A long time ago some ancients needed to make 5 weights that could weigh any amount of weight up to 121 grams.

The ancients though, had only a 1 gram weight and 120 grams of clay(which they would later make into weights. So with only a scale, 120 grams of clay and a 1 gram weight how would you create all the other weights needed to weigh any amount up to 121 grams?

What is the minimum amount of weighings required to do this?

The 1-g weight can be used to measure out 1 g of clay, then that, together with the first 1-g weight can weigh out another 2 g of clay, which can be combined with the 1 go of clay to make a 3-g weight to keep.

The 1-g weight and the new 3-g weight can weigh out 4 g of clay, then that plus the 1-g weight can weigh out another 5 g of clay, which can be combined with the 4 g to make a new 9-g weight.

The 1-g, 3-g and 9-g can weigh out 13 g of clay, and then the 1-g and 13-g can be used to weigh out 14 g of clay, which combined with the 13 can make a new 27-g weight.

At this point if by some methodology these ancients actually KNEW they had 120 g of clay to begin with, and hadn't lost anything in handling, the rest of the clay would be 81 grams. Otherwise they'd have to go through another iteration and use the 1-,3-,9- and 27-g weights to weigh out 40 g of clay and then the 1-g and 40-g weights could weigh out the last 41 g to combine with the 40 g to make 81 g.

I don't know if there might be additional shortcuts, but this is 6 weighings if the ancients knew they had 120 g of clay to begin with, or 8 weighings if they didn't.

Posted by Charlie on 2003-03-24 03:22:10