수학동아 - 폴리매스

주니어 폴리매스 문제

수학 문제를 푸는 매력에 빠져 보세요!

[KAIST 과학영재교육연구원] K1. 사각형에 숫자 채우기

수학동아 2018.08.01 01:28 조회 3004

페이스북 트위터 카카오톡 카카오스토리 네이버블로그 네이버밴드

문제 다음 모양으로 총 $\LARGE m+n+1$ 개의 사각형이 이뤄져 있을 때, 각 사각형마다 $\LARGE 1$ 에서 $\LARGE m+n+1$ 까지의 숫자를 채우려고 한다. 각 칸의 숫자를 그 위 칸과 왼쪽 칸보다 작게 채우는 방법의 개수는 얼마인가?

다음글 k2. 바이러스 감염을 막아라!

문제 찜하기 목록보기

댓글 작성하기

구머 Lv.6 2018.08.01 19:01 비밀댓글

비밀 댓글이 등록 되었습니다.

댓글 작성하기 댓글수3
- 주니어멘토 Lv.1 2018.08.02 23:22
  
  안녕하세요! 주니어폴리매스 멘토입니다. 문제의 서술에 혼동의 여지가 있었네요.
  
  각 칸의 숫자가 위 칸과 왼쪽 칸보다 작다는 이야기는 바로 변을 공유한 상태로 이웃하고 있는 사각형들에게만 해당되는 조건입니다!
  
  좋아요0
- 구머 Lv.6 2018.08.03 22:39
  
  그냥 제가 잘못읽었네요)_(
  
  좋아요0
- 나짱 Lv.1 2018.08.10 18:48
  
  겁나쉬움
  
  좋아요0
쓰다만연습장 Lv.5 2018.08.02 08:34

요고는 요롷게 하는 게 어떨까...요?

혹시 오류가 있거나, 문제점이 있으면 언제든지 대댓글로 올려주세요!!

댓글 작성하기 좋아요1 댓글수2
- 주니어멘토 Lv.1 2018.08.02 20:18
  
  안녕하세요! 주니어폴리매쓰 담당 멘토입니다.
  
  m=5, n=3인 경우에 대한 예시를 하나 들어주셨는데요. 예시로 들어준 숫자의 배치는 맞습니다. 유일한 방법은 아닙니다.
  (올려주신 것에서 6과 7의 자리를 바꾸어도 조건을 만족하는 것을 확인하실 수 있습니다.)
  
  다만, 문제에서 요구하는 것은 그런 배치가 몇 가지 가능한 지 구하는 것입니다.
  
  한번 그쪽으로 노력해보시면 좋을 것입니다! ^^
  
  좋아요1
- 수학짱 Lv.1 2018.08.02 23:14
  
  m이 있는 자리에 1이상 m+1이하인 수도 들어갈 수 있지 않나요?
  
  좋아요0
21세기오일러 Lv.11 2018.08.02 23:13

n이1일때와 n이2일때는 찾았습니다

댓글 작성하기 좋아요0 댓글수0
21세기오일러 Lv.11 2018.08.02 23:16

n이1이면m+1개입니다.

댓글 작성하기 좋아요0 댓글수0
21세기오일러 Lv.11 2018.08.02 23:16

n이2면

1부터m+1까지의수의 합입니다.

댓글 작성하기 좋아요0 댓글수1
- 수학짱 Lv.1 2018.08.02 23:17
  
  왜 그런 거죠???
  
  좋아요0
21세기오일러 Lv.11 2018.08.02 23:31

조합과 관련이있는것 같네요.

m+n개중n개를 크기에맞게 고르니까요.

댓글 작성하기 좋아요0 댓글수0
구머 Lv.6 2018.08.04 02:53

1)n개의.가로줄에 m+n+1이 들어감

남은 n-1개 가로칸에 들어갈 숫자 정해서 넣고, 남은 m개의 세로칸과 1개의 꼭짓점에 들어갈 수 있는 배열의 가짓수는 꼭짓점에 남은 m+1개의 수 중 가장 큰 수만 안들어가면 되므로 총 가짓수:(m+n)C(n-1) m

2)1번 반대쪽

I.S.W, (m+n)C(m-1) n

둘이 더하면 됩니다.

