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문제

추석 연휴를 맞아 홍준이는 자신의 집에서 파티를 열었다. 파티에는 어린 아이들도 많이 왔다. 다음 날 홍준이는 자신의 침대에 여러 얼룩들이 있는 것을 보았다. 그 얼룩들은 침대 시트 밑의 매트릭스까지 스며들었다. 매트릭스 전체를 청소하는 비용이 매우 비싼 관계로 홍준이는 작은 매트릭스 청소 도구를 구매하려한다.

매트릭스의 윗 면은 M × N 크기의 직사각형 격자로 볼 수 있다. 즉, 매트릭스의 수직 성분 개수는 M 개이며 수평 선분 개수는 N 개다. 매트릭스의 각 격자를 좌표로 나타낼 수 있는데, 매트릭스의 왼쪽-위 격자의 좌표는 (1, 1)이며, 오른쪽-아래 격자의 좌표는 (m, n)이다. 한 얼룩은 격자 하나를 덮는다.

홍준이가 구매하려는 매트릭스 청소 도구는 연속하고 직사각형 변에 평행한 3 × 3 영역에 있는 모든 얼룩을 지워주는 도구다. 도구는 하나 사면 한 번만 얼룩을 지울 수 있는 소비성이 있는 도구다. 홍준이를 위해 매트릭스 크기와 얼룩에 대한 정보가 주어졌을 때, 필요한 청소 도구의 최소 개수를 구하자.

<그림 1>은 가능한 상황의 한 예다. 매트릭스의 크기는 6 × 11이며, 각 얼룩의 위치 좌표는 (4, 3), (6, 5), (3, 6)과 (4,7)이다. 위와 같은 경우에 필요한 청소 도구의 최소 개수는 2개다.

입력

입력은 여러 개의 테스트케이스로 구성되어 있다.

각 테스트케이스의 첫 줄에는 매트릭스의 크기를 나타내는 두 자연수 m과 n이 주어진다. (3 ≤ m ≤ 10, 3 ≤ n ≤ 1,000). 둘째 줄에는 얼룩의 개수를 나타내는 정수 c가 주어진다. (0 ≤ c ≤ mn) 그리고 다음 c개의 줄에 각 얼룩의 좌표가 주어진다. 얼룩의 좌표는 서로 다르게 주어지며, 모두 매트릭스 위에 위치한다.

출력

각 테스트케이스 별로 얼룩을 모두 지우기 위해 필요한 청소 도구의 최소 개수를 의미하는 하나의 정수를 출력한다.

