What four, positive, dissimilar numbers have a sum that equals its product?

Discussion in 'Community Discussion' started by psycoswimmer, Mar 5, 2008.

  1. psycoswimmer macrumors 65816

    psycoswimmer

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    #1
    All in the title really.

    My little sister's math teacher gave them this as a "fun" problem, and I can't get it. I tried google but the answers there didn't help me much.

    Basically, what I need is something like "1+1+1+1=1*1*1*4" but following the right criteria.

    Any math geniuses to the rescue? :D
     
  2. Abstract macrumors Penryn

    Abstract

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  3. siurpeeman macrumors 603

    siurpeeman

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  4. psycoswimmer thread starter macrumors 65816

    psycoswimmer

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    #4
    I was thinking the same thing, Abstract. :D

    Thanks for that siurpeeman, but I need 4, not 3. :eek:
     
  5. swiftaw macrumors 603

    swiftaw

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    #5
    Do they have to be integers? Or any positive numbers?

    For example, 1, 2, 3, 6/5 works

    If you allow repeats then the integers 1,1,2,4 work
     
  6. siurpeeman macrumors 603

    siurpeeman

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    #6
    oops, i misread that. math is my thing, not english. :)

    1*(1/4)*(19/4)*32 = 38

    1+(1/4)+(19/4)+32 = 38

    further, you could make up your own numbers for three of them and solve for the last.

    example (we'll use swiftaw's solution):

    1 + 2 + 3 + x = 1*2*3*x

    6 + x = 6x

    6 = 5x

    x = 6/5 or 1.2
     
  7. swiftaw macrumors 603

    swiftaw

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    #7
    Exactly, there are an infinite number of solutions if you are dealing with the positive real numbers.

    However, if you restrict to positive integers only, I believe that the only solution is 1,1,2,4, although I don't have a rigorous proof to support that claim.

    Edit: I now have a mathematical proof that 1,1,2,4 is the only set of 4 positive integers for which their sum and product are equal.
     
  8. psycoswimmer thread starter macrumors 65816

    psycoswimmer

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    #8
    Thanks for the answers! It's really appreciated. :)
     
  9. Zwhaler macrumors 603

    Zwhaler

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    #9
    How about: -2, 5, -3, 0.

    It works...

    edit: forgot they had to be positive.
     
  10. orpheus1120 macrumors 65816

    orpheus1120

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    #10
    There are many answers to this question, since this is a quad algebraic equation with only 2 known conditions. Answers are infinite and rather simple, although positive integer answers are kinda tough.

    One possible answer is 0.89 (8/9), 1, 2 and 5.
     
  11. orpheus1120 macrumors 65816

    orpheus1120

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    #11
    You need positive numbers. 0 is not positive, neither are the rest.
     
  12. swiftaw macrumors 603

    swiftaw

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    #12
    To summarize what I said above.

    4 Positive real numbers: Infinite correct answers
    4 Positive Integers: 1 correct answer = 1,1,2,4
     
  13. orpheus1120 macrumors 65816

    orpheus1120

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    #13
    4 DISSIMILAR POSITIVE numbers.
     
  14. swiftaw macrumors 603

    swiftaw

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    #14
    There are no set of 4 different positive integers for which sum = product.
     
  15. elcid macrumors 6502

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    #15
    I think he was just saying that was the only answer for integers, not that it was right for the problem.
     
  16. orpheus1120 macrumors 65816

    orpheus1120

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    #16
    I am only stating the title of this thread, which if I'm not wrong is the question of this problem. You need 4 different positive numbers.

    If we take the question as it's phrased, then these numbers don't have to be integers, and I think it's safe to think so considering it's a problem for young children.
     
  17. swiftaw macrumors 603

    swiftaw

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    #17
    I guess, to me, the question as phrased makes very little sense because if it is just find 4 unique positive numbers that have sum = product, then that is trivial as there are an infinite number of solutions.

    If the numbers have to be positive integers, then we also have a problem because there is no set of 4 positive integers that has sum = product and has all 4 numbers being different because the ONLY set of 4 positive integers that has sum = product is 1,1,2,4.
     
  18. psycoswimmer thread starter macrumors 65816

    psycoswimmer

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    #18
    Thanks for the input since I left.

    Swiftaw provided me with both a correct answer (which I was looking for), and, on a separate note, the only way to answer it with integers.

    As it turns out, my sister told me the wrong information the first time around and her teacher said that there could be a repeat of one number. In this case, swiftaw's "1,1,2,4" answer is the correct answer.

    I'm still having her use a fraction one though to be different. :D
     
  19. orpheus1120 macrumors 65816

    orpheus1120

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    #19
    This question is purposely phrased in this way to test the analytical and reasoning abilities of children to solve this problem. It stands to reason that since this is a quad (4 unknown) algebraic equation, by assuming any 3 values to any 3 of the unknowns, the 4th can be found, hence the infinite possibilities. The path towards this analytical process is the reason for such an abstract yet meaningful problem. It's the process that the teacher is interested in teaching, not solving the problem.
     
  20. Sherman Homan macrumors 6502

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  21. psycoswimmer thread starter macrumors 65816

    psycoswimmer

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    #21
    If you didn't use variables to solve this problem, how would go about solving it?

    My sister went to the same school I went to, had the same math teacher, etc. and I know that they don't learn variables until next year. So the "1+2+3+x=1*2*3*x" method wouldn't go over well with them. So I'm guessing that she just wanted them to do guess and check? I don't see how that's helping them analyze and reason as orpheus1120 suggested.
     
  22. Dros macrumors 6502

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    #22
    You could start trying 1 x 2 x 3 x 4, and note that the product is 24 and the sum is 10. So it has to involve smaller numbers, which means something has to be repeated. And soon you'll see 1 1 2 4 is it, as noted above.
     
  23. Hls811 macrumors 6502a

    Hls811

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    #23
    1,1,2,4

    this was in this past Sundays Parade Magazine
     
  24. psycoswimmer thread starter macrumors 65816

    psycoswimmer

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    #24
    Ahh, this could be where she got it from.

    I kind of like these problems though. My school might be starting a half-credit class for these types of problems and I might sign up.
     
  25. c073186 macrumors 6502a

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    #25
    I still don't see how 1,1,2,4 is the answer if they are supposed to be dissimilar.
     

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