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문제

m × n 직사각 그리드(rectangular grid)는, x-좌표의 범위가 0부터 n-1까지인 정수이고 y-좌표의 범위가 0부터 m-1까지 정수인 평면상의 점들에 대응하는 정점들을 가지고, 두 정점에 대응하는 두 점 사이의 거리가 1 일 때에만 그 둘을 잇는 에지가 있는 그래프다. 예를 들어, 4 × 6 직사각 그리드가 그림 1 에 있다. 이 그리드는 m개 행 각각에 n개의 정점이 있고, n개 열 각각에 m 개의 정점이 있다. 행 i와 열 j에 있는 정점을 (i,j)라고 나타낸다. 여기서 0 ≤ i ≤ m-1이고 0 ≤ j ≤ n-1이다.

m × n 직사각 그리드의 모든 행 i ∈ {0, … , m-1}에 대하여 두 정점 (i, 0)과 (i, n-1)을 잇는 에지를 추가하고, 또한 모든 열 j ∈ {0, … , n-1} 에 대하여 두 정점 (0, j) 와 (m-1, j) 을 잇는 에지를 추가하면, 그림 2 에 보인 것과 같이 각 행은 길이 n인 사이클을 이루고 각 열은 길이 m인 사이클을 이루게 된다. 이렇게 만들어진 그래프를 종종 m × n 토로이드 그리드(toroidal grid) 라고 부르는데, 왜냐하면 이 그래프를 토러스(torus)에 에지가 교차하지 않도록 그릴 수 있기 때문이다.

주어진 m × n 토로이드 그리드에 대하여, 모든 정점을 정확히 한번씩 지나는 사이클을 찾는 프로그램을 작성하시오. 문제에서 요구하는 사이클은 그래프에 있는 서로 다른 mn개 정점들의 열 (v1, v2, … , vmn)로 나타낼 수 있는데, 이때 모든 k ∈ {1, … , mn-1}에 대하여 vk와 vk+1은 인접하며 또한 vmn과 v1도 인접하여야 한다.

입력

표준입력에서 데이터를 읽는다. 입력은 T개의 의 테스트 데이터로 구성되어 있다. 입력의 첫째 줄에 입력 데이터의 개수를 나타내는 정수 T가 주어진다. 각 테스트 데이터는 두 정수 m과 n을 포함하는 한 줄로 구성되어 있는데, 입력 그래프가 m × n 토로이드 그리드임을 가리킨다. 여기서 3 ≤ m, n ≤ 100이다.

출력

표준출력으로 데이터를 출력한다. 각 테스트 데이터에 대해, 출력의 첫째 줄에는 조건을 만족하는 해가 존재하는지 아닌지를 나타내는 정수를 출력해야 한다. 만약 해가 존재하면, 그 정수는 1 이어야 한다; 그렇지 않으면 -1 이다. 첫째 줄이 1 일 경우에만 추가로 문제에서 요구한 사이클의 정점 열(sequence)을 mn 개의 줄에 출력한다. 복수의 해가 가능하면, 그 중 임의의 하나를 출력하면 된다. 어떤 줄에도 공백 문자(빈칸이나 탭)는 허용되지 않는다.

