## Teaser 2980: Egyptian weights and measures

**From The Sunday Times, 3rd November 2019** [link]

We were wondering why ancient Egyptians chose to represent arbitrary fractions as sums of distinct unit fractions of the form 1/n (thus 5/7 = 1/2 + 1/5 + 1/70). One of us recalled long ago watching our greengrocer use four brass weights of 1/2, 1/4, 1/8, 1/16 lb to weigh any number of ounces up to 15 by stacking some of them on one side of her balancing scales. We wanted to make a metric equivalent, a set of distinct weights of unit fractions of a kilo, each weighing a whole number of grams, to weigh in 10g steps up to 990g.

Naturally, we wanted to use as little brass as possible, but we found that there were several possible such sets. Of these, we chose the set containing the fewest weights.

List, in increasing order, the weights in our set.

[teaser2980]

## Jim Randell 5:03 pm

on1 November 2019 Permalink |Possible weights are the divisors of 1000.

This Python program looks for subsets that permit all the required weights to be made, and then selects those sets with the minimal possible weight. It runs in 501ms.

Run:[ @repl.it ]Solution:There are 10 weights in the set: 2g, 5g, 8g, 10g, 25g, 40g, 50g, 100g, 250g, 500g.The total weight of the set is 990g (which is obviously the minimum possible total to be able to weigh up to 990g).

There are 4 viable sets of weights that have a total weight of 990g:

When I initially read the puzzle I solved it allowing weights to be placed on both sides of the scales, and I found two sets of 7 weights which can be used to achieve the required values, where both sets weigh 990g in total:

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