shows that the weight is on the opposite side. This arrangement facilitates efficient calculations and gives room for instinctive calculations.

For instance, if a customer asks for 7 grams of herbs, I can explore how to arrange the weights of 1g, 3g, 9g, and 27g. By placing 9 grams on the right side of the scale, I need to determine the appropriate weights to balance the herbs on the left side. This scenario can be expressed as:

Herbs + 3g + 1g = 9g.

It’s important to note that the 3g and 1g are not part of the herbs but are additional weights of placements used for balancing, and rather than viewing this simply in terms of addition and subtraction to derive 5g, I must consider the placement of the weights on the scale. As we are not just trying to find the total, we must logically acknowledge the nature of the opposing weights.

Thus, with the equation:

Herbs + 3g + 1g = 9g, it becomes clear that the weights of 3g and 1g are effectively subtracted from 9g on the right side. Consequently, I can reorganize the equation to reflect

Herbs = 9g - 3g - 1g.

From a straightforward arithmetic standpoint, this suggests that the weight of the herbs would be calculated as 5g. However, it’s crucial to remember that the puzzle of balancing weights on the scale focuses on the ternary system and not the principle of addition and subtraction we are conversant with. Since both weights are on the same side as the herbs, they contribute negatively to the right side. So, realigning back to our initial equation, we can express it as:

Herbs + 3g + 1g = 9g, where both 3g and 1g are positioned as -1.

This leads us to conclude that the effective weight of the herbs is ultimately 7 grams. This example beautifully illustrates how the unique design of a two-pan scale allows for intuitive balancing, as this system works because of the powers of 3,1g,3g, 9 g, and 27g, and you can measure numbers from 1 to 40 grams perfectly.

The one-pan scale operates differently from the two-pan scale, as it relies solely on addition. In this setup, all weights are placed on the same side as the item being weighed, leaving the other pan empty, which means there’s no need to account for subtraction from an opposing weight. The one-pan scale utilizes binary representations, consisting of weights of 1g, 2g, 4g, 8g, and 16g, allowing it to measure any quantity from 1 to 31.

The principle of binary digits (0 and 1) comes into play, where each weight can either be included (1) or excluded (0). For instance, to weigh 7 grams, I would combine the weights of 1g, 2g, and 4g. This is a straightforward illustration of binary arithmetic, as the sum of these weights gives the desired measurement of 7 grams.

I have observed in my classroom that algebra can often be quite challenging for my students. To make these concepts more relatable, I’ve discovered that using one-pan and two-pan scales can help clarify the ideas in a hands-on manner. For example, I can take my students outside to set up a balance scale using a rope and two pans tied to a tree branch. On one side, I place a pack of salt along with a 5-gram weight, while on the other side, I add a 9-gram weight. This physical setup allows us to visualize the balance and connect it to symbolic equations. I explain that if we let the weight of the pack of salt be represented by x , we can formulate the equation:

 x + 5 = 9

By solving this equation, students can figure out the original weight of the pack of salt.

To extend this exploration, I present a new scenario where we have four packs of salt plus a 5-gram weight that balances out to 25 grams. This gives us the equation:

4x + 5 = 25

This approach encourages students to reason through their answers and justify their thinking rather than solely focusing on calculations. By developing relatable scenarios, students become more engaged and grasp algebraic concepts within a practical context. Furthermore, allowing them to create their own scenarios solidifies their understanding by linking mathematics to everyday objects and experiences, thereby bridging the gap between their familiar world and abstract reasoning.

The concept behind the puzzle is closely related to number theory and number bases because, much like number theory, which focuses on the study of whole numbers, their patterns, and ways to break them into smaller components, students can see these reflections in the puzzle. For instance, in the two-pan puzzle, every number up to 40 can be represented using the numbers 1, 3, 9, and 27, which form a recognized pattern of powers of 3. In contrast, the one-pan puzzle utilizes the pattern 1, 2, 4, 8, and 16 to represent numbers up to 31. To achieve balance on the scale, specific combinations of these numbers are required, just like the combination of prime numbers to get factors.

Additionally, the one-pan system follows a binary representation using digits 0 and 1, while the two-pan system embraces a balanced ternary system with representations of thought. +1, -1, and 0.

Exploring puzzles and their connection to the number system truly highlights the beauty of mathematics, and this is a big part of why I'm passionate about teaching it, because I love playing with numbers. While it's one thing to convey these concepts from a textbook perspective, it’s entirely different to witness my students apply these ideas in real time. Seeing them engage with hands-on activities that help them grasp the patterns of numbers and bases is incredibly fulfilling. It’s rewarding to observe how this foundational knowledge will allow them to understand and solve puzzles, bringing the abstract concepts they learn in the classroom to life.