pigeonhole principle

Twelve basketball players, whose uniforms are numbered 1 through 12, stand around the center ring on the court in an arbitrary arrangement. Show that some three consecutive players have the sum of their numbers at least 20.

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  • 1 decade ago
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    if no three consecutive players have the sum of their numbers at least 20,

    then the maximum value is 19

    Generally speaking, for a circular arrangement, the total no of count of three consecutive players is 12

    the maximum total values is 12*19= 228

    however, from the other point of view, in the total combination of three consecutive players, each no [1,2...12] is counted three times

    total values =78*3=234

    since 228 < 234

    it is a contradiction

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