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문제

스도쿠는 가로 9칸, 세로 9칸으로 이루어져 있는 표에 1부터 9까지의 숫자를 채워 넣는 퍼즐이다. 퍼즐을 푸는 방법은 각 가로줄과 세로줄에 1에서 9까지의 숫자를 한 번만 넣고, 3*3칸의 작은 격자에도 1에서 9까지의 숫자를 겹치지 않게 넣어야 한다.

스도쿠를 푸는 방법에는 여러 가지 방법이 있다. 그 중 한가지 방법은 Cross-hatching이다.

Cross-hatching방법을 이용할 때는, 먼저 숫자를 하나 골라야 한다. 그런 뒤에, 그 숫자가 등장하는 가로줄이나 세로줄 또는 3*3 박스에 선을 긋는다. 이렇게 선을 그었을 때, 3*3박스에 고른 숫자가 들어갈 수 있는 칸이 한 칸일 때, 그 숫자를 채우는 것이다.

아래 그림 중 왼쪽 그림은 매우 조금만 숫자가 채워진 스도쿠이고, 오른쪽 그림은 cross-hatching 방법을 이용해서 4가 들어갈 수 있는 곳을 찾은 한가지 예이다.

부분적으로 숫자가 채워진 스도쿠 퍼즐이 주어졌을 때, cross-hatching방법만을 이용해서 스도쿠를 푸는 프로그램을 작성하시오. 더이상 이 방법을 이용해서 풀 수 없을 때 까지 스도쿠를 풀어야 한다. 다른 방법을 이용하면 안된다.

입력으로 주어지는 스도쿠 퍼즐이 규칙을 지키지 않는 경우가 입력으로 주어질 수 있다. (예, 한 가로줄에 1이 2개 있는 경우, 숫자 1이 어떤 가로줄, 3*3박스, 세로줄에 있을 수 있는 곳이 전혀 없는 경우) 이런 경우도 판별해야 한다.

입력

입력은 9개줄로 이루어지며, 각 줄은 9개 문자로 이루어져 있다. 문자는 1~9 숫자 또는 '.'이다. 마침표는 빈 칸을 나타낸다.

출력

올바른 스도쿠 퍼즐이고, 모순이 일어나지 않는다면, 입력과 같은 형식으로 cross-hatching만을 이용해서 푼 결과를 출력한다. 그렇지 않다면 "ERROR"를 출력한다. (쌍따옴표는 출력하지 않는다)

예제 입력 1

..9......
.....4...
.......4.
.........
.4.......
.........
.........
.........
.........

예제 출력 1

4.9......
.....4...
.......4.
.........
.4.......
.........
.........
.........
.........
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(\uc608, \ud55c \uac00\ub85c\uc904\uc5d0 1\uc774 2\uac1c \uc788\ub294 \uacbd\uc6b0, \uc22b\uc790 1\uc774 \uc5b4\ub5a4 \uac00\ub85c\uc904, 3*3\ubc15\uc2a4, \uc138\ub85c\uc904\uc5d0 \uc788\uc744 \uc218 \uc788\ub294 \uacf3\uc774 \uc804\ud600 \uc5c6\ub294 \uacbd\uc6b0) \uc774\ub7f0 \uacbd\uc6b0\ub3c4 \ud310\ubcc4\ud574\uc57c \ud55c\ub2e4.<\/p>\r\n","input":"<p>\r\n\t\uc785\ub825\uc740 9\uac1c\uc904\ub85c \uc774\ub8e8\uc5b4\uc9c0\uba70, \uac01 \uc904\uc740 9\uac1c \ubb38\uc790\ub85c \uc774\ub8e8\uc5b4\uc838 \uc788\ub2e4. \ubb38\uc790\ub294 1~9 \uc22b\uc790 \ub610\ub294 &#39;.&#39;\uc774\ub2e4. \ub9c8\uce68\ud45c\ub294 \ube48 \uce78\uc744 \ub098\ud0c0\ub0b8\ub2e4.<\/p>\r\n","output":"<p>\r\n\t\uc62c\ubc14\ub978 \uc2a4\ub3c4\ucfe0 \ud37c\uc990\uc774\uace0, \ubaa8\uc21c\uc774 \uc77c\uc5b4\ub098\uc9c0 \uc54a\ub294\ub2e4\uba74, \uc785\ub825\uacfc \uac19\uc740 \ud615\uc2dd\uc73c\ub85c cross-hatching\ub9cc\uc744 \uc774\uc6a9\ud574\uc11c \ud47c \uacb0\uacfc\ub97c \ucd9c\ub825\ud55c\ub2e4. \uadf8\ub807\uc9c0 \uc54a\ub2e4\uba74 &quot;ERROR&quot;\ub97c \ucd9c\ub825\ud55c\ub2e4. (\uc30d\ub530\uc634\ud45c\ub294 \ucd9c\ub825\ud558\uc9c0 \uc54a\ub294\ub2e4)<\/p>\r\n","hint":"","original":"0","problem_lang_code":"\ud55c\uad6d\uc5b4"},{"problem_id":"2955","problem_lang":"1","title":"CROSS","description":"<p>In the game of Sudoku, the objective is to place integers between 1 and 9 (inclusive) into a 9x9 grid so that each row, each column, and each of the nine 3x3 boxes contains all nine numbers. The starting board is partially filled in so that it is possible to logically deduce the values of the other cells. Sudoku puzzles range in difficulty, and complex analysis methods are required to solve the hardest puzzles. In this problem, however, you will implement one of the simplest methods, cross-hatching.&nbsp;<\/p>\r\n\r\n<p>In cross-hatching, we select one of the nine numbers and, for each of its occurrences in the grid, cross out the corresponding row, column and 3x3 box. Now look for any 3x3 boxes where there is only one possible placement for the number and place it there.&nbsp;<\/p>\r\n\r\n<p>The first image below shows a very sparsely filled in Sudoku grid. However, even in this grid it is possible to deduce using cross-hatching that the number in the top left cell is 4, as illustrated in the second image.&nbsp;<\/p>\r\n\r\n<p><img alt=\"\" src=\"\/upload\/images\/sdk.png\" style=\"height:174px; width:407px\" \/><\/p>\r\n\r\n<p>You will be given a partially filled-in grid. Your task is to repeatedly apply the cross-hatching method for different numbers until no more deductions can be made about any number.&nbsp;<\/p>\r\n\r\n<p>The initial placement of the numbers in the grid may be invalid. It is also possible that there will be no available cell for a number in a 3x3 box. In both cases, you are to report an error<\/p>\r\n","input":"<p>Input will consist of 9 lines, each containing exactly 9 characters. Each character will either be a digit between 1 and 9, or a period (&#39;.&#39;) denoting an empty cell.&nbsp;<\/p>\r\n","output":"<p>If the input is valid and there is no contradiction while solving, you should output the grid in the same format it was given in, with cells filled in if their value can be deduced using cross-hatching. Otherwise, output &quot;ERROR&quot; (quotes for clarity).&nbsp;<\/p>\r\n","hint":"","original":"1","problem_lang_code":"\uc601\uc5b4"}]