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Ducks and Chickens

  1. Jun 25, 2008 #1
    A farmer had a whole mess of chickens and ducks. He had twice as many ducks as he had chickens. One day he went to market and sold 413 of his ducks. When he came back home, he discovered that 19 of chickens had died. He now had half as many ducks as chickens.

    The challenge is to find a solution without using algebra. Furthermore, a solution that would be accessible to an extremely talented 4th grader, but, with only a fouth grade education.

    I do not know if such a solution exists. I'll outline my "best shot" in a subsequent post. It uses no algebra, but it uses too many things that might be viewed as "tricks" by a talented fourth grader. This means that it does a poor job of statifying the "accessible to a fourth grader" criteria. Another way of stating the problem is to find a solution that might be more accessible to a fourth grader than mine.

    The problem was allegedly assigned to the son of a member in my church. He posed the problem to a group of about 10 adults in the church who could not solve it (without algebra). I was home sick that day. My wife bought the problem home and I got it, but, it took me quite a while.

    I was not sure where to post this problem. I considered the "number theory" thread. Because of its similarity to the "three sailors, one monkey and a whole bunch of coconuts" problem that is described in the book "Algebra" by Saunders and MacLane. I considered other forums. This one seemed the most likely to have readers interested in this kind of thing.

    I'll wait a while before I post my solutions. Maybe some of you will better me first and I won't have to.

    I don't know how to use hidden text. What is the sequence of symbols that I type to get it started?

  2. jcsd
  3. Jun 25, 2008 #2
    [color=#e1e1e2]Hidden text[/color]
  4. Jun 25, 2008 #3


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    A challenge indeed.
  5. Jun 25, 2008 #4


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    I have the answer, and I did it without algebra:
    :There is no solution. Why? Because no question was asked!:
  6. Jun 25, 2008 #5
    Would asking for the answer on a forum on the internet constitute "not using algebra"?
  7. Jun 25, 2008 #6
    Oh, that is good! ROTFL!

    Of course you know that's not what I meant, but it certainly meets the criteria as i spelled them out. You also realize, I hope, that it is not so much an "answer" but an "understanding" that I am searching for.

    Of course, I meant a solution that reasons out the answer using things that are commonly understood as techniques of reasoning. Things like adding, subtracting, multilying, dividing, approximatiing, iterating, intelligent or informed guessing, etc. I don't mean to include "brute forcing it," i.e., trying all possible

    My answer to your question reminds me of the conclusion of the highest-level sudoku players that if they can't find the solution by using "logic," they go on to the next puzzle. In other words they don't count "guessing a number for a square and proceeding until a contradiction arises" as an interesting solution to a sudoku puzzle.

  8. Jun 25, 2008 #7
    Jimmy, A thousand thanks. DJ
  9. Jun 25, 2008 #8
    Dave, I don't understand your comment.

    [Edit added later: Oh, Dave, I think I understand know. You are asking the question, "What is the question?" Hence the multiple question marks. But why three and not two question marks? Any reason? --- Also, I did not change what i orginally wrote below in my reply to your quote. -- It still seems relevant.]

    The problem was certainly a challenge for me. In fact it makes me wonder if "old age senility" is beginning to set in.

    Confidentially, that is part of the reason I am asking the question. I have strong suspicions that there is a better answer than the one I found, but, I have been unable to find it after hours of calculationing with paper and pencil and days of cognating.

    Actually, I never use pencil, only pens. Vanity. :)


    For this problem, I refused to use algebra to "find the answer" before I solved the problem. And I spent an hour so so trying to solve it in my head before "I put the pencil to the paper."
    Last edited: Jun 25, 2008
  10. Jun 25, 2008 #9
    Actually, that's very perceptive of you. In fact, I just ran into that problem as I attempted to estimate the number of computations required in a "brute force" solution.

    But, I failed to see my omission of an explicit question as a flaw in my problem statement. Thank you for pointing that out.

    The question that I had in mind is:

    How many ducks and how many chickens did the farmer have before he went to the market?

    However, I if I were teaching the class, I would give full credit for a solution that showed how to find how many ducks and chickens the farmer had after he came home from the market.

    I also appreciated the increased excitement created by your use of "hidden text." When I saw that you claimed that you had a solution, I selected your "hidden text" with great anticipation. When I saw what you had discovered, I found it quite amusing. Not exactly ROTFL, but definitly "sitting in my chair laughing."



    Since you are so perceptive, I will point out that the form of my question is carefully put to lead the solver in the the direction of a solution. That is why there is that strange redundancy in the question -- namely, once you know the original number of chickens, asking for the number of ducks is (almost) redundant.

    In other words, the form of the question is actually kind-of-like a hint for how to solve it. On the other hand, the "hint" is based on the solution that I found. That's why I added the part about "full credit" for the numbers after the farmer comes home from the market.
    Last edited: Jun 25, 2008
  11. Jun 25, 2008 #10


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    Actually, you're over-analyzing it.

    "??? A challenge indeed."

    Simply means:

    "What??? I can't even imagine any way to solve such a problem without algebra!"

    That comment has nothing to do with any of my subsequent comment(s).
  12. Jun 26, 2008 #11
    Ducks and Chickens - Suboptimal Solution

    Here's the solution I found. My impression is that there exists a simpler solution. Or, at least one with fewer "tricks." See what you think.

    [Currently this only contains the start of the solution that I found.

    Trick Number One: Solve a simpler, but similar, problem.

    In this problem, the number of chickens that died is much smaller than the number of ducks that were sold, so go ahead and set the number of chickens that died equal to zero.

    Trick Number Two: Look at the problem from the point of view of the farmer. Be imagiinative. Flesh out the story.

    Well the farmer doesn't know much about math, but he does now that before he went to market he had twice as many ducks as chickens, and now he's got half as many ducks as he has chickens.

    So, the farmer can arrange the three numbers that he wants in increasing order as follows:

    1) Number of ducks after market trip
    2) Number of chickens
    3) Number of ducks before market trip

    Trick Number Three. Try substuting the numbers in the problem with the smallest numbers that make sense to you. Start of with the number one if you can.

    Well the number of ducks after the market trip is the smallest number, so let's assume that is 1. ...
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