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Do you own any of the other varieties of ant-nose coins?
I own two of the nose coins, not the ant coin.
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What is the rarest type (I'm guessing it might be the three-character type)?
There are several that are even rarer than this one. They have single-character inscriptions such as "Xing" (crossroads), "Jin" (metal), and a few others even rarer than those that I can't think of off the top of my head. The three-character type (the ant coin) is the second most common.
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Also, these are thought to have hung on strings, right? Or is the hole just a decorative carry-over from an older tradition of stringing cowrie shells? I ask because I wonder if their hanging down from a string through the holes is what gave rise to the assumption (of Coole's) that they should be oriented the other way.
Very interesting question. I would say plausible, except for the fact that the vast majority of these coins do not have a hole that goes all the way through, making the vast majority incapable of being strung together. Coole's reasoning was the assumption that the hole (or very deep divot) was part of the inscription.
You can find my discussion of this here:
https://goccf.com/t/234645&whichpage=2#1971645