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[국가수리과학연구소] 국4. 겹치지 않는 경로는 몇 가지?

수학동아 2017.05.21 04:15 조회 2022

페이스북 트위터 카카오톡 카카오스토리 네이버블로그 네이버밴드

국가수리과학연구소 4번

겹치지 않는 경로는 몇 가지?

문제 출제자 : 이석형 수리과학연구소 연구원

직선을 홀수 n개 긋습니다.

이때 어떤 두 직선도 평행하지 않고, 어떤 세 직선도 한 점에서 만나지 말아야 합니다.

그러면 한 직선 위에는 반드시 교점이 n-1개 있습니다.

이제 한 교점에서 다른 교점으로 움직이는 이동을 생각합니다.

점 P와 점 P를 지나는 직선 ℓ의 경우, 점 P가 직선 ℓ의 교점 중 i번째라면 점 P에서 직선 ℓ의 n-i번째 교점으로 이동합니다.

(몇 번째 교점인지는 직선의 어느 쪽에서 세어도 상관없습니다.)

이런 이동을 몇 번이고 활용해서 점 P에서 갈 수 있는

모든 점의 집합을 “경로”라고 할 때,

모든 교점은 절대 겹치지 않는 서로 다른 경로 몇 개로 나눌 수 있습니다.

이 때 서로 다른 경로는 최대 몇 개, 최소 몇 개일까요?

n=7, n=9일 때를 해결하는 것부터 시작해 보세요.

※아래 나온 예시 그림은 n=5일 때 나올 수 있는 직선 배치 중 일부입니다.

각 경우에 흰색 점들이 경로 하나를 이루고, 빨간색 점들이 다른 경로 하나를 이룹니다.

n=5일 때 직선을 배치할 수 있는 방법은 이외에도 2가지가 더 있는데,

이때에도 경로는 2가지로만 나뉩니다.

즉, n=5일 때 서로 다른 경로는 최대 2개, 최소 2개인 셈입니다.

이전글 국5. 혼자 하는 카드게임의 승률은?

다음글 국3. 점과 숫자 뽑기

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에프매스 2017.05.23 00:05

[mathdonga.blog.me에서 옮겨 온 댓글입니다]

benedict07242017.03.31. 21:03

왜 댓글이 없죠?

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에프매스 2017.05.23 00:07

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[mathdonga.blog.me에서 옮겨 온 댓글입니다]

benedict07242017.03.31. 21:20

최솟값= $\frac{(n-1)}{2}$ =최댓값 아닌가요?

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- 에프매스 2017.05.23 00:10
  
  수돌이2017.04.01. 00:26
  
  그러게요... 최댓값, 최솟값을 구하라는 건 낚시일수도..
  
  좋아요0
에프매스 2017.05.23 00:13

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수돌이2017.04.01. 13:39

//추측//
각각의 경로의 점들이 모두 지나는 직선이 존재한다.
//추측//
그 직선 위에서 각각의 경로의 점들은 정확히 두 개 있다.

이 둘을 증명하면 최솟값= $\frac{(n-1)}{2}$ =최댓값입니다.

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에프매스 2017.05.23 00:17

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benedict07242017.04.01. 21:33

일단 별 모양일 때는 n번째 점에서만 왔다갔다 하므로 $\frac{(n-1)}{2}$ 입니다.그래서 최솟값 $\leq \frac{(n-1)}{2}\leq$ 최댓값이라는 것은 증명할 수 있습니다.

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에프매스 2017.05.23 00:36

benedict07242017.04.01. 21:36

증명하기는 못 할 것 같지만 반례는 못 찾겠네요.

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에프매스 2017.05.23 00:37

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엔곰2017.04.01. 22:42

점 P에서 Q로 간다면 점 Q에서 P로 간다.
그리고 이는 항상 일대일 대응.

n=3~9의 경우에서는
$\frac{(n-1)}{2}$ 의 반례를 못찾겠네요.