예제 입력 1

2
3 4
3 3

예제 출력 1

1
(0,0)
(0,1)
(1,1)
(1,2)
(0,2)
(0,3)
(1,3)
(2,3)
(2,2)
(2,1)
(2,0)
(1,0)
1
(2,2)
(2,0)
(1,0)
(0,0)
(0,1)
(0,2)
(1,2)
(1,1)
(2,1)
[{"problem_id":"10252","problem_lang":"0","title":"\uadf8\ub9ac\ub4dc \uadf8\ub798\ud504","description":"<p>m &times; n \uc9c1\uc0ac\uac01 \uadf8\ub9ac\ub4dc(rectangular grid)\ub294, x-\uc88c\ud45c\uc758 \ubc94\uc704\uac00 0\ubd80\ud130 n-1\uae4c\uc9c0\uc778 \uc815\uc218\uc774\uace0 y-\uc88c\ud45c\uc758 \ubc94\uc704\uac00 0\ubd80\ud130 m-1\uae4c\uc9c0 \uc815\uc218\uc778 \ud3c9\uba74\uc0c1\uc758 \uc810\ub4e4\uc5d0 \ub300\uc751\ud558\ub294 \uc815\uc810\ub4e4\uc744 \uac00\uc9c0\uace0, \ub450 \uc815\uc810\uc5d0 \ub300\uc751\ud558\ub294 \ub450 \uc810 \uc0ac\uc774\uc758 \uac70\ub9ac\uac00 1 \uc77c \ub54c\uc5d0\ub9cc \uadf8 \ub458\uc744 \uc787\ub294 \uc5d0\uc9c0\uac00 \uc788\ub294 \uadf8\ub798\ud504\ub2e4. \uc608\ub97c \ub4e4\uc5b4, 4 &times; 6 \uc9c1\uc0ac\uac01 \uadf8\ub9ac\ub4dc\uac00 \uadf8\ub9bc 1 \uc5d0 \uc788\ub2e4. \uc774 \uadf8\ub9ac\ub4dc\ub294 m\uac1c \ud589 \uac01\uac01\uc5d0 n\uac1c\uc758 \uc815\uc810\uc774 \uc788\uace0, n\uac1c \uc5f4 \uac01\uac01\uc5d0 m \uac1c\uc758 \uc815\uc810\uc774 \uc788\ub2e4. \ud589 i\uc640 \uc5f4 j\uc5d0 \uc788\ub294 \uc815\uc810\uc744 (i,j)\ub77c\uace0 \ub098\ud0c0\ub0b8\ub2e4. \uc5ec\uae30\uc11c 0 &le; i &le; m-1\uc774\uace0 0 &le; j &le; n-1\uc774\ub2e4.<\/p>\r\n\r\n<p>m &times; n \uc9c1\uc0ac\uac01 \uadf8\ub9ac\ub4dc\uc758 \ubaa8\ub4e0 \ud589 i &isin; {0, &hellip; , m-1}\uc5d0 \ub300\ud558\uc5ec \ub450 \uc815\uc810 (i, 0)\uacfc (i, n-1)\uc744 \uc787\ub294 \uc5d0\uc9c0\ub97c \ucd94\uac00\ud558\uace0, \ub610\ud55c \ubaa8\ub4e0 \uc5f4 j &isin; {0, &hellip; , n-1} \uc5d0 \ub300\ud558\uc5ec \ub450 \uc815\uc810 (0, j) \uc640 (m-1, j) \uc744 \uc787\ub294 \uc5d0\uc9c0\ub97c \ucd94\uac00\ud558\uba74, \uadf8\ub9bc 2 \uc5d0 \ubcf4\uc778 \uac83\uacfc \uac19\uc774 \uac01 \ud589\uc740 \uae38\uc774 n\uc778 \uc0ac\uc774\ud074\uc744 \uc774\ub8e8\uace0 \uac01 \uc5f4\uc740 \uae38\uc774 m\uc778 \uc0ac\uc774\ud074\uc744 \uc774\ub8e8\uac8c \ub41c\ub2e4. \uc774\ub807\uac8c \ub9cc\ub4e4\uc5b4\uc9c4 \uadf8\ub798\ud504\ub97c \uc885\uc885 m &times; n \ud1a0\ub85c\uc774\ub4dc \uadf8\ub9ac\ub4dc(toroidal grid) \ub77c\uace0 \ubd80\ub974\ub294\ub370, \uc65c\ub0d0\ud558\uba74 \uc774 \uadf8\ub798\ud504\ub97c \ud1a0\ub7ec\uc2a4(torus)\uc5d0 \uc5d0\uc9c0\uac00 \uad50\ucc28\ud558\uc9c0 \uc54a\ub3c4\ub85d \uadf8\ub9b4 \uc218 \uc788\uae30 \ub54c\ubb38\uc774\ub2e4.