G'morning again David, 

 DW> If he had just used the words "closed figures" instead of
"circles",
 DW> there would have been no difficulty.

Exactly right - but thank goodness he didn't go for four spheres 
in 3D space !

Set and group theory don't require set geometric locations for the 
members they handle,  even though each constraint may be treated 
as if it were orthoganl to all its fellows.

 DW> Alternatively, if he had given the
 DW> total area *or* the triple-overlap area, but *not* both, the question
 DW> would have been consistent, although probably not soluble by the kids
 DW> to whom it was given.

Err ... only if he omitted the terms `circle' and `cm^2' to talk 
instead about group membership...

 DW> But, by stating "circles" and giving both areas,
 DW> he unknowingly postulated an impossibility.

Right on.

 DW> I did the same a few times, when I was a math teacher. It's an easy
 DW> trap to fall into.

The road to elucidation is paved with misunderstandings.

I enjoyed a lecture where my psychometrics professor went through 
the formal proof for calculating a coefficient of correlation.

The class stared at the workings on the board until a timid voice 
asked if a line near the beginning was not indeed wrong.

Mrs Bernadelli stared, nodded, apologised and corrected the line, 
and then stood beaming ... expectantly.

Emboldened,  others scoured the proof for other mistakes until a 
triumphant yell pointed out a second mistake.

"Of course there's two mistakes", bubbled the prof, "You have to 
make an equal number of them to get to the right answer !"

And in my own teaching,  I now shudder at the times I knowingly 
corrected initial mistakes to make a final point....

:-)

--- Maximus/2 3.01