//9월 18일: 비밀댓글->공개로 바꿈

댓글 작성하기 좋아요0 댓글수2
- 뉴턴의 사과 Lv.1 2018.08.04 03:02
  
  제가 글쓰고 있는동안에 그머님이 벌써 정답 올린 것 같네요;;
  
  좋아요1
- 주니어멘토 Lv.1 2018.08.08 21:38
  
  안녕하세요! 주니어폴리매스 담당 멘토입니다.
  
  구머님께서 정답을 맞춰주셨습니다!
  
  (i) 가장 큰 수가 꺾인 지점에 들어갈 수 없다는 점, (ii) 가로 줄과 세로 줄이 적당히 독립적이라는 점 등을 잘 활용하여 풀면 되는 문제인데요.
  
  해당하는 설명을 좀 더 풀어서 쓴다면 가장 완벽한 풀이가 될 것입니다.
  
  풀이 감사드립니다. 다른 분들도 직접 좋은 풀이를 더 발견해보시거나, 이 풀이를 참고하셔서 이해하시면 될 것 같습니다 ^^
  
  좋아요0
뉴턴의 사과 Lv.1 2018.08.04 03:01

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Function: require_once

일단 n개가 늘어서 있는 줄에 왼쪽에서부터 각각 $N_{1}, N_{2}, \cdots, N_{n-1}, N_{n}$ 이라 하고 m개가 늘어서 있는 줄에는 위에서부터 각각 $M_{1}, M_{2}, \cdots, M_{m-1}, M_{m}$ 이라 하고 나머지 하나는 $K$ 라고 합시다. 그러면 이 문제는 $N_{1}>N_{2}>\cdots >N_{n-1}>N_{n}, M_{1}>M_{2}>\cdots >M_{m-1}>M_{m}$ , $K<N_{1}, K<M_{1}$ 을 만족하는 각각이 1부터 n+m+1의 값을 가지는 $N_{1}, N_{2}, \cdots, N_{n-1}, N_{n}$ , $M_{1}, M_{2}, \cdots, M_{m-1}, M_{m}$ , $K$ 를 찾는 문제로 변형할 수 있네요.

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- 쓰다만연습장 Lv.5 2018.08.04 03:36
  
  ......생각했던 것보다 많이 어렵네요......
  
  좋아요0
- 뉴턴의 사과 Lv.1 2018.08.04 03:52
  
  글이 장황해서 그렇지 사실상 문제는 그다지 어려운 내용이 아닙니다. 풀이는 어려울수도 있지만...
  
  좋아요0
21세기오일러 Lv.11 2018.08.04 05:56

문제를 바꾸면

n+m개의자연수를 뽑아서

큰 수가 앞에오도록 n개를배열하는 방법을 구하시오

로 바꿀 수 있습니다.

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- 21세기오일러 Lv.11 2018.08.04 05:58
  
  이유는 가장큰수는무조건 n열과 m열이만나는가운데에만 있을수있고나머지수를 n열에다배열하면 m열에배열하는 방법은 한 가지만있기 때문입니다.
  
  좋아요1
- 수학짱 Lv.1 2018.08.04 06:44
  
  가장 큰 수는 들어갈수 없지 않나요?
  
  왼쪽 수보다 작아야 하니까요
  
  좋아요0
- 21세기오일러 Lv.11 2018.08.04 20:52
  
  죄송합니다.
  
  문제를 ㄱ자모양으로 봤네요
  
  좋아요0
happy Lv.2 2018.08.05 03:50 비밀댓글

비밀 댓글이 등록 되었습니다.

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- 주니어멘토 Lv.1 2018.08.08 21:34 비밀댓글
  
  비밀 댓글이 등록 되었습니다!
파인만 Lv.1 2018.08.10 22:39 비밀댓글

비밀 댓글이 등록 되었습니다.

