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Math Help - Problem Solving

  1. #1
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    Problem Solving

    Let a1,...,a7 be an arbitrary arrangement of the numbers 1,...,7. Show that (a1-1)(a2-2)...(a7-7) is even.
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  2. #2
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    Assume the contrary, i.e. (a_1 - 1)(a_2 - 2) \cdots (a_7 -7) is odd.

    Then all of the factors (a_1 - 1), (a_2 - 2) etc. must be odd. Do you see why?

    Since a_1 -1 is odd, a_1 must be even.
    Since a_2 -2 is odd, a_2 must be odd.
    ....

    Continue along these lines and see if you can arrive at an impossible situation, i.e. a contradiction.
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