What is the only state capital that shares no letters with the state it is capital of?
You are right:
Pierre is a city located in South Dakota, so, these two names share no letters.
The city serves as the capital of the state. Pierre became the state capital in 1889, when South Dakota gained statehood. Pierre is located almost in the very middle of the state territory, which makes it a functional capital. Very few people live here nowadays (around 13 thousand).