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Homework Help: Arranging numbers problem (induction)

  1. Oct 15, 2011 #1

    Here is a problem I am trying to do ..

    Show that it is possible to arrange the numbers 1, 2, . . . , n
    in a row so that the average of any two of these numbers
    never appears between them.

    Here is my attempt. The base case will consist of first two numbers.

    Base Case: [tex]n=2[/tex]

    The arrangment 1,2 doesn't have average of 1 and 2 between them.
    So this proves [tex]P(2)[/tex]

    Induction case: Let [tex]n\geqslant 2[/tex] be arbitrary.
    Suppose [tex]P(n)[/tex]. To prove [tex]P(n+1)[/tex],
    consider numbers [tex]1,2,\cdots,n,n+1[/tex]

    Now I am negating the goal and seeking the contradiction.
    So assume that with any configuration of these [tex]n+1[/tex]
    numbers, there always is at least a pair of numbers such that
    their average is between them. Now due to the inductive hypothesis,
    we know that for first [tex]n[/tex] numbers, there is a configuration
    such that for any pair of numbers, their average is not between them.
    So lets put these [tex]n[/tex] numbers in that particular
    configuration. And now put [tex]n+1[/tex] at the end of this
    list. But due to our assumption, for the list of all these
    numbers (including [tex]n+1[/tex]), there is no desired
    configuration possible. So there must exist a number
    [tex]1\leqslant i < n [/tex] such that the average of [tex]i[/tex]
    and [tex]n+1[/tex] is between these two numbers. That average


    Now after this point I don't know how to seek the contradiction.
    Any hints will be helpful. There is one hint given in the problem.

    Hint: Show that it suffices to prove this fact when n
    is a power of 2. Then use mathematical induction to
    prove the result when n is a power of 2.

    But I dont know how to use it, so I started with another approach.

  2. jcsd
  3. Oct 15, 2011 #2


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    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    I think that you have to come up with a scheme for the arrangement.
  4. Oct 15, 2011 #3
    Sammy , can you give some hints for this arrangement ?
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