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Function: mkdir

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

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

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

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
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[자유 기사] 이 작은 루빅스 큐브에 어마어마한 수가 숨겨져 있다?
황금 열쇠 2020.05.09 21:33

루빅스 큐브, 다들 한번 쯤은 보셨죠?

루빅스 큐브는 1974년, 에르뇨 루빅(Rubik Ernő) 이란 사람이 발명 했습니다. 그 후 루빅스 큐브는 엄청난 인기를 끌었고, 현재까지 그 인기가 지속되었습니다. 

그런데 이 작은 루빅스 큐브에 어마어마한 수가 숨겨져 있다는 사실, 아시나요? 바로 루빅스 큐브를 섞을 수 있는 경우의 수 입니다!

 

먼저 경우의 수를 계산해보기 전에 간단한 루빅스 큐브 조각의 명칭을 알아볼게요. 루빅스 큐브는 총 26개의 조각으로 이루어져 있습니다.

그리고 조각은 총 3가지 종류가 있습니다. 

 

센터: 스티커가 한 개 있는 조각입니다.       엣지: 스티커가 두 개 있는 조각입니다.       코너: 스티커가 세 개 있는 조각입니다.

                                              

 

자 그럼 지금부터 루빅스 큐브를 섞을 수 있는 경우의 수를 계산해 보겠습니다. 

먼저 센터 블럭은 큐브를 아무리 돌려도 움직이지 않는다는 걸 알수 있습니다. 따라서 센터 블럭은 계산할 필요가 없겠죠?

계산식은 이렇게 됩니다.

 

(엣지의 방향의 경우의 수) X (코너의 방향의 경우의 수) X (엣지의 위치의 경우의 수) X (코너의 위치의 경우의 수)

 

먼저 조각들의 "방향"의 경우의 수를 구해봅시다.

엣지 조각을 보면 조각마다 스티커가 2개가 있습니다. 그럼 엣지 조각 12개마다 2가지 경우가 있겠죠. 그래서 엣지 조각의 방향의 경우의 수는 2^{_{12}}가지 입니다.

그리고 마찬가지로 코너 조각은 스티커가 3개씩 있습니다. 따라서 코너 조각 8개마다 3가지 경우가 있으므로 코너 조각의 방향의 경우의 수는 3^{_{8}}가지 입니다.

 

이번엔 조각들의 "위치"의 경우의 수를 구해봅시다.

엣지 조각은 첫번째 조각은 있을수 있는 위치가 12곳, 두번째 조각은 하나를 뺀 11곳, 세번째 조각은 또 하나를 뺀 10곳, ......이므로 총 경우의 수는 

12x11x10x...x2x1=12!가지 입니다.

코너 조각도 첫번째 조각은 있을수 있는 위치가 8곳, 두번째 조각은 하나를 뺀 7곳, 세번째 조각은 또 하나를 뺀 6곳, ......이므로 총 경우의 수는 

8x7x6x...x2x1=8!가지 입니다. 

 

이 모든 수를 곱하면 루빅스 큐브 총 경우의 수가 나옵니다. 2^{_{12}} X 3^{_{8}} X 12! X 8!을 계산해보면 519,024,039,293,878,272,000‬이 나옵니다. 하지만 이건 조각을 분해했다가 다시 끼웠을 경우까지 포함한 값이죠. 일반적으로는 맞출수 없는, 조각이 돌아간 경우도 포함됐단 말입니다. 그럼 실제로 큐브를 돌려서 섞을수 있는 경우의 수를 구하려면 몇개의 값을 바꿔야 합니다.

 

엣지 조각의 방향의 경우의 수는 엣지가 한개만 뒤집혀 있으면 안되므로, 11개의 엣지의 방향이 정해지면 나머지 한개는 자동적으로 방향이 정해지게 됩니다. 그래서 2^{_{12}}가지가 아니라 2^{_{11}}가지 입니다. 

 

코너 조각의 방향의 경우의 수는 코너가 한개만 돌아가 있으면 안되므로, 7개의 코너의 방향이 정해지면 나머지 한개는 자동적으로 방향이 정해지게 됩니다. 그래서 3^{_{8}}가지가 아니라 3^{_{7}}가지 입니다.

