Wednesday, December 1, 2021

Puzzle: Market Vendor

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 allows them to weigh out any of these amounts of herbs -- without using the herbs or any other object as an auxiliary weight.


I assume that x must be 1 so that if y, a, and b were even or odd, I could easily make them one larger or smaller.


I will skip a 2g weight, since it can be represented by 3-1.  So, let weight y=3.

So far, we have x=1, y=3.

Since 4 can be represented by x+y, we will skip this number.
5 can be represented by 9-1-3, so let a = 9.
We continue...
  • 6=9-3
  • 7=9-3+1
  • 8=9-1
  • 9=9
  • 10=9+1
  • 11=9+3-1
  • 12=9+3
  • 13=9+3+1
14 can be represented by introducing b=27.  So, 14=27-9-3-1
We now have the only combination of four numbers that can be used to measure weights of 1-14g.
We also see that 27+9+3+1 = 40, so good sign.
We continue to see if this holds true...
  • 15=27-9-3
  • 16=28-9-3+1
  • 17=27-9-1
  • 18=27-9
  • 19=27-9+1
  • 20=27-9+3-1
  • 21=27-9+3
  • 22=27-9+3+1
  • 23=27-3-1
  • 24=27-3
  • 25=27-3+1
  • 26=27-1
  • 27=27
  • ... (we can stop here since we have proved number 1-13 using only x, y, and a and since we can represent 27 with just the 27g weight)
  • example: 35 = 27 + 8 (8 was proved to be 8=9-1, so we can write 35 = 27+9-1)
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Some observations
  • numbers 1-4 can be representation by x=1 and y=3
  • numbers 5-13 can be represented by x=1, y=3, and a=9
    • the numbers (1-4) created from x and y can be added to a to create a wider range of numbers (5-13)
  • numbers 14-40 can be represented by x=1, y=3, a=9, and b=27
    • the numbers (1-4) created from x and y and numbers (5-13) from x, y, and a can be added to b to create a wider range of numbers (14-40)
  • the four weights found are ascending powers of base 3 (exponent=0, 1, 2, 3)
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To extend this problem, I would question whether this pattern of ascending powers of base 3 can be translated to herb amounts greater than 40g and whether this problem would use base 3 as well with greater powers.

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