Question 789195
{{{drawing(300,300,-15,15,-10,20,
circle(-5,0,8),circle(5,0,8),circle(0,8.66,8),
locate(-7,-1,B),locate(6,-1,R=673),locate(-0.5,-1,BR),
locate(-0.8,3.5,BPR),locate(-0.3,11,P),
locate(-5,6,BP),locate(4,6,PR)
)}}}

There are 7 variables for 7 groups of readers:
B = number of people who read only the Bugle
P = number of people who read only the Planet
R = number of people who read only the Recorder
BP = number of people who read only the Bugle and the Planet
BR = number of people who read only the Bugle and the Recorder
PR = number of people who read only the Planet and the Recorder
BPR = number of people who read all 3 papers
 
There are 6 equations:
BP + BR + PR = 196 because "196 read exactly two of the three papers"
R = 673 because "673 read only the recorder"
B + R + BR = 926 because "926 of those who read a paper did not read the planet"
B + BPR + PR = 153 because "153 of the planet readers read at least one of the other two papers
B = BR + 171 because "Of those bugle readers who did not read the planet, there were 171 more who read the bugle alone than who read the recorder"
B + BPR = 210 because "210 of the bugle readers read either the bugle alone or else both of the other papers"
 
With more variables than equations, we cannot solve the system of linear equations for all variables.
However, there is an inconsistency.
There is one subset of equations where we can solve for 3 of those variables.
Among the people who do not read the Planet, we have 3 groups:
those who read only the Recorder,
those who read only the Bugle,
and those who read just the Bugle and the Recorder.
We can find the size of those 3 groups.
{{{system(R=673,B+R+BR=926,B=171+BR)}}} --> {{{system(R=673,B=212,BR=41)}}}
So, if B=212, then B + BPR = 210 would require a negative BPR.
{{{drawing(300,300,-15,15,-10,20,
circle(-5,0,8),circle(5,0,8),circle(0,8.66,8),
locate(-9.5,-1,B=212),locate(6,-1,R=673),locate(-2,-1,BR=41),
locate(-0.8,3.5,BPR),locate(-0.3,11,P),
locate(-5,6,BP),locate(4,6,PR)
)}}}