Solving Questions

Date: 08/28/2002 at 10:00:12
From: Pierre
Subject: Logical Riddle

Dear Sirs,

I got this challenging question in my Math course:

In a poll of 37 students, 16 felt confident solving quantitative 
comparison questions, 20 felt confident solving multiple choice 
questions, and 18 felt confident solving gridded response questions. 
Of all the students, 4 were confident solving gridded response and 
quantitative comparison questions, 5 were confident solving multiple 
choice and quantitative comparison questions, and 6 were confident 
with gridded response and multiple choice questions. Of those 
students, only 1 felt confident with all three types of questions.
What number of students felt confident solving only multiple choice 
questions and no others?

I tried to solve this riddle, and I got 10. My teacher told me that 
the answer is actually 8. I just can't figure this out. Would you 
please tell me which one is the right answer, and explain to me how to 
solve such questions?

Thanks for any help.

Date: 08/28/2002 at 12:31:49
From: Doctor Peterson
Subject: Re: Logical Riddle

Hi, Pierre.

I am always uneasy with these puzzles, because they are never worded 
clearly enough for me. Which numbers are meant to be inclusive? The 
first three numbers are clearly inclusive (that is, 16 like AT LEAST 
QC, but possibly others), since otherwise the answer would be given 
directly. Then the next three numbers seem likely to be inclusive as 
well, so that 4 like both GR and QC, AND possibly also MC. That 
implication is further supported by the wording "of those students 
[the 4, 5, and 6?], only 1 ...." So my assumption, probably like 
yours, is that the 1 is included in each of those numbers, so that 
there are 3, 4, and 5 who like ONLY two types of questions. That 
makes it easy to fill in a Venn diagram, and we quickly see that the 
circle for MC, whose total size is 20, has 1 shared with both QC and 
GR, 4 with QC alone, and 5 with GR, leaving 20-(1+4+5) = 10 for MC 
alone. So I agree with your answer...

... except that if we then add up the total, we find that 40 students 
like something, out of 37 students in all! So our assumptions do not 
fit all the facts. Looking for something to reinterpret, I consider 
the possibility that the single student who liked everything might 
NOT be included in the 4, 5, and 6. Taking it that way, the number 
for MC alone is 20-(1+5+6) = 8, agreeing with your teacher's answer.

I think the problem is very poorly stated. You might like to try 
rewording it to make everything clear. That's a good exercise in 
communication, which mathematicians, pollsters, and puzzle-makers 
should pay attention to as well! (Except when a puzzle is deliberately 
vague to mislead you, which this one should not be.)

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/