The Market Scales Puzzle

I did not end up solving this problem. Here is what I tried:

First, I looked at all the different ways 4 numbers could be placed on 4 scales, and it seems like there is only one way to make 40 unique numbers when you start with 4. 

Then I thought about Egyptian multiplication. I tried doubling to make 40 and the numbers that were used are 1, 2, 4, 8, 16, 32. Then I tried to find a combination of 4 of these that could be used. I knew 1 would have to be one of the numbers so I could make odd (I wrote even on my page but meant odd) numbers. And I knew 32 would have to be one so I could make large enough numbers. 32+16=48 which is larger than 40 so assuming I was right that each way of adding and subtracting the numbers (that doesn’t produce a negative number) needs to be 1 – 40, 16 could not be one of the numbers. So, I looked at all the ways 1, 2, 3, 4, and 32 could be used to make 1 – 40 to see if I could figure out what the final 4 numbers would be.

I wasn’t able to make 16 however with these 5 numbers and I did not find a obvious way that 4 numbers could make 1 – 4 using this.

Some ways I could have went wrong are that it seems that either I was wrong that there are only unique ways to combine 4 numbers or, I should not be using the doubles as the numbers. If it is true that there are only 40 ways to combine 4 numbers then the largest number would be a+b+c+d so a+b+c+d would need to equal 40 but there is no combination of 1 ,2 ,4, 8, 16 and 32 that includes 4 numbers and adds up to 40. Beyond this there are lots of combinations of these numbers that repeat (16-8=8 but you already have 8 on its own same with 8-4=4, 4-2=2, and 2-1=1). So either my guess that I should be using doubles was wrong or my logic in determining how 4 numbers a, b, c, d could be combined was wrong.

Despite this I did find this to be an interesting exercise, and I feel that I understand the Egyptian way of adding doubles to create any number better. Specifically adding the doubles to get 1 – 15 made it really clear how they are essentially binary.