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  • Category: Quiz & Puzzles
    Points:5

    A real mathematical challenge for the members

    Yesterday I raised a thread purely for entertainment purpose. Today I am going to table a real mathematical challenge for the members of the site.

    Here is the challenge:-

    The mathematics-loving king announces a mathematics competition for the subjects of his kingdom. He displays an equation on the arch of his palace and challenges his subjects to solve the equation. He promises that anyone who will solve the equation first, will get the princess for marriage.

    The equation is:
    (x^3+y^3+z^3) * (x+y+z) =42

    [x, y, z are positive integers greater than zero; * means multiplication]

    Any member who solves the equation will not only get the hand of the princess in marriage, but also a beautiful virtual gift from my side.
  • #774157

    Points:2
    I am not a wizard at solving such problems. The author must be knowing the solution and I want him to explain the same if all of the members fail to do so.
    " Be Good and Do Good "

  • #774158

    Points:2
    I want to know from the author the following two clarifications.
    1. X, Y and Z values are only positive integers or can be fractions also.
    2.* indicates multiplication or it is addition.
    drrao
    always confident

  • #774159

    Points:1
    I have written exactly what the king wrote on the arch of his palace.
    (a) Those who have forgotten Noakhali, how can they protest Sandeshkhali?
    (b) Have no fear of perfection - you'll never reach it. ---------- Salvador Dali

  • #774162

    Points:2
    My observation:
    Just by deduction if we put x=1, y=1, and z=2 then we get the value of L.H.S. as (1+1+8)*(1+1+2) and that comes to 40.
    Any increase in x or y or z will take the value much above 42 and we would not get any solution.
    Knowledge is power.

  • #774163

    Points:1
    Dr. Rao Sir (#774158):

    As I consider myself a very liberal person, I will accept the fractions as answer, although the king only asked for integers. But please note that * denotes multiplication.
    (a) Those who have forgotten Noakhali, how can they protest Sandeshkhali?
    (b) Have no fear of perfection - you'll never reach it. ---------- Salvador Dali

  • #774164

    Points:2
    This problem cannot be solved if the values of x, y and z are positive numbers or fractions. As the author himself mentioned that he is a very liberal person, he should also be liberal enough to accept negative numbers.

  • #774177

    Points:1
    The king, along with the princess, is eagerly waiting for the solution.
    (a) Those who have forgotten Noakhali, how can they protest Sandeshkhali?
    (b) Have no fear of perfection - you'll never reach it. ---------- Salvador Dali

  • #774178

    Points:3
    I could only find this solution in the internet,
    [(-80538738812075974)^3 + (80435758145817515)^3 + (12602123297335631)^3] = 42

    That means, from the internet, I could find the answer for (x^3+y^3+z^3) = 42 and not for (x^3+y^3+z^3) * (x+y+z) =42.

    If the author thinks that I found a partial solution, he can give me a virtual gift. But I don't need the princess as I am already married.

  • #774182

    Points:3
    I am sorry. I couldn't solve the challenge posed by Partha Sir. We should be able to divide 42 into three numbers and each of that numbers should have a proper cube root.
    If * is an addition symbol we can get a solution. Otherwise, I am not able to see an answer. However, mathematical experts may come out with a solution.
    If Partha Sir is having the answer I kindly request him to enlighten us with the answer.
    drrao
    always confident

  • #774183

    Points:3
    As far as we know that solving the equations having three unknown variables as is evident in this case, we need to have three equations of these variables in the three different forms no matter they are in any form involving their additions, subtractions, multiplications or in the fractional forms. Trial and error methods would not provide us the exact solution.
    However, it is the turn of Partha Sir to offer us much awaited right solution in this case.

  • #774185

    Points:1
    I am sure that by now, all the Mathematics-loving members of ISC have realized that the shrewd king is not at all ready to hand over the beautiful princess to anyone. But he immensely enjoys the fun when his subjects waste their time to try to solve this insoluble equation.

    Now let us come to the equation. There are three variables but only one equation. So, the equation cannot be attempted by any conventional method. We have to try something unconventional.

    First, let us consider positive integers as directed by the king. Let us go to the right hand side of the given equation. In the right hand side of the equation, the value is given 42. 42 can be factorized in the following manner:-

    42=1*42
    42=2*21
    42=3*14
    42=6*7

    (a) If (x^3 + y^3 + z^3) = 1, then (x + y + z) must be equal to 42, since their product should be 42. However, it is impossible to find three positive integers whose sum is 42 (when the sum of their cubes is 42).
    (b) If (x^3 + y^3 + z^3) = 2, then (x + y + z) is equal to 21 since their product should be 42. Again, it is impossible to find three such positive integers whose sum is 21 (when the sum of their cubes is 2)
    (c) If (x^3 + y^3 + z^3) = 3, then (x + y + z) is equal to 14 since their product should be 42. Again, it is impossible to find three such positive integers whose sum is 14 (when the sum of their cubes is 3).
    And so on ...............................

    Checking all these possibilities as per the factorization of 42, we find there is no such set of three positive integers which satisfies the given equation.

    So, no princess for anyone.