예제 입력 1

2
6 11
4
4 3
6 5 
3 6
4 7 
7 7
4 
3 2
5 6
6 4
7 6

예제 출력 1

2
2

힌트

[{"problem_id":"10257","problem_lang":"0","title":"\uc5bc\ub8e9","description":"<p>\ucd94\uc11d \uc5f0\ud734\ub97c \ub9de\uc544 \ud64d\uc900\uc774\ub294 \uc790\uc2e0\uc758 \uc9d1\uc5d0\uc11c \ud30c\ud2f0\ub97c \uc5f4\uc5c8\ub2e4. \ud30c\ud2f0\uc5d0\ub294 \uc5b4\ub9b0 \uc544\uc774\ub4e4\ub3c4 \ub9ce\uc774 \uc654\ub2e4. \ub2e4\uc74c \ub0a0 \ud64d\uc900\uc774\ub294 \uc790\uc2e0\uc758 \uce68\ub300\uc5d0 \uc5ec\ub7ec \uc5bc\ub8e9\ub4e4\uc774 \uc788\ub294 \uac83\uc744 \ubcf4\uc558\ub2e4. \uadf8 \uc5bc\ub8e9\ub4e4\uc740 \uce68\ub300 \uc2dc\ud2b8 \ubc11\uc758 \ub9e4\ud2b8\ub9ad\uc2a4\uae4c\uc9c0 \uc2a4\uba70\ub4e4\uc5c8\ub2e4. \ub9e4\ud2b8\ub9ad\uc2a4 \uc804\uccb4\ub97c \uccad\uc18c\ud558\ub294 \ube44\uc6a9\uc774 \ub9e4\uc6b0 \ube44\uc2fc \uad00\uacc4\ub85c \ud64d\uc900\uc774\ub294 \uc791\uc740 \ub9e4\ud2b8\ub9ad\uc2a4 \uccad\uc18c \ub3c4\uad6c\ub97c \uad6c\ub9e4\ud558\ub824\ud55c\ub2e4.<\/p>\r\n\r\n<p><img alt=\"\" src=\"\/upload\/images2\/stains.png\" style=\"float:right; margin:0px 0px 10px 10px\" \/><\/p>\r\n\r\n<p>\ub9e4\ud2b8\ub9ad\uc2a4\uc758 \uc717 \uba74\uc740 M &times; N \ud06c\uae30\uc758 \uc9c1\uc0ac\uac01\ud615 \uaca9\uc790\ub85c \ubcfc \uc218 \uc788\ub2e4. \uc989, \ub9e4\ud2b8\ub9ad\uc2a4\uc758 \uc218\uc9c1 \uc131\ubd84 \uac1c\uc218\ub294 M \uac1c\uc774\uba70 \uc218\ud3c9 \uc120\ubd84 \uac1c\uc218\ub294 N \uac1c\ub2e4. \ub9e4\ud2b8\ub9ad\uc2a4\uc758 \uac01 \uaca9\uc790\ub97c \uc88c\ud45c\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\ub294\ub370, \ub9e4\ud2b8\ub9ad\uc2a4\uc758 \uc67c\ucabd-\uc704 \uaca9\uc790\uc758 \uc88c\ud45c\ub294 (1, 1)\uc774\uba70, \uc624\ub978\ucabd-\uc544\ub798 \uaca9\uc790\uc758 \uc88c\ud45c\ub294 (m, n)\uc774\ub2e4. \ud55c \uc5bc\ub8e9\uc740 \uaca9\uc790 \ud558\ub098\ub97c \ub36e\ub294\ub2e4.<\/p>\r\n\r\n<p>\ud64d\uc900\uc774\uac00 \uad6c\ub9e4\ud558\ub824\ub294 \ub9e4\ud2b8\ub9ad\uc2a4 \uccad\uc18c \ub3c4\uad6c\ub294 \uc5f0\uc18d\ud558\uace0 \uc9c1\uc0ac\uac01\ud615 \ubcc0\uc5d0 \ud3c9\ud589\ud55c 3 &times; 3 \uc601\uc5ed\uc5d0 \uc788\ub294 \ubaa8\ub4e0 \uc5bc\ub8e9\uc744 \uc9c0\uc6cc\uc8fc\ub294 \ub3c4\uad6c\ub2e4. \ub3c4\uad6c\ub294 \ud558\ub098 \uc0ac\uba74 \ud55c \ubc88\ub9cc \uc5bc\ub8e9\uc744 \uc9c0\uc6b8 \uc218 \uc788\ub294 \uc18c\ube44\uc131\uc774 \uc788\ub294 \ub3c4\uad6c\ub2e4. \ud64d\uc900\uc774\ub97c \uc704\ud574 \ub9e4\ud2b8\ub9ad\uc2a4 \ud06c\uae30\uc640 \uc5bc\ub8e9\uc5d0 \ub300\ud55c \uc815\ubcf4\uac00 \uc8fc\uc5b4\uc84c\uc744 \ub54c, \ud544\uc694\ud55c \uccad\uc18c \ub3c4\uad6c\uc758 \ucd5c\uc18c \uac1c\uc218\ub97c \uad6c\ud558\uc790.<\/p>\r\n\r\n<p>&lt;\uadf8\ub9bc 1&gt;\uc740 \uac00\ub2a5\ud55c \uc0c1\ud669\uc758 \ud55c \uc608\ub2e4. \ub9e4\ud2b8\ub9ad\uc2a4\uc758 \ud06c\uae30\ub294 6 &times; 11\uc774\uba70, \uac01 \uc5bc\ub8e9\uc758 \uc704\uce58 \uc88c\ud45c\ub294 (4, 3), (6, 5), (3, 6)\uacfc (4,7)\uc774\ub2e4. \uc704\uc640 \uac19\uc740 \uacbd\uc6b0\uc5d0 \ud544\uc694\ud55c \uccad\uc18c \ub3c4\uad6c\uc758 \ucd5c\uc18c \uac1c\uc218\ub294 2\uac1c\ub2e4.<\/p>\r\n","input":"<p>\uc785\ub825\uc740 \uc5ec\ub7ec \uac1c\uc758 \ud14c\uc2a4\ud2b8\ucf00\uc774\uc2a4\ub85c \uad6c\uc131\ub418\uc5b4 \uc788\ub2e4.<\/p>\r\n\r\n<p>\uac01 \ud14c\uc2a4\ud2b8\ucf00\uc774\uc2a4\uc758 \uccab \uc904\uc5d0\ub294 \ub9e4\ud2b8\ub9ad\uc2a4\uc758 \ud06c\uae30\ub97c \ub098\ud0c0\ub0b4\ub294 \ub450 \uc790\uc5f0\uc218 m\uacfc n\uc774 \uc8fc\uc5b4\uc9c4\ub2e4. (3 &le; m &le; 10, 3 &le; n &le; 1,000).