math challenge

Week of March 25 - 31, 1996

King Harold's Jewels

Harold is about to be crowned king of MathLand. He is selecting the jewels for his crown and wants only the finest, heaviest stones. He has only a finely-calibrated balance scale to assist him in his task.

Harold has two diamonds; one a trifle heavier than the other.

How can Harold determine which is the heaviest?

Oops, that's too easy. All he has to do is put one diamond in one tray, and the other in the second tray. It will be pretty easy to see which one is the heaviest.

a) But suppose Harold has three blue sapphires, b1; b2; and b3. The weight of
b1 < b2 < b3. How will Harold determine which sapphire is which, and what is the minimum number of weighings he will need?

b) Harold figured that out and now wants to select his rubies and emeralds. He has two red rubies (r1 and r2) and two green emeralds (g1 and g2). The weight of r1 = the weight of g1. The weight of r2 = the weight of g2. The weight of r1 < r2. How will Harold determine the weight of r1, r2, g1, and g2, and what is the minimum number of weighings he will need to do so?

c) Harold has four amethysts and is hoping that their reputed good-luck qualities will help him. Two of the amethysts are equally light. The other two amethysts are the same weight compared to each other, but are a little heavier than the two lighter amethysts. Harold does not know the weight of any of the amethysts. How will Harold determine which amethysts are the light ones and which are the heavy ones using the minimum number of weighings?

See the solutions