(1) ...aquadium...is worth 3x more than haxite...
(2) ...The Chief-Miner oversaw the operation, rubbing his haxite necklace for good luck.
(3) ...the miners got 14 aquadium crystals and 26 haxite chunks.
(Directive) "Split both gems' numbers in three and fill up the ore carts!"
(Problem) How did they split up equal worth into all three ore carts?
The three relevant facts, the directive and the problem appear in the quote above. Note that the problem, which is to divide value, is actually different from the directive, which is to divide numbers. Nevertheless, let's assume that's what the sub-chief meant.
My first thought was to convert everything to haxite value and find out how much each prince gets. Thus, 14 x 3 + 26 = 42 + 26 = 68. Since 68 is not evenly divisible by 3, the problem is unsolvable without more information.
Based on the apparently passing reference found in #2, I assume the "real" answer has to do with "borrowing" the Chief Miner's necklace to increase the total value to 69 haxite and divide up the bootie, with the supposition that the Chief Miner gets his necklace back from the miners' work tomorrow. But this resolution suffers from some problems:
1. No gem that I have ever heard of has a single monolithic value for each stone or crystal. This assumption would require that haxite and aquadium gemstones are all perfectly uniform in their natural state, a condition that is neither stated in the problem nor reasonable to assume. It is more reasonable to assume that statement #1 means that aquadium is three times as valuable as haxite
per unit weight, or some equivalent treatment.
2. A "haxite necklace" may well consist of more than just a single stone pendant, just as a "pearl necklace" might be a string of pearls and not just a single pearl. Based on the wording of the problem, we cannot safely assume that the "haxite necklace" is a single stone.
3. We do not know that the Chief Miner was even present when the division was made, or that he would be willing to give up a valuable (we assume) necklace just so that his underling could divide things up.
4. Somehow, introducing outside value into the pot to be divided seems to violate the spirit of what's trying to be done. Sure, if you can't divide things out evenly, you can always throw some of your own money in to make things work out. You might even do this if you're a parent and your kids are fighting. But this doesn't seem like a reasonable course of action for a manual laborer.
A better solution, I think, is to choose the largest aquadium gemstone and the largest haxite gemstone and split each of them in two. A fraction of a gemstone is still a gemstone, and doing this would give you a number of each gemstone that is evenly divisible by three, thus allowing the Chief Miner's original instructions to be followed to the letter.
