pigeonhole principle

Question

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.

Answer 1

最愛解答

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