They are the....
And here's your host....
John Davidson!
Broadcast History
Airdates: January 15, 1986-September 8, 1989 ("The New Hollywood Squares" during first season)
Host: John Davidson
Announcer: Shadoe Stevens, Richard Stevens
Substitute Host: Jim J. Bullock, Alf, Shadoe Stevens
Music: Stormy Sacks
Origination: NBC Studios, Burbank (1986), ABC Television Center, Los Angeles (1987), Fox Television Center, Los Angeles (1987-1988), Universal Studios, Hollywood (1989)
Main Game
The Squares returned in 1986, with a new host, John Davidson and basically the same gameplay.
John explained it this way:
"The
object of the show is to win Tic Tac Toe. Three squares across, down or diagonally,
or to
acquire as many Squares they can." As before also, it was up to them figure
out if the celebrity was giving the 
correct
answer, or an incorrect one. In season one, every game was
worth $500 until time ran out. When time ran out, each Square was worth $50. In
season two and beyond, the first 
two
games were worth $500, with each additional worth $1000. When time ran out, $100
was awarded for each Square. Also, a 5 Square win could still win a game if no
Tic Tac Toe was possible. Game 2 was always the "Secret Square" game,
which usually had a trip as a bonus
prize. The player with the most money went to the bonus round.
Bonus Round
In the bonus round, John
Davidson showed the player a crystal bowl, which had 5 keys in it. The player
picked a 
key,
and then would pick one of five cars. The player would then take the key they
picked, and
hoped it would start that car. If they won, they retired. If they lost, that car
would be eliminated, and the player would try their key the next day (if they
won) to one of the four cars. Later in the run, the player picked
one of the 9 squares, which all had keys, to fit one of the 5
cars.
Facts
-Joan Rivers was the Center Square for most of the run. Also, Jim J. Bullock was a regular, as was announcer Shadoe Stevens, who frequently occupied a Square, as well as announcer.
-The New Hollywood Squares was one of the first game shows to go on the road, with trips to Hollywood, FL and New York City.
-John Davidson was a semi-regular on the original Hollywood Squares
The newest version of the Squares, hosted by Tom Bergeron!