-> Probably... it's now obvious to me that the problem-setter MUST  
-> have used set algebra to pose the question,  because there's no  
-> easy way to construct a possible answer AND THEN measure it's  
-> outcomes. 
  
No, and yes. The problem, as set, included a sketch (*not* an accurate 
drawing) of three overlapping circles, with their centres apparently at 
the vertices of an equilateral triangle. The triple-overlap region, in 
the middle, was shaded. The wording of the problem was pretty much as I 
described it previously. 
  
It is possible to "solve" the problem by a little algebra based on this 
sketch. If the area of each of the "single-cover" regions is X, and the 
area of each of the three "double-cover" regions is Y, then, since the 
area of the "triple-cover" region is 3, we can write two equations: 
  
X + 2Y + 3 = 20    (area of each circle = 20) 
3X + 3Y + 3 = 36   (total area covered = 36) 
  
Dividing the second equation by 3, then subtracting it from the first 
gives: 
  
Y + 2 = 8 
  
So Y = 6, and the total "double-cover" area, which is 3Y, is therefore 
18. That's the answer. No set theory is involved. 
  
All the other work that the kids in that class have recently been doing 
involves simple geometrical stuff. Certainly not set theory. 
  
-> I've had a look at all of the integral variations set logic  
-> permits with a total of 36 and size 20 circles,  and checked with  
-> enough physical rendering to show that none of them can actually  
-> be drawn - because of the above differences between set logic and  
-> geometric restraint. 
  
How can set logic say that something is impossible, when it can, in 
fact, be drawn geometrically? 
  
-> I predict that this teacher probably copied the problem from  
-> somewhere else in good faith,  and that the real author worked  
-> from set logic using integers to create a monster that cannot  
-> exist on a flat surface as might be claimed. 
 
-> I also predict that the original problem-setter expected the  
-> solution to be by way of the algebra,  especially when the  
-> difficulty of measuring geometric solutions is taken into account. 
  
I am sure that the original problem setter (who may or may not be my 
stepson's teacher - I'm not sure) just scribbled the sketch, and 
invented a couple of numbers, 3 and 36, such that the simple 
geometrical/algebraic solution I showed above could be done. It simply 
didn't occur to him that the figure could not be accurately drawn. 
  
Oh well.... 
  
                              dow 
--- Platinum Xpress/Win/WINServer v3.0pr5