Before creating
the 5 weights, The ancients had to create a 1 gram weight to measure out all their other weights.
So the ancients had a bunch of super fine sand which was made entirely of the same material, and every grain was its most basic material. They had 121 grams of this sand. They also had a tube which could hold exactly 121 grams of sand, no more, no less.
How do the ancients weigh out 1 gram of sand?
The problem goes to great lengths to assure us that the sand is a homogenous mixture with a constant density. It seems to me that if you marked off the tube in 1/121 increments, the sand between each tick would weight 1 gram. It would be difficult to calculate where the ticks belong. If the number were 128, you could just divide by 2, 7 times. If you could accurately divide the tube into 11 sections, and then divide each section by 11 again, you would have a 1 gram mark.
Do we have a balance that will allow us to split the collection of sand in half? I'm not sure if that knowledge would help.
Viable approach
