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### 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

... 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/
```
Associated Topics:
High School Logic
High School Puzzles
Middle School Logic
Middle School Puzzles

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