**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)