10.2-Response - Market scale puzzle

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The weights I'm using are 1g, 2g, 5g, and 10g. I chose these weights because they are similar to the denominations of currency in my country. By arranging the weights on the scale's pans in different ways, various amounts can be measured. For example:

Herb weight

Left pan

Right pan

1

1

herbs

2

2

herbs

3

1+2

herbs

4

5

1+herbs

5

5

herbs

6

1+5

herbs

7

2+5

herbs

8

1+2+5

herbs

9

10

1+herbs

10

10

herbs

11

10+1

herbs

12

10+2

herbs

13

10+1+2

herbs

…

…

…

For larger amounts, multiple 10g weights can be stacked, and the remaining grams can be measured using the methods from the table.

Additionally, using a two-pan scale expands the flexibility of weight combinations. For instance, using 1g, 3g, 9g, and 27g allows for the measurement of almost any weight between 1 and 40g by placing the weights on either side of the scale. If only one pan can be used, 1g, 2g, 4g, 8g, and 16g are needed to measure up to 31g.

This puzzle can aid students in understanding mathematics better. It can prompt them to think about different base systems, such as binary or ternary, and use hands-on activities to comprehend concepts in number theory, like powers and mod operations. Furthermore, it encourages problem-solving, as students can figure out different combinations of weights on their own.

This way, abstract math concepts can be connected to real-life situations, making learning more engaging and practical.

To extend this puzzle for students, we could use base-3 or base-2 representations, so students can better understand number systems and how they apply to real-world situations.


Ps: This reminds me of my childhood in the countryside, where we used a steelyard (磅秤) to weigh grains and even measure body weight. It only needed a few weights to measure a wide range of different items.

 





10.2 edited

I thought I totally understood the questions now, and there are 4 weights: 1g, 3g, 9g, 27g. 

Herb weight

Left pan

Right pan

1

1

herbs

2

3

1+herbs

3

3

herbs

4

3+1

herbs

5

9

3+1+herbs

6

9

3+herbs

7

9+1

3+herbs

8

9

1+herbs

9

9

herbs

10

9+1

herbs

11

9+3

1+herbs

12

9+3

herbs

13

9+3+1

herbs

14

27

9+3+1+herbs

15

27

9+3+herbs

16

27+1

9+3+herbs

17

27

9+1+herbs

18

27

9+herbs

19

27+1

9+herbs

20

27+3

9+1+herbs

21

27+3

9+herbs

22

27+3+1

9+herbs

23

27

3+1+herbs

24

27

3+herbs

25

27+1

3+herbs

26

27

1+herbs

27

27

herbs

28

27+1

herbs

29

27+3

1+herbs

30

27+3

herbs

31

27+3+1

herbs

32

27+9

3+1+herbs

33

27+9

3+herbs

34

27+9+1

3+herbs

35

27+9

1+herbs

36

27+9

herbs

37

27+9+1

herbs

38

27+9+3

1+herbs

39

27+9+3

1+herbs

40

27+9+3+1

herbs

Comments

  1. Susan GerofskyOctober 2, 2024 at 10:46 AM

    Thanks Nanxi! Interesting that you initially chose weights of 1, 2, 5 and 10 because they were already familiar from the monetarty system. However those weights don't actually work to measure beyond 18 grams, as far as I can see! You are not allowed to add additional 10 gram weights -- only four weights in total. You did seem to see that weights of 1, 3, 9 and 27 would work, but I'm puzzled by your statement that these would measure "almost any" weight from 1 -40 grams! How did you arrive at these weights, and have you tried them out? I think your hedging (using the word "almost") might mean that you have a bit more work to do here -- but lots of good ideas for starters!

    ReplyDelete

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