[math-fun] Death Valley marathon
Either way, this problem is suitable for a kindergarten entrance exam.
With arithmetic like 5 minus 8/3 = 7/3? You've got to be kidding! If instead of $8 to be divided there were $24 to be divided, so that we could avoid fractions, this problem might be suitable for discussion in grade school. But I think there are economic/ethical issues of what constitutes fairness, in addition to purely mathematical issues. Without a clearer discussion of this, I don't think the puzzle has a single "right" answer. (Though I suspect that the 7-to-1 split is preferable. Maybe consideration of related scenarios with more extreme numbers could make this clearer.) If I ran an elite grade school and wanted to use this problem or some variant as the basis for admission, I'd change the story so that the third party pays $24 instead of $8, so that the use of fractions could be avoided. I'd ask the students how they thought the money should be divided, and why. If their answer involved a $15 and $9 split, I'd explain the reasoning behind a $21 and $3 split; and if their answer involved a $21 and $3 split, I'd explain the reasoning behind a $15 and $9 split. The kids I'd most want to admit to the school would be those who could see that both answers make a kind of sense, and who find this cognitive dissonance both frustrating and enjoyable. ("Hey, that's weird; hey, this is fun!") Speaking of which: Anyone out there care to help me resolve my own cognitive dissonance? Because I'm still unsure which way of splitting the money should be called "fair"! (I have a feeling that there's an Arrow-like theorem lurking here that says that certain seemingly compelling characteristics that fairness should have are actually incompatible.) Jim
On Fri, 10 Aug 2007, James Propp wrote:
Either way, this problem is suitable for a kindergarten entrance exam.
Let me make sure I remember this correctly: A supplies 5L, B supplies 3L, and C supplies $8. Each drinks 8/3L. The water is arguably worth $8 / 8/3L = $3/L. A drank 8/3L of his 15/3 liters, so he needs to be reimbursed for the 7/3L he gave the reporter: $7 B drank all but 1/3L, so he deserves $1. Just to check my reasoning, rather than each person drinking his "own" water, we could give A an IOU for $15 and B an IOU for $9 when they pour their water into the pot. A pays for 8/3L = $8 and is owed $7, B is owed $1. C pays his $8. (Sorry, that seemed obvious to me too, but I wanted to make sure) I suppose we could also make a socialistic argument, since by splitting the water equally they seemed to be accepting some sort of a "from each according to his ability, to each according to his needs" approach. In that case, I'd argue that A, B and C each deserve $8/3. Of course, the problem stipulates that C leaves (perhaps this is symbolic of him living his life to the end without the need for money, since society has provided for all his needs, or perhaps that having experienced brotherhood he no longer desires material goods), in which case the remaining A and B each get $4. The desert setting makes this all the more likely: the author places us in a Death Valley marathon, at the low point of western civilization. All necesseties are critically scarce, yet the people are driven to compete to be first across a meaningless finishing line. Ironic that the water they need so much is their only source of shade. The journalist then fits even better as the outsider bringing enlightenment. Of course, in this case the problem leaves us in an open-ended state. Will A and B be tempted by the ironic capitalist notes? Since there are two of them, and given the eastern motif, it seems not unlikely to me that one would play the yin, perhaps using the notes merely as paper on which to write a poem, and one the yang, eventually being overcome by greed. However, a more optimistic author might have both A and B learn from C's example and give up desire, exemplifying how good flows outward and in so doing, grows. Of course, we'd then need to know how A and B's actions affected D, E, F and G to determine whether the author intended the growth to be of the form 2^n, 2n, or something more exotic. Here, then, is my final answer to the question: A desert burns hot Three travelers share their thirst Enlightening all
Hi Jim I graduated from kindergarten 80 years ago. But I'm afraid don't see "reasoning behind a $15 and $9 split". In all the economic systems I know of a person pays the same price for buying or selling a good independent of whether he is rich or poor. What did you have in mind? At 10:27 AM 8/10/2007, you wrote:
Either way, this problem is suitable for a kindergarten entrance exam.
With arithmetic like 5 minus 8/3 = 7/3? You've got to be kidding!
If instead of $8 to be divided there were $24 to be divided, so that we could avoid fractions, this problem might be suitable for discussion in grade school. But I think there are economic/ethical issues of what constitutes fairness, in addition to purely mathematical issues. Without a clearer discussion of this, I don't think the puzzle has a single "right" answer. (Though I suspect that the 7-to-1 split is preferable. Maybe consideration of related scenarios with more extreme numbers could make this clearer.)
If I ran an elite grade school and wanted to use this problem or some variant as the basis for admission, I'd change the story so that the third party pays $24 instead of $8, so that the use of fractions could be avoided. I'd ask the students how they thought the money should be divided, and why. If their answer involved a $15 and $9 split, I'd explain the reasoning behind a $21 and $3 split; and if their answer involved a $21 and $3 split, I'd explain the reasoning behind a $15 and $9 split. The kids I'd most want to admit to the school would be those who could see that both answers make a kind of sense, and who find this cognitive dissonance both frustrating and enjoyable. ("Hey, that's weird; hey, this is fun!")
Speaking of which: Anyone out there care to help me resolve my own cognitive dissonance? Because I'm still unsure which way of splitting the money should be called "fair"! (I have a feeling that there's an Arrow-like theorem lurking here that says that certain seemingly compelling characteristics that fairness should have are actually incompatible.)
Jim
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Please let's stay with water. If you think the 5-3 split makes sense with water then you shouldn't have to argue it by appealing to three story buildings. Here is Salimin's original argument The two friends are paid $8. The guy who contributed 5 l gets $5, and the guy who contributed 3 l gets $3. The mistake is in the word contributed. The first guy didn't contribute 5 liters. He happened to have 5 liters. He contributed 7/3 of a liter, and similarly for the other guy. dg At 10:27 AM 8/10/2007, you wrote:
Either way, this problem is suitable for a kindergarten entrance exam.
With arithmetic like 5 minus 8/3 = 7/3? You've got to be kidding!
If instead of $8 to be divided there were $24 to be divided, so that we could avoid fractions, this problem might be suitable for discussion in grade school. But I think there are economic/ethical issues of what constitutes fairness, in addition to purely mathematical issues. Without a clearer discussion of this, I don't think the puzzle has a single "right" answer. (Though I suspect that the 7-to-1 split is preferable. Maybe consideration of related scenarios with more extreme numbers could make this clearer.)
If I ran an elite grade school and wanted to use this problem or some variant as the basis for admission, I'd change the story so that the third party pays $24 instead of $8, so that the use of fractions could be avoided. I'd ask the students how they thought the money should be divided, and why. If their answer involved a $15 and $9 split, I'd explain the reasoning behind a $21 and $3 split; and if their answer involved a $21 and $3 split, I'd explain the reasoning behind a $15 and $9 split. The kids I'd most want to admit to the school would be those who could see that both answers make a kind of sense, and who find this cognitive dissonance both frustrating and enjoyable. ("Hey, that's weird; hey, this is fun!")
Speaking of which: Anyone out there care to help me resolve my own cognitive dissonance? Because I'm still unsure which way of splitting the money should be called "fair"! (I have a feeling that there's an Arrow-like theorem lurking here that says that certain seemingly compelling characteristics that fairness should have are actually incompatible.)
Jim
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participants (3)
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David Gale -
James Propp -
Jason Holt