Question

Why the weights in the weight box are in the ratio 5:2:2:1?

Answer 1

This average comes out to be 2.1 in the 5:2:2:1 case and, wait for it, 2.0 in the 4:3:2:1 case. 

Answer 2

This is interesting. There may well be a simple historical reason why this is so, but if I had to guess from a purely mathematical perspective…

First, observe that you can make every weight from 1 to 10 (units) using the above four weights.

1=11=1

2=22=2

3=1+23=1+2

4=2+24=2+2

5=55=5

6=1+56=1+5

7=2+57=2+5

8=1+2+58=1+2+5

9=2+2+59=2+2+5

10=1+2+2+510=1+2+2+5

Can we do the same thing simply using 4:3:2:1 instead? It turns out we can. So what gives?

Let us calculate the average number of weights required to make up a particular sum in both the cases. This average comes out to be 2.1 in the 5:2:2:1 case and, wait for it, 2.0 in the 4:3:2:1 case. We still don’t have an answer!

Knowing what we do, my best bet would be that the 5-unit weight is more commonly used than the 4-unit weight, and so it is more desirable to have the 5-unit weight as a ‘basic’ weight rather than making it up from smaller weights (1+4). We can, though, afford to have the 4-unit weight as a combination of (2+2).

Math is fun.

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