[問題]OG DS-Q083

[marenda]

OG DS-Q083. The participants in a race consisted of 3 teams with 3 runners on each team. A team was awarded 6-n points if one of its runners finished in nth place, where 1<= n <=5. If all the runners finished the race and if there were no ties, was each team awarded at least one point?

1) No team was awarded more than a total of 6 points.
2) No pair of teammates finished in consecutive places among the top five places.
Answer : A

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OG解答上提到: Statement (1) ........Suppose that two teams got the maximum of 6 points each , accounting for a total of 12 of the 15 points. Clearly, the third team would have gotten the other 3 points. 為什麼要這樣假設呢?

偶知道statement(2)的原因，就是卡在statement(1) …
腦筋打結了~也許這個問題很傻啦，但希望有好心人可指點一下偶呀… 3Q

[giyoshi]

我記得這跟以前學過的一個抽屜定理還是鴿籠定理有點像....
什麼定理不重要....總之,現在假如有一個題目是3枝筆要放進兩個抽屜是不是保證一個抽屜有兩枝....應該是吧!!

現在再想回題目....

(1)No team was awarded more than a total of 6 points

總共會有5個place
1st=6-1=5 point
2nd=6-2=4 point
...
5th=6-5=1point

總共有多少分?1+2+3+4+5=15
每隊都<=6分(by statement(1))
所以才會試著先給一隊6分 第二隊6分 發現還有3分可以分配給第三隊
所以滿足

如果statement(1)改成
No team was awarded more than a total of 8 points
就insufficient了
因為第一隊8分 第二隊7分都得到了 第三隊就沒分數可以分配了

[marenda]

歐~~了解~
就是要反過來推如果Points 都先被拿走的情況下
再去想會剩多少就是了

謝謝你舉的例子~ 好理解多了 請喝茶