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에프매스 2017.05.23 00:37

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엔곰2017.04.01. 22:53

k번째 점과 n-k번째 점은 항상 같은 경로고, 이 경로들이 k가 다를 시에 항상 다르면 최소, 최대 모두 $\frac{(n-1)}{2}$ 라는 게 증명되겠군요.

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- 에프매스 2017.05.23 00:38
  
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  수돌이2017.04.01. 23:29
  
  그 직선을 지나지 않는 경로가 있다면 경로의 수가 $\frac{(n-1)}{2}$ 보다 많을 수도 있지 않나요?
  
  좋아요0
- 에프매스 2017.05.23 00:40
  
  엔곰2017.04.01. 23:42
  
  수돌이
  
  그 직선을 지나지 않는 경로가 없음을 보여야겠죠.
  
  좋아요0
- 에프매스 2017.05.23 00:47
  
  수돌이2017.04.01. 23:49
  
  엔곰
  
  그렇죠그렇죠
  (드디어 토론이 시작되었다!)
  
  좋아요0
에프매스 2017.05.23 00:48

동석2017.04.01. 23:30

모든 경로가 지나는 직선이 있음을 보여야 하겠네요

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에프매스 2017.05.23 00:48

수돌이2017.04.02. 01:29

[추측 실패]
경로의 수가 1인 경우도 존재합니다. 그 예시를 찾았습니다.
http://blog.naver.com/dillon0108/220972903453

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- 에프매스 2017.05.23 00:49
  
  동석2017.04.02. 01:36
  
  오 마이 갓
  
  좋아요0
- 에프매스 2017.05.23 00:49
  
  benedict07242017.04.02. 10:17
  
  이런
  
  좋아요0
- 에프매스 2017.05.23 00:50
  
  Nicholas2017.05.12. 21:07
  
  그런데 수돌이님 블로그의 반례는 n=7일 때 아닌가요? n은 직선의 개수이니깐요. 'n=9일때' 라고 되어있길래 직선 개수를 세어보니 7개인 것 같아서...
  
  좋아요0
- 에프매스 2017.05.23 00:50
  
  수돌이2017.05.12. 21:39
  
  Nicholas오 그렇군요! 수정했습니다.
  
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- 에프매스 2017.05.23 00:51
  
  Nicholas2017.05.12. 22:13
  
  수돌이저도 새로운 반례를 찾은 줄 알았는데,수돌이님의 반례랑 같은 거더라구요.안타까웠네요...ㅎㅎ
  
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- 에프매스 2017.05.23 00:52
  
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  수돌이2017.05.12. 22:16
  
  Nicholas그리고 새로 찾은 반례로 $\frac{(n-1)}{2}$ 를 넘어가는 경우도 있습니다.
  n=9일때 별처럼 그리면 (문제에서 예시의 경우는 n=5일 때의 별입니다)
  경로의 개수가 총 6개가 됩니다 ^^
  
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- 에프매스 2017.05.23 00:52
  
  Nicholas2017.05.12. 22:38
  
  수돌이오호~ 그렇군요! 반례를 최대한 찾으면 증명에 큰 도움이 될 듯 하네요. 함께 열심히 해서 증명해 볼 수 있으면 좋겠어요. ㅎㅎㅎ
  
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- 에프매스 2017.05.23 00:53
  
  수돌이2017.05.12. 22:39
  
  Nicholas화이팅!
  
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에프매스 2017.05.23 00:53

서제원2017.04.07. 09:25

위상기하학의 쾨니히스베르크 다리문제로 치환하여 오일러경로와 해밀턴회로 그래프로 풀 수 있을 듯 한데...

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양파양파 Lv.1 2018.01.06 01:29

'점 P'와 '점 P'를 어떻게 잇죠?

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- 수학자 Lv.2 2018.02.21 23:37
  
  점 P와 "점 P를 지나는 직선 l"로 구분해서 읽으면 됩니다.
  
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Undefined Lv.1 2019.06.02 06:56

최댓값의 하한을 구하는 데에는 수학적 귀닙법을 쓸 수 있을 것 같습니다. 교점이 있는 곳 외부에 두 직선을 그으면 최소 1만큼은 증가하게 됩니다.

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