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함께 풀고 싶은 문제
창의력을 기를 수 있는 수학 문제 또는 퍼즐을 내는 곳입니다.
[창의 퍼즐] 콜라츠의 추측 (3n+1추측)의 변형 (나도 아직 풀고 있음)
1버트 9슈타인 2020.03.12 21:54 조회 569
문제 푼 사람 0명
공유하기
페이스북 트위터 카카오톡 카카오스토리 네이버블로그 네이버밴드

무작위 수 n을 고른 다음 다음조건을 만족하며 1이될때까지 할때  모든수가 1이(안)된다를 증명하라 

 when n is odd n+1, when n is even n/2

이 문제 어떠셨나요?

글쎄요

0

어려워요

0

문제 찜하기 목록보기
댓글 8
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  •  
    로보카폴리 Lv.11 2020.03.12 21:55

    5가 아니라 3아닌가요?

    댓글 작성하기 좋아요0 댓글수1
  •  
    켋 Lv.7 2020.03.12 23:34

    반증 자체는 간단합니다. 어떤 n에 관해 콜라츠 추측을 유한 번 반복했을때 1이 되기 전에 n이 되는 자연수 n을 하나만 찾으면 됩니다. 

    댓글 작성하기 좋아요0 댓글수0
  •  
    리프 Lv.7 2020.03.12 23:40

    2^n-x 꼴로 표현한 후  x를 이진법으로 나타내서 값이 어떻게 변화하는지를 관찰하면 위 문제는 해결됩니다. (귀찮으니 다른 분께 맡기죠)

    댓글 작성하기 좋아요0 댓글수0
  •  
    수티브 잡스 Lv.6 2020.03.13 07:59

    아 또 귀찮다 어떡하지

    여러분 저대신 풀어주실분 모집합니다

     

    댓글 작성하기 좋아요0 댓글수1
    •  
      콜라츠 Lv.1 2022.01.02 00:53

      전 순환하는 반례는 없다는건 풀었습니다

      좋아요0
  •  
    진리를 찾아서 Lv.5 2022.01.14 18:49 비밀댓글
    확인요청중
    비밀 댓글이 등록 되었습니다.
    댓글 작성하기 댓글수0
  •  
    우까 Lv.4 2022.03.16 10:04
    확인요청중

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    대충 쓰겠습니다.

    짝수는 어차피 2로 나누어지니까 홀수가 됩니다

     

    홀수만 증명하면 되겠네요. 

    n은 홀수라 합시다.

    n->n+1->\frac{n+1}{2^k}

     

    됩니다. 어차피 n 홀수니까 n이 \frac{n+1}{2^k}보다 큽니다.

     

    다시 말해서 규칙에 넣으면 수가 작아지게 되고 유한번 내에 2가 되어 1이 됩니다. 끝

    댓글 작성하기 좋아요0 댓글수0

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