 

(엣지 조각의 위치의 경우의 수) X (코너 조각의 위치의 경우의 수)도 실제로는 이 값의 반 입니다. 왜냐하면 엣지 조각 2개의 위치만 바뀌는 것은 불가능하고, 코너 조각 2개의 위치만 바뀌는 것은 불가능하지만 엣지 조각과 코너조각의 위치가 2개 다 바뀌는 것은 가능합니다. 그래서 4로 나누는게 아니라 2로 나누는 것 입니다.

 

 

따라서 최종 경우의 수는 2^{_{11}} X 3^{_{7}} X 12! X 8! X \frac{1}{2} = 43,252,003,274,489,856,000가지가 나옵니다.

이게 어느정도냐 하면 전 세계인구 75억 명이 모두 섞기 15초, 미리보기 15초, 맞추기 30초로 큐브 하나를 1분안에 맞춘다고 합시다. 이렇게 10000년 동안 잠도 안 자고 해도 모든 경우가 나올수 없습니다.

다르게 설명해보면 이 모든 큐브의 경우의 수를 만들어 일렬로 나열하면 길이가 217"광년"이 됩니다.

 

작은 큐브에서 이렇게나 무궁무진한 경우의 수가 나올수 있다니 놀랍죠?

 

 

 

 

대표 이미지 출처: https://ko.wikipedia.org/wiki/%EB%A3%A8%EB%B9%85%EC%8A%A4_%ED%81%90%EB%B8%8C

센터 블럭, 엣지 블럭, 코너 블럭 이미지 출처: https://cafe.naver.com/cubemania/307568

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수학동아 기자의 한마디
수학동아 기자 2020.05.12
전 세계 인구가 1000년 동안 도전해도 정복하기 어려울 만큼 경우의 수가 많을 줄이야!
갑자기 큐브가 무시무시해 보입니다.

큐브를 돌리는 과정을 설명하기 위해서 조각의 종류부터 소개한 점 칭찬합니다. 기사에서는 이렇게 용어가 분명해야 하죠!
경우의 수를 계산하는 과정에서는 궁금한 점이 많네요.
'방향'과 '위치'가 어떻게 다른지 모르겠어요. 그래서일까요? 11개 엣지, 7개 코너의 방향이 정해지면 나머지 한 개는 자동으로 정해진다는 부분도 모호하네요.
이런 내용은 큐브 그림이나 사진에 화살표를 써서 어디가 뒤집히고 돌아간다는 뜻인지 보여주면 어떨까요?

기사를 잘 다듬어서 루빅스 큐브의 매력이 널리 알려졌으면 좋겠네요!
이 기사 어떠셨나요?

유익해요

4

웃겨요

0

신기해요

6

어려워요

1

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  •  
    왕토끼 Lv.1 2020.09.03 00:30

    좋은 기사 감사드려요~~

    article.reply.showForm('2581'); $('#is_clear').parent().hide()">댓글 작성하기 좋아요0 댓글수0
  •  
    PF_ Lv.8 2021.02.19 00:59

    그런데 26개의 조각이 아니라 27개의 조각이 아닌가요?

    article.reply.showForm('2793'); $('#is_clear').parent().hide()">댓글 작성하기 좋아요1 댓글수1
  •  
    피카파이 Lv.8 2021.03.12 06:38

    저는 큐브가 그렇게 어렵지 않든데요(1분)

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  •  
    초딩파이 Lv.2 2021.09.06 02:56

    우와! 54개의 네모들이 움직일수있는 방법이 정말 많네요!

    좋은 정보 감사함다!

    이거 다찾으시느라 조금 힘드셨겠어요

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    초밥 Lv.3 2022.02.13 03:58

    기사 넘 유익하게 읽고가여~

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    초3이지만 초5 Lv.3 2022.07.09 23:00

    오 저 엄청 잘맞추는데 무섭네요 ㅋㅋ

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    전개미 Lv.1 2022.08.01 23:29 비밀댓글
    비밀 댓글이 등록 되었습니다.
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