    Those who have unlimited time, would try to solve the equation with positive fractions, but with no result. A great Mathematics-lover of this site has even attempted to solve it with negative fractions!!!

    Umesh-Sir very quickly understood that the equation can't be solved using his own unconventional method.

    Although, those members who attempted the equation, won't get the princess, they will all get virtual gift from me for their valiant efforts.
    (a) Those who have forgotten Noakhali, how can they protest Sandeshkhali?
    (b) Have no fear of perfection - you'll never reach it. ---------- Salvador Dali

  • #774187

      

    Points:4
    Wasn't this a prank by the author, especially after coming up with a fun thread on mathematics and indicating that this one is a real mathematical challenge? I personally do not feel that this kind of practices (whether intentional or otherwise) is in good taste.
    Now the question is as to what for are the virtual gifts being presented by the author who asked a question well knowing that there is no answer. In fact, I feel, ethically, the members who participated deserve to be tendered an apology.
    'Educating the mind without educating the heart is no education at all'.
    -Aristotle

  • #774188

    Points:1
    Doesn't the respected Lead Editor know that 'No solution' is an accepted answer in Mathematics? At least two members have stated that there is no solution.

    The Lead Editor may not like many things. But these type of pranks have given birth to many new branches in Mathematics.
    (a) Those who have forgotten Noakhali, how can they protest Sandeshkhali?
    (b) Have no fear of perfection - you'll never reach it. ---------- Salvador Dali

  • #774190

      

    Points:3
    Partha,

    Humour and pranks are all very well, but I do agree with Saji that if there is no solution, do not refer to it as a 'real mathematical challenge'. It is totally unfair to incite members who think it really is a challenge and spend hours trying to find the solution. Nobody has unlimited time. Time is finite and should not be wasted, not to mention effort in trying to solve a completely unsolvable math puzzle.
    When you make a commitment, you create hope. When you keep a commitment you create trust! ~ John C. Maxwell

  • #774191

    Points:1
    Oof! No solution is a solution in higher Maths.
    (a) Those who have forgotten Noakhali, how can they protest Sandeshkhali?
    (b) Have no fear of perfection - you'll never reach it. ---------- Salvador Dali

  • #774192

      

    Points:1
    Oof! You are not understanding. Why irritate members? Avoid no-solution "challenges".
    When you make a commitment, you create hope. When you keep a commitment you create trust! ~ John C. Maxwell

  • #774196

    Points:2
    Again I learned a point from Partha Sir that No solution is an answer to some mathematical problems. I never know this before. Anyhow, sometimes we have to spend a lot of time learning small points also. It is like digging a hill to catch hold of a mouse.
    drrao
    always confident

  • #774199

    Points:1
    Dr. Rao Sir: You have stated "Again I learned a point from Partha Sir that No solution is an answer to some mathematical problems." I am little bit astonished. I am perplexed.

    I am giving a set of two equations involving two variables.
    2x-4y=7
    2y=5/7+x

    Although there are two equations and two variables, you will not be able to solve this set of equations and find the value of x and y.

    The (mathematical) reason why this set of equations cannot be solved is known to a student of Class-XI Science. Isn't it?

    This is a preliminary example. There are many such examples in Higher Maths. where it is perfectly normal when there is no solution to a particular problem.
    (a) Those who have forgotten Noakhali, how can they protest Sandeshkhali?
    (b) Have no fear of perfection - you'll never reach it. ---------- Salvador Dali

  • #774202

    Points:1
    I find two Mathematics-loving members are eligible to get virtual gifts from me for successfully finding the answer to the question. The answer is that the values of x, y and z cannot be determined within the given set of conditions.
    These two members are Umesh Sir and Bhuvan Sir. While I could send a virtual gift to Umesh Sir following the instructions of Vandana Madam, I am not being able to send a virtual gift to Bhuvan Sir. I don't know the reason behind it. My apology to Bhuvan Sir because I cannot send the virtual gift to him.
    I request the Editors to send a virtual gift to Bhuvan Sir on my behalf and close the thread. There is nothing more to discuss here.
    (a) Those who have forgotten Noakhali, how can they protest Sandeshkhali?
    (b) Have no fear of perfection - you'll never reach it. ---------- Salvador Dali

  • #774203

      

    Points:2
    Partha,

    I just now replied to your Social Hub message that you could not send a virtual gift to Bhuvan. Refer- New Virtual Gifts Protocols. In case you are unable to send it within the next few days as per that policy, let us know which gift, and we'll do the needful.
    When you make a commitment, you create hope. When you keep a commitment you create trust! ~ John C. Maxwell

  • #774205

    Points:0
    Partha Sir, thank you for your explanation.
    drrao
    always confident

  • #774208

    Points:1
    Thanks Partha sir for your virtual gift. Even if you are unable to send the gift, I can assume that I have received it.

  • #774230

    Points:1
    As I am still unable to send any gift to Bhuvan Sir, I request the Lead Editor to send a bunch of flowers to him on my behalf and close this thread.
    Let us all wait for another challenge.
    (a) Those who have forgotten Noakhali, how can they protest Sandeshkhali?
    (b) Have no fear of perfection - you'll never reach it. ---------- Salvador Dali


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