Well, that's the theory.
In practice, they ran out of well-known paintings years ago, and now it's got to the stage where they may as well just publish the painting as a whole and ask people what it's called. Here's what last week's detail was:
Here's the painting it came from:
Now if you know the painting, no way are you not going to get it from that detail. They might as well show you the whole painting and say "What's this called and who painted it?".
Even that wouldn't help me much. I've never seen this painting before, and I doubt I'll ever see it again. It has a title I've never heard of (Allegory of Smell), is by an artist I've never heard of (Jusepe Ribera) and it hangs in a gallery I've never heard of (the Abello Collection, Madrid). When I looked up Allegory of Smell in Google and omitted the references to Brueghel (who did a painting of the same name) I got 9/13 matches — and three of those referred to Wagner, the composer. Still, I did get to see what the other paintings in the Allegory of... series looked like, as one of the links had them all as thumbnails. The first of these, Allegory of Touch, actually seems familiar to me, but I suspect it's only because they've used it in The Independent on Sunday's quiz at some point.
Oh well, the prize for correctly identifying the picture from the detail is a bottle of champagne, so it's useless to me anyway as I don't drink...