Which numbers are assigned to Jennings during his stay at the foster homes in the book They Cage the Animals at Night?

The answer to your question is fairly simple:  12, 17, and 27.  These numbers, however, require a bit of explanation. Jennings is placed in quite a few different group homes and foster homes.  Many of these places give Jennings a number in order to identify him.  In the first group home, Jennings number is 27.

She pushed me through one of the dark doorways. ... 'Your number is twenty-seven.  Don't forget it!'"

We see Jennings following that particular number for a while.  For example he lifts "the pair of pajamas off of the hook number twenty-seven" and quickly begins to identify himself as that number instead of by his name which reveals his real identity.  In fact, Jennings even refers to himself as the number when speaking to the other children.

"Who are you?"  A voice jolted me.  There were four boys standing around me. ... "Twenty-seven," I said without thinking.  They roared with laughter before I could correct myself.  "Jennings.  My name is Jennings," I said over their laughter.

Here we can see how important it is for a person (and especially a child) to continue understanding his or her own identity even when he or she is referred to only as a number.  Depending on the group home or foster home Jennings finds himself in, he is always referred to as number twelve, seventeen, or twenty-seven.  It is a way to organize the children in the homes, but it is also a way to make them feel insignificant and unimportant.