G'morning David, 

 DW> "Three circles are drawn on a flat surface. Each circle encloses an
 DW> area of 20 cm^2. They overlap so that the total area that is enclosed
 DW> by one or more circles is 36 cm^2. The area that is enclosed by all
 DW> three circles is 3 cm^2. What is the total area that is enclosed by
 DW> exactly two circles?"

 DW> The more I play with it, the more interesting it gets. Any thoughts
 DW> about it, from anyone, would be welcome.

Mmmm - the use of `circles' and `cm^2' is misleading for this 
problem can be the province of combinatories, set theory or 
Boolean algebra....

I went at it like this....

Given     A = B = C = 20,
And (A and B and C) = 3,
Ergo = A and B = A and C = B and C

 Then 36  = A+B+C-(A and B)-(A and C)-(B and C)
            +(A and B and C)
          = 60 - 3(A and B) +3
(A and B) = (60 + 3 - 36)/3
          = 9
(A and B) - (A and B and C) = 9 -3 = 6

Ergo,  the `total area enclosed by exactly two circles'
is 18 (cm^2),  giving you the following distribution of 
coverages...

Amount Covered By...
3 circles =  3 cm^2
2 circles = 18 cm^2
1 circle  = 15 cm^2
~~~~~~~~~~~~~~~~~~~
Total     = 36 cm^2

This all fits the logic of the problem;  I haven't tried to see if 
one can actually draw a true-to-scale Venn diagramme with real 
circles of it....

Any help ?

:-)

--- Maximus/2 3.01