[rec.puzzles] Results

mayne@sun16.scri.fsu.edu (William (Bill) Mayne) (04/03/91)

[Additional cross post to rec.humor because of the political
interpretation at the end.]

This is a summary of the results of my survey of methods net readers
used to solve the following simple problem. Thanks to all who have
responded. No more responses needed. If anyone is seriously interested
I'll collect the responses and email them. Use "request" in your
subject line to be sure it gets my attention. Since the survey is over
I may not read all future solutions. I don't want to hear any more about
apple! :-) There have been far too many responses for me to reply to each,
and they are still coming in even as I write this. This summary answers the
questions of several respondents.

>*** PROBLEM ***
>
>  Mary has an even number of apples. Twice the number
>  of apples that Mary has plus the number of apples that
>  John has is some (unknown) constant C. Suppose Mary throws
>  half of her apples away. What should be done with John's
>  apples so that twice the number of apples that Mary has
>  plus the number of apples that John has is still C?

[rec.humor readers may want to skip to the end now.]

First some explanation: The source of the problem was an article by
David Gries in CACM, March 1991, entitled "Teaching Calculation and
Discrimination: A More Effective Curriculum". A solution illustrating
formal methods is given there. Yes, the problem *appears* trivial.  The
point is not what answer you give, but how you interpret and solve it.

As I expected many net readers had seen the article. (Several even sent
me email about it.) That is why I specified I wanted responses from
people who hadn't read or heard the problem before. I knew that would
limit the potential respondents and bias the survey, but that was
unavoidable. I did not credit the source or give more explanation
because I wanted, as much as possible, not to bias the answers
further.

When I read the problem I thought of two different interpretations,
and the main point of my survey was to see how many people would think
of each of them. The actual methods used were of secondary interest.
I asked that partly to avoid suggesting that people go out of their
way to find alternative interpretations. But reading the many
descriptions of methods was interesting, too. I have already gotten
more than 100 responses. I haven't actually tabulated them, but
summarize from general impressions.

There are at least two different interpretations of the problem, and
two ways to try to solve either interpretation, though there are a lot
of different ways to describe either approach.

Interpretation 1: Give John some more apples (or as a few said, have
him pick them, possibly taking some from those Mary threw away.) This
is the obvious interpretation. Only two people besides myself rejected
it or mentioned alternatives. Of those solving this interpretation
there were two approaches.

Approach 1: A large majority used some variation of the following
algebraic approach:

  M = number of apples Mary has
  J = number of apples John has

  2*M+J=C (initial condition)

  Mary throws half of hers away. To compute the number John must have
  after we do something (J'), solve 2*M+J=M+J' for J'=J+M. So give
  John as many apples as Mary had to begin with.

  A variation on this was to have the original count of Mary's apples
  be 2*M, a way of encoding the fact that she had an even number.
  Few used this. At least one commented that the fact that the number
  of apples Mary had being even was only to make the problem work
  without splitting apples.

  Another variation is to solve for the change in the number of John's
  apples directly, i.e. use J+X for the new number of John's apples and
  solve for X.

Approach 2: Think of the problem in terms of changes. (One person
  mentioned his engineering background and thought of it as a signal/gain
  problem.) When Mary throws away M/2 apples this decreases the 
  function of interest by M, so that is how many apples we must give
  John. Most of the 20% or so who thought of the problem in this way
  jumped directly to the solution without need of algebra.

  This approach has the advantage (IMO) that it generalizes well.
  Mary could throw away any number of apples, or any fraction of
  her apples, and the answer "twice as many as Mary threw away" will
  still be correct.

Either "as many as Mary had" or "twice as many as Mary threw away"
is right. The choice of which way to describe the answer correlated
to the approach used to find it.

A few mentioned both approaches, or started with (1) but then realized
the shortcut and jumped to (2). One detailed explanation got all the
way to the end using the first approach and then said "Aha!" and gave
the second approach.

About 4 or 5 got the problem wrong, giving John only as many apples as
Mary threw away. This was the only incorrect answer given.

Interpretation 2: The question asks what should be done "with John's
apples", so it is not given that we must (or are even allowed to) give
him more. I don't think of this as playing with words. It is a word
problem, after all, inviting interpretation. To limit it to the first
interpretation, making it more a test of algebra than a word problem,
it could have just said "How many apples should be given to John...?"

Of the two who thought of this one said that since the number of John's
apples must increase, the only way he can accomplish this by doing
something with his apples is to eat them, plant the seeds, and wait for
the trees to bear fruit. On a test I'd count that as wrong (unless the
calculation of the number needed was given), but give bonus points for
creativity!

The other solution is to take some of John's apples away and give them
to Mary. A little thought or calculation shows that the number required
is twice the number Mary threw away, or the number which she had to
start with. One way to express the equation to be solved for the number
to take from John (X) is: 2*M+J=2(M/2+X)+(J-X) then X=M. For the more
general version, where Mary throws away Y, use 2*M+J=2(M-Y+X)+(J-X)
then X=2*Y.

This solution of course only works if John has at least that many
apples. [Honest, I thought of this as a legitimate answer before
I read the other solution, I wasn't trying to be a smart [deleted]. :-)]

The one respondent who gave this answer saw the problem in rec.puzzles.
This didn't surprise me. I expected that the rec.puzzles forum was more
likely to predispose readers to look for unusual solutions.

End of survey results. But to close with a little humor:

I tried asking my wife, who is a psychologist (counseling type, not a
scientist) to solve the problem. She picked the obvious (first)
interpretation, but had so much trouble figuring out how many apples to
give John that I interrupted to tell her she'd already answered the
real point of my question by showing her interpretation. She then
thought of labeling the approaches Democratic and Republican. For some
reason I can't understand she thought that the second solution was
Republican (maybe because I thought of it and she didn't). But I would 
reverse this. The second solution is Democratic.

John and Mary's Apples, A Political Interpretation:

Democrat (as described by a Democrat): John should help the less
fortunate Mary by giving her some of his apples after her loss.

Democrat (as described by a Republican): Mary threw some of her apples
away, so the state should take some away from John, who saved his, to
make up for her throwing some away, plus more.

Republican (as described by a Republican): Increase the total supply of
apples in the problem, in this case by giving John more. We might give
some to Mary as part of a "safety net", but the problem specified a
solution involving John. Besides, we don't want to reward Mary for
throwing apples away.

Republican (as described by a Democrat): John is the fat cat who has
all of his original supply of apples. Give him even more. The rich get 
richer. Too bad about Mary.

Bill Mayne (mayne@scri16.scri.fsu.edu)