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- 주니어멘토 Lv.1 2018.08.13 19:20 비밀댓글
  
  비밀 댓글이 등록 되었습니다!
과학천재 수학바보 Lv.1 2018.08.11 06:06

m하고n은 x와y처럼 방정식에서 정하는것이죠?

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- 주니어멘토 Lv.1 2018.08.13 19:24
  
  안녕하세요! 주니어폴리매쓰 담당 멘토입니다.
  
  여기서 m과 n은 정해져있는 숫자는 아니지만, 상수라고 생각하시면 되는데요.
  
  예를 들어서 m각형의 내각의 크기를 합을 구하라는 문제가 있다면 답은 180*(m-2)도가 되겠죠? 답에도 m이라는 문자가 들어있도록 푸시면 됩니다!
  
  좋아요1
ln(-1)=πi Lv.3 2018.08.18 00:30

A PHP Error was encountered

Severity: Warning

Message: mkdir(): Permission denied

Filename: libraries/Common.php

Line Number: 202

Backtrace:

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 202
Function: mkdir

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 236
Function: getLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view_comment_list.php
Line: 90
Function: parseLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view.php
Line: 343
Function: view

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
Line: 558
Function: view

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

A PHP Error was encountered

Severity: Warning

Message: file_put_contents(/DATA/upload/polymath/latex/a6521b4863c9adb7fbf669aa8fbdcfb1.gif): failed to open stream: No such file or directory

Filename: libraries/Common.php

Line Number: 213

Backtrace:

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 213
Function: file_put_contents

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 236
Function: getLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view_comment_list.php
Line: 90
Function: parseLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view.php
Line: 343
Function: view

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
Line: 558
Function: view

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

정답은 $2^{m-n}-(m-n)$ 일 것 같은데 이걸 설명하기에는 사각형 칸 만드는 방법을 몰라서 나중에 증명하겠습니다.

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- 김우현 기자 Lv.5 2018.08.19 19:49
  
  여백이 부족한 건 아니겠죠!!
  
  (이해할 사람만 이해하기!)
  
  좋아요2
- ln(-1)=πi Lv.3 2018.08.19 23:16
  
  A PHP Error was encountered
  
  Severity: Warning
  
  Message: mkdir(): Permission denied
  
  Filename: libraries/Common.php
  
  Line Number: 202
  
  Backtrace:
  
  File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
  Line: 202
  Function: mkdir
  
  File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
  Line: 236
  Function: getLatexImg
  
  File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view_comment_list.php
  Line: 200
  Function: parseLatexImg
  
  File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view.php
  Line: 343
  Function: view
  
  File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
  Line: 558
  Function: view
  
  File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
  Line: 315
  Function: require_once
  
  A PHP Error was encountered
  
  Severity: Warning
  
  Message: file_put_contents(/DATA/upload/polymath/latex/b4203234c7ba19c05c0a8855583cf9b3.gif): failed to open stream: No such file or directory
  
  Filename: libraries/Common.php
  
  Line Number: 213
  
  Backtrace:
  
  File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
  Line: 213
  Function: file_put_contents
  
  File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
  Line: 236
  Function: getLatexImg
  
  File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view_comment_list.php
  Line: 200
  Function: parseLatexImg
  
  File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view.php
  Line: 343
  Function: view
  
  File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
  Line: 558
  Function: view
  
  File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
  Line: 315
  Function: require_once
  
  실수로 조건을 작성하지 않았네요! $m> n$ 입니다.
  
  좋아요0
냐뇨기 Lv.1 2018.08.24 21:41 비밀댓글

비밀 댓글이 등록 되었습니다.

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21세기오일러 Lv.11 2018.10.03 04:39

그런데 구머님께서 풀었으면 해결딱지 붙여야 하는 것 아닌가요?

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zxcvbnm Lv.1 2019.01.06 20:42

근데 이미 문제 다 푼 것 같은데 해결 딱지 안 붙이나요?

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