G'morning David

 DW> But how did you calculate these figures?

I expressed the problem as...

36 = 20 - 3x + y,  or  20 = z + 3x - 3

...and ran them through my MEDIATOR - a Media Calculator I wrote 
sometime ago when I was supposed to be the local guru on media 
audience analysis, formation and planning for mass communications 
purposes.

 DW> But it also looks to me like you *assumed* that the centres
 DW> of the circles are at the vertices of an equilateral triangle, and
 DW> there is nothing in the problem that states that this must be so.

When I got to try rendering the logical answers on paper,  and 
only to the extent necessary to judge the fit.

 DW> all you've proved is that it's impossible for the circles to be in the
 DW> equilateral-triangle arrangement. Some other arrangement may be
 DW> possible.

Mmmm -  well, as far as I can see,  NOT for all the possible 
Diophantine variations...


Thrice  0  3  6  9 12
Twice  24 18 12  6  0
Once   12 15 18 21 24

Total  36 36 36 36 36

And, of course, circles are stated,  and fit in with the mind- 
state attending the Venn diagramme model.

 DW> The question is: Can *any* arrangement of the circles exactly fulfill
 DW> the problem's description of the situation? If so, what? And if not,
 DW> can we prove it?

I suspect not on any flat surface as specified - and I lack the 
topological skills to even suggest a non-surface that might permit 
it.

So, given that each possible logical integral arrangement and the 
two alternative non-integral arrangements cannot be drawn as 
specified on a flat surface,  may we not pronounce the matter 
proven - subject only to a model answer disproving the 
pronouncement ?

Which leads to further thoughts...

:-)
 
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