This sounds like the show "Incredible Games". It was broadcast between 1994-1995 (two series) by the BBC. It included a variety of games, performed by contestants between the ages of 11 and 14. The show itself was set in a fictional skyscraper, with a talking lift.
The aim of the show was to complete each game (set on a "floor" of the skyscraper), gradually rising up the tower, until finally reaching the penthouse, where prizes could be won. If a player lost a game, they were sent to the "basement" of the tower (the laundry room), and the other players had to take the lift back down to rescue them.
Two of the more famous games were the alphabet soup game, where giant letters had to fished out of swimming pool (supposedly a soupbowl) to make words on the "fridge", and the "Dark Knight game". In this game, the contestants had to move from square to square on a chess board until they reached the end. The catch was that the "dark knight" was attempting to stop them. Fortunately for the players, he could only see where they had been, not where they were.