He’s 81, he’s broke, and he has Alzheimer’s — yet a California man somehow managed to travel 3,000 miles to Manhattan in a journey that has baffled cops on both coasts.

Jude Hunter, who doesn’t even have a wallet, vanished on Jan. 10 during a routine trip to his doctor in Oakland.

How he ended up a week later at St. Luke’s Hospital, in Morningside Heights, is a mystery.

Hunter never mentioned an interest in New York — although the day before he left, he talked about someday visiting London, said Ignacio Tan, the administrator of Hunter’s Newark, Calif., group home.

“If he needed something, he just had a few dollars in his pocket, and that’s it,” Tan said.

A social worker yesterday was trying to figure out a way to get Hunter home to California.