1. The Two-Pan Scale: Weighing 1 to 40 Grams with Just 4 Weights
If you have a traditional two-pan balance scale, weights can go on either side, or even be split between both pans. That means for any weight we want to measure (x), we can add our standard weights to either the side with the herbs or the side with the counterweights.
Mathematically, this means each weight can have three states:
Placed on the same side as the herbs (acting like a negative value relative to the total).
Left off the scale entirely (0).
Placed on the opposite side of the herbs (positive value).
Because each weight has 3 possibilities (-1, 0, +1), we are essentially working in Base 3 ( ternary)!
To cover every integer from 1 up to 40, our four weights need to be the first four powers of 3:
1 gram (3^0)
3 grams (3^1)
9 grams (3^2)
27 grams (3^3)
If you add them all up, 1 + 3 + 9 + 27 = 40. Because of the balanced ternary system, every single integer from 1 to 40 can be represented uniquely using combinations of addition and subtraction with these four weights.
For example: 2 grams: Put 3g on the herb side, 1g on the opposite side (3 - 1 = 2).
5 grams: Put 9g on the opposite side, and 3g + 1g on the herb side (9 - (3 + 1) = 5).
40 grams: Put all four weights on the opposite side (27 + 9 + 3 + 1 = 40).
2. The One-Pan Scale: Weighing Up to 31 Grams with 5 Weights
What if we change the rules? On a modern one-pan scale (or a two-pan scale where you're only allowed to put weights on the empty pan alongside the herbs), you can't subtract weights anymore. Every weight you use must add to the total.
Now, each weight only has two states: included (1) or excluded (0). This brings us straight into Base 2 (binary)!
To cover every whole number up to 31 using powers of 2, we need:
3 grams (3^1)
9 grams (3^2)
27 grams (3^3)
If you add them all up, 1 + 3 + 9 + 27 = 40. Because of the balanced ternary system, every single integer from 1 to 40 can be represented uniquely using combinations of addition and subtraction with these four weights.
For example: 2 grams: Put 3g on the herb side, 1g on the opposite side (3 - 1 = 2).
5 grams: Put 9g on the opposite side, and 3g + 1g on the herb side (9 - (3 + 1) = 5).
40 grams: Put all four weights on the opposite side (27 + 9 + 3 + 1 = 40).
2. The One-Pan Scale: Weighing Up to 31 Grams with 5 Weights
What if we change the rules? On a modern one-pan scale (or a two-pan scale where you're only allowed to put weights on the empty pan alongside the herbs), you can't subtract weights anymore. Every weight you use must add to the total.
Now, each weight only has two states: included (1) or excluded (0). This brings us straight into Base 2 (binary)!
To cover every whole number up to 31 using powers of 2, we need:
1 gram (2^0)
2 grams (2^1)
4 grams (2^2)
8 grams (2^3)
16 grams (2^4)
Add those up: 1 + 2 + 4 + 8 + 16 = 31. Every number from 1 to 31 can be made in one unique way using a subset of these weights, exactly like converting numbers to binary!
3. Connecting to Ancient Egyptian Multiplication
How does this tie back to ancient Egyptian math? The ancient Egyptians didn't use a multiplication table the way we do today. Instead, they used a system of doubling (duplatio) and halving, which is fundamentally binary decomposition.
To multiply two numbers, say 13 times 12, an Egyptian scribe would break 13 down into powers of 2 (8 + 4 + 1), double 12 accordingly, and add the results.
2 grams (2^1)
4 grams (2^2)
8 grams (2^3)
16 grams (2^4)
Add those up: 1 + 2 + 4 + 8 + 16 = 31. Every number from 1 to 31 can be made in one unique way using a subset of these weights, exactly like converting numbers to binary!
3. Connecting to Ancient Egyptian Multiplication
How does this tie back to ancient Egyptian math? The ancient Egyptians didn't use a multiplication table the way we do today. Instead, they used a system of doubling (duplatio) and halving, which is fundamentally binary decomposition.
To multiply two numbers, say 13 times 12, an Egyptian scribe would break 13 down into powers of 2 (8 + 4 + 1), double 12 accordingly, and add the results.
1 —> 12
2 —> 24
4 —> 48
8 —> 96
Since 8 + 4 + 1 = 13, they’d just add 96 + 48 + 12 = 156.
The one-pan scale puzzle is essentially physical binary math. It shows that any quantity can be built by repeatedly doubling units (1, 2, 4, 8, 16), which is the exact algorithmic backbone of how ancient scribes computed multiplication thousands of years ago without calculators.
4. Extending the Puzzle for Students
If I were using this in a classroom, I wouldn't just hand them the answers. I'd set it up as an interactive challenge. I will give small groups physical counters or actual gram weights and a simple balance. Ask them: "Can you figure out how to weigh 5 grams using only 1, 3, and 9?" Let them discover the negative/subtraction trick on their own when they realize putting weights on the item side helps.
Puzzles like this are awesome because they trick you into doing rigorous number theory. Whether we’re balancing weights on a scale, thinking in binary or ternary, or doubling numbers like an ancient Egyptian scribe, it’s all the exact same underlying mathematical structure - just manifested in different ways!
2 —> 24
4 —> 48
8 —> 96
Since 8 + 4 + 1 = 13, they’d just add 96 + 48 + 12 = 156.
The one-pan scale puzzle is essentially physical binary math. It shows that any quantity can be built by repeatedly doubling units (1, 2, 4, 8, 16), which is the exact algorithmic backbone of how ancient scribes computed multiplication thousands of years ago without calculators.
4. Extending the Puzzle for Students
If I were using this in a classroom, I wouldn't just hand them the answers. I'd set it up as an interactive challenge. I will give small groups physical counters or actual gram weights and a simple balance. Ask them: "Can you figure out how to weigh 5 grams using only 1, 3, and 9?" Let them discover the negative/subtraction trick on their own when they realize putting weights on the item side helps.
Puzzles like this are awesome because they trick you into doing rigorous number theory. Whether we’re balancing weights on a scale, thinking in binary or ternary, or doubling numbers like an ancient Egyptian scribe, it’s all the exact same underlying mathematical structure - just manifested in different ways!
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