Response to "The market scales puzzle & connections with ancient Egyptian mathematics: Due Monday September 28"

 The Two-Pan Scale (Balanced Ternary) 

The four weights are 1 g, 3 g, 9 g, and 27 g.

Measurements:

1- The One pan scale

2 grams: Place the 3 g weight on the opposite pan and the 1 g weight on the herb pan (3 – 1 = 2).

5 grams: Place 9 g on the opposite pan, and 1 g + 3 g on the herb pan (9 – 3 – 1 =5).

14 grams: Place 27 g on the opposite pan, and 9 g + 3 g + 1 g on the herb pan (27 – 9 – 3 – 1 = 14).

40 grams: Place all weights on the opposite pan (27 + 9 + 3 + 1 = 40).

2-The One-Pan Scale

What if weights can only be placed on one pan?

On a one-pan scale, a weight can only be placed on the side opposite the item being measured. This removes the subtraction option each weight can only be included (+1) or excluded (0). To measure up to 31 grams using 5 weights, we use powers of 2: 1 g, 2 g, 4 g, 8 g, and 16 g.

Connections to Ancient Egyptian Mathematics and Number Bases

Ancient Egyptian Multiplication & Binary representation

Ancient Egyptian multiplication relies on a method of doubling and adding. To multiply two numbers. This algorithm resembles the one-pan scale problem. Expressing any number as a unique sum of powers of 2 which is precisely base-2 (binary)

Extending the Puzzle for Deep Classroom Learning

To help students build structural understanding rather than just finding answers by trial and error, these extensions should be considered:

 Greedy Algorithm Failure vs. Powers Construction:

 Ask students: “If we start with a 1g weight, what is the largest second weight we can pick so that no gaps are left in our measurable range?”

 Exploring Minimal Weight Sets:

Challenge students to derive a general formula for n weights

Hands-On Representation Cards:

Provide students with cards labelled +1, -1, 0 for powers of 3, or 1, 0 for powers of 2. Have them “code” numbers physically before writing equations

Negative Weights & Ternary Arithmetic:

 Ask students how balanced ternary handles subtraction without explicit minus signs, connecting physical scale balance directly to algebraic expressions.



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