A market vendor sells dried cooking herbs in whole-number amounts
from 1 to 40 grams. The vendor has an old-fashioned two-pan weigh scale,
and has exactly four weights of different amounts that allow them to
weigh out any of these amounts of herbs -- without using the herbs or any other
object as an auxiliary weight.
- What
must the values of the four weights be? Why?
One of the weights should be 1 g.
My first insight was to find the number of combinations for
4 integers that can add 40, but I realized that the number of combinations is
huge. I need to add the constraint that I can place the weights on both sides of
the scales. Every time the scale should be calibrated to zero with both sides weighing
the same, including the herbs.
If the weights are on the side of the herbs, they rest, but
if they are on the opposite side of the herbs, they add. Because the minimum number
of herbs to sell is 1gr, there should be one weight of 1gr. Then I need to find
how to calibrate the scale to sell 2 gr. I could use the 1gr. along the herbs
and place a 3gr on the other side. It could be a 2g weight too, but remember that
we have that single weight to balance any even number.
If I need to sell 3 gr, I would use the 3-gr weight
If I need to sell 4 gr,
Note: I think the other weights should be odd because I have a 1 g weight to balance.
I would need more time to find the rest of the weights. For now, I am sure, 1 is 1gr and the second should be 3gr.
- What
if it was a one-pan scale -- i.e., a scale where you could only put the
weights on one of the pans, rather than both? What five weights would you
need to weigh up to, say, 31grams?
25gr, 15gr, 10gr, 5gr,1gr is like counting in base 5
After answering this second part, I can re-think the
previous problem. If I use base 3, the weight would be 1gr, 3gr, 9gr, 27gr
- How
could you extend this puzzle to help your students understand mathematics
more deeply?
I don’t know yet, but these problems have taught me that it
is important to dedicate time to think and think and think.
- How
does this puzzle connect with ideas about number theory and bases you are
already familiar with?
The problem pushed me to go and think about
alternatives away from the decimal system. This part was difficult because it
is what we use daily. I like the connection between the different
base systems and the diversity of paths to find the solution to a problem.
I like how you first reasoned through the logic of balance and recognized that one weight must be 1 gram, and then gradually realized how using both pans changes the system. The way you connected your thinking to different number bases (first base 5, then base 3) shows flexibility and genuine mathematical curiosity.
ReplyDeleteI also really appreciated your last comment — that sometimes the most important part is to stop, think, and think again. That kind of reflection is exactly what these puzzles are meant to inspire.