&nbsp;\ub458\uc9f8 \uc904\uc5d0\ub294 \uc5bc\ub8e9\uc758 \uac1c\uc218\ub97c \ub098\ud0c0\ub0b4\ub294 \uc815\uc218 c\uac00 \uc8fc\uc5b4\uc9c4\ub2e4. (0 &le; c &le; mn)&nbsp;\uadf8\ub9ac\uace0 \ub2e4\uc74c c\uac1c\uc758 \uc904\uc5d0 \uac01 \uc5bc\ub8e9\uc758 \uc88c\ud45c\uac00 \uc8fc\uc5b4\uc9c4\ub2e4. \uc5bc\ub8e9\uc758 \uc88c\ud45c\ub294 \uc11c\ub85c \ub2e4\ub974\uac8c \uc8fc\uc5b4\uc9c0\uba70, \ubaa8\ub450 \ub9e4\ud2b8\ub9ad\uc2a4 \uc704\uc5d0 \uc704\uce58\ud55c\ub2e4.<\/p>\r\n","output":"<p>\uac01 \ud14c\uc2a4\ud2b8\ucf00\uc774\uc2a4 \ubcc4\ub85c \uc5bc\ub8e9\uc744 \ubaa8\ub450 \uc9c0\uc6b0\uae30 \uc704\ud574 \ud544\uc694\ud55c \uccad\uc18c \ub3c4\uad6c\uc758 \ucd5c\uc18c \uac1c\uc218\ub97c \uc758\ubbf8\ud558\ub294 \ud558\ub098\uc758 \uc815\uc218\ub97c \ucd9c\ub825\ud55c\ub2e4.<\/p>\r\n","hint":"","original":"0","problem_lang_code":"\ud55c\uad6d\uc5b4"},{"problem_id":"10257","problem_lang":"1","title":"Stains","description":"<p>During the ChuSeok holidays you had quite a few guests, including some very young boys. After they went back to their respective homes, you found some stains on your bed. The stains are deep and actually go into the mattress. It is quite expensive to clean the whole mattress, so you have decided to buy a few of the clean-mattress-squares, which can clean some area of the mattress. The surface of the mattress can be divided into an m &times; n grid, i.e., a rectangle with m unit squares vertically and n unit squares horizontally. A coordinate system is defined so that the unit square at the top-left corner has coordinate (1, 1) and the unit square at the bottom-right corner has (m, n). In each square there may be a stain. Each of your clean-mattress-squares is able to clean up all stains in a 3 &times; 3 area. Your mission is to minimize the number of clean-mattress-squares to be used while, of course, removing all of the stains.&nbsp;<\/p>\r\n\r\n<p><img alt=\"\" src=\"\/upload\/images2\/stains.png\" style=\"float:right; height:276px; width:441px\" \/>Figure 1 is an example where the total mattress is a 6 &times; 11 grid and there are four stains located at unit squares whose coordinates are (4, 3), (6, 5), (3, 6), and (4, 7), respectively. In this case, two clean-mattress-squares will suffice.<\/p>\r\n\r\n<p>Given the size of the mattress and the coordinates of the unit squares where stains are located, you are to write a program that computes the minimum number of clean-mattress-squares to be used.&nbsp;<\/p>\r\n","input":"<p>Your program is to read from standard input. The input consists of T test cases. The number of test cases T is given in the first line of the input. Each test case starts with integers m and n, giving the vertical and horizontal size of the mattress, respectively, where 3 &le; m &le; 10 and 3 &le; n &le; 1,000. In the next line, a nonnegative integer c is given, indicating the number of unit squares where stains are located. In the next c lines, the coordinates of the stains are given one per line. It always holds that 0 &le; c &le; mn. Also, the coordinates given are all different and are all on the mattress.&nbsp;<\/p>\r\n\r\n<p>&nbsp;<\/p>\r\n","output":"<p>Your program is to write to standard output. Print exactly one line for each test case. The line should contain a nonnegative integer, which is the minimum number of clean-mattress-squares needed to remove all of the stains.<\/p>\r\n","hint":"","original":"1","problem_lang_code":"\uc601\uc5b4"}]