Answer: Teeth is to hen as nest is to MARE.
According to sources, the expression "as scarce as hen’s teeth" is a colorful idiom meaning “nonexistent” because hens do not have teeth. The original meaning of "mare's nest" is something illusory; a hoax or fraud.
The expression “to find a mare’s nest” means to come upon something fraudulent or deceptive, like a hoax. You expected something real, but you found a mare’s nest.
Every language has its own collection of wise sayings. They offer advice about how to live and also transfer some underlying ideas, principles and values of a given culture / society. These sayings are called "idioms" - or proverbs if they are longer.
These combinations of words have (rarely complete sentences) a "figurative meaning" meaning, they basically work with "pictures". Here’s a list of commonly used idioms and sayings (in everyday conversational English), can help to speak English by learning English idiomatic expressions.
Of interest, idioms, aka proverbs, have long infiltrated and colored our language, often confusing children and foreigners alike. But what of them? Many, still lodged in the agricultural age, are outdated and in need of some modernization.
Check out a few popular phrases or idioms that belong to a different era at The Huffington Post.