<\/p>\r\n\r\n<p style=\"text-align:center\"><img alt=\"\" src=\"\/upload\/images2\/gg1.png\" style=\"height:243px; width:643px\" \/><\/p>\r\n\r\n<p>\uc8fc\uc5b4\uc9c4 m &times; n \ud1a0\ub85c\uc774\ub4dc \uadf8\ub9ac\ub4dc\uc5d0 \ub300\ud558\uc5ec, \ubaa8\ub4e0 \uc815\uc810\uc744 \uc815\ud655\ud788 \ud55c\ubc88\uc529 \uc9c0\ub098\ub294 \uc0ac\uc774\ud074\uc744 \ucc3e\ub294 \ud504\ub85c\uadf8\ub7a8\uc744 \uc791\uc131\ud558\uc2dc\uc624. \ubb38\uc81c\uc5d0\uc11c \uc694\uad6c\ud558\ub294 \uc0ac\uc774\ud074\uc740 \uadf8\ub798\ud504\uc5d0 \uc788\ub294 \uc11c\ub85c \ub2e4\ub978 mn\uac1c \uc815\uc810\ub4e4\uc758 \uc5f4 (v<sub>1<\/sub>, v<sub>2<\/sub>, &hellip; , v<sub>mn<\/sub>)\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\ub294\ub370, \uc774\ub54c \ubaa8\ub4e0 k &isin; {1, &hellip; , mn-1}\uc5d0 \ub300\ud558\uc5ec v<sub>k<\/sub>\uc640 v<sub>k+1<\/sub>\uc740 \uc778\uc811\ud558\uba70 \ub610\ud55c v<sub>mn<\/sub>\uacfc v<sub>1<\/sub>\ub3c4 \uc778\uc811\ud558\uc5ec\uc57c \ud55c\ub2e4.<\/p>\r\n","input":"<p>\ud45c\uc900\uc785\ub825\uc5d0\uc11c \ub370\uc774\ud130\ub97c \uc77d\ub294\ub2e4. \uc785\ub825\uc740&nbsp;T\uac1c\uc758 \uc758 \ud14c\uc2a4\ud2b8 \ub370\uc774\ud130\ub85c \uad6c\uc131\ub418\uc5b4 \uc788\ub2e4. \uc785\ub825\uc758 \uccab\uc9f8 \uc904\uc5d0 \uc785\ub825 \ub370\uc774\ud130\uc758 \uac1c\uc218\ub97c \ub098\ud0c0\ub0b4\ub294 \uc815\uc218 T\uac00 \uc8fc\uc5b4\uc9c4\ub2e4. \uac01 \ud14c\uc2a4\ud2b8 \ub370\uc774\ud130\ub294 \ub450 \uc815\uc218 m\uacfc n\uc744 \ud3ec\ud568\ud558\ub294 \ud55c \uc904\ub85c \uad6c\uc131\ub418\uc5b4 \uc788\ub294\ub370, \uc785\ub825 \uadf8\ub798\ud504\uac00 m &times; n \ud1a0\ub85c\uc774\ub4dc \uadf8\ub9ac\ub4dc\uc784\uc744 \uac00\ub9ac\ud0a8\ub2e4. \uc5ec\uae30\uc11c 3 &le; m, n &le; 100\uc774\ub2e4.<\/p>\r\n","output":"<p>\ud45c\uc900\ucd9c\ub825\uc73c\ub85c \ub370\uc774\ud130\ub97c \ucd9c\ub825\ud55c\ub2e4. \uac01 \ud14c\uc2a4\ud2b8 \ub370\uc774\ud130\uc5d0 \ub300\ud574, \ucd9c\ub825\uc758 \uccab\uc9f8 \uc904\uc5d0\ub294 \uc870\uac74\uc744 \ub9cc\uc871\ud558\ub294 \ud574\uac00 \uc874\uc7ac\ud558\ub294\uc9c0 \uc544\ub2cc\uc9c0\ub97c \ub098\ud0c0\ub0b4\ub294 \uc815\uc218\ub97c \ucd9c\ub825\ud574\uc57c \ud55c\ub2e4. \ub9cc\uc57d \ud574\uac00 \uc874\uc7ac\ud558\uba74, \uadf8 \uc815\uc218\ub294 1 \uc774\uc5b4\uc57c \ud55c\ub2e4; \uadf8\ub807\uc9c0 \uc54a\uc73c\uba74 -1 \uc774\ub2e4. \uccab\uc9f8 \uc904\uc774 1 \uc77c \uacbd\uc6b0\uc5d0\ub9cc \ucd94\uac00\ub85c \ubb38\uc81c\uc5d0\uc11c \uc694\uad6c\ud55c \uc0ac\uc774\ud074\uc758 \uc815\uc810 \uc5f4(sequence)\uc744 mn \uac1c\uc758 \uc904\uc5d0 \ucd9c\ub825\ud55c\ub2e4. \ubcf5\uc218\uc758 \ud574\uac00 \uac00\ub2a5\ud558\uba74, \uadf8 \uc911 \uc784\uc758\uc758 \ud558\ub098\ub97c \ucd9c\ub825\ud558\uba74 \ub41c\ub2e4. \uc5b4\ub5a4 \uc904\uc5d0\ub3c4 \uacf5\ubc31 \ubb38\uc790(\ube48\uce78\uc774\ub098 \ud0ed)\ub294 \ud5c8\uc6a9\ub418\uc9c0 \uc54a\ub294\ub2e4.<\/p>\r\n","hint":"","original":"0","problem_lang_code":"\ud55c\uad6d\uc5b4"},{"problem_id":"10252","problem_lang":"1","title":"Grid Graphs","description":"<p>The m &times; n rectangular grid is a graph whose vertices correspond to the points in the plane with x-coordinates being integers in the range 0, &hellip; , n-1 and y-coordinates being integers in the range 0, &hellip; , m-1, and two vertices are joined by an edge whenever the corresponding points are at unit distance apart. For example, a 4 &times; 6 rectangular grid is shown in Figure 1. The grid has n vertices appearing in each of m rows and m vertices in each of n columns. The vertex in row i and column j is denoted by (i, j), where 0 &le; i &le; m - 1 and 0 &le; j &le; n - 1.<\/p>\r\n\r\n<p>If we add an edge joining two vertices (i, 0) and (i, n-1) of the m &times; n rectangular grid for every row i &isin; {0, &hellip; , m-1} and moreover, add an edge between two vertices (0, j) and (m-1, j) for every column j &isin; {0, &hellip; , n-1}, then each row forms a cycle of length n and each column forms a cycle of length m, as illustrated in Figure 2. The resulting graph is often called an m &times; n toroidal grid, because it can be drawn on a torus without edge crossings.<\/p>\r\n\r\n<p style=\"text-align:center\"><img alt=\"\" src=\"\/upload\/images2\/gg2.png\" style=\"height:239px; width:643px\" \/><\/p>\r\n\r\n<p>Given an m &times; n toroidal grid, you are to write a program to find a cycle that visits every vertex exactly once. Here, the required cycle may be represented as a sequence, (v<sub>1<\/sub>, v<sub>2<\/sub>, &hellip; , v<sub>mn<\/sub>), of mn distinct vertices of the graph such that v<sub>k<\/sub> and v<sub>k+1<\/sub> are adjacent for all k &isin; {1, &hellip; , mn-1} and moreover, v<sub>mn<\/sub> and v<sub>1<\/sub> are adjacent.<\/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 consists of a single line containing two integers, m and n, where 3 &le; m, n &le; 100, indicating that the input graph is an m &times; n toroidal grid.<\/p>\r\n","output":"<p>Your program is to write to standard output. For each test case, the first line of the results must contain an integer indicating whether there exists a feasible solution. If yes, the integer must be 1; otherwise -1. If the first line is 1 and only 1, then it must be followed by mn lines, describing a sequence of vertices of the required cycle. In case multiple solutions are possible, just output any one of them. No whitespace characters (blanks and\/or tabs) are allowed inside a line.<\/p>\r\n","hint":"","original":"1","problem_lang_code":"\uc601\uc5b4"}]