시간 제한 메모리 제한 제출 정답 맞은 사람 정답 비율
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문제

안드로이드는 스마트폰을 잠글때 패턴을 이용한다. 보통 잠근 패턴은 3×3 그리드에 노드 9개로 이루어져 있다. 상현이는 보다 더 높은 보안을 위해 아래와 같은 잠금 패턴을 제공하는 앱을 만드려고 한다.

상현이가 만들 앱의 잠금 패턴의 크기는 2×2부터 m×m까지로 다양하며, 항상 인접한 노드만 연결할 수 있다. 잠금 패턴은 그래프로 나타낼 수 있다. 아래 그림은 2×2, 3×3, 4×4 잠금 패턴을 그래프로 나타낸 것이다. 꼭지점에 해당하는 정점은 총 3개와 연결되어 있고, 나머지 정점은 최대 8개와 연결될 수도 있다. 항상 아래 그림과 같이 인접한 점과만 연결할 수 있다.

잠금 패턴은 항상 스패닝 트리를 이루어야 한다. 스패닝 트리는 그래프 상의 간선의 집합으로, 닫힌 루프를 포함하지 않아야 한다. 또, 임의의 두 점 사이의 경로가 존재해야 하고, 모든 m2 = n개의 정점과 n-1개의 edge로 이루어져 있어야 한다. 잠금 패턴 상에서 스패닝 트리는 여러 가지 경우가 나올 수 있다.

그래프 G의 스패닝 트리의 개수를 구하기 위해 모든 정점에 번호 v1, ..., vn을 붙인다. 이제 행렬 T = [tij]를 아래와 같이 만들 수 있다.

  • i = j인 경우에는 vi와 연결된 간선의 개수
  • i ≠ j인 경우에 vi와 vj 사이에 간선이 있으면 -1, 없으면 0

스패닝 트리의 개수는 다음과 같이 구할 수 있다.

cofactor of aij = (-1)i+jMij 

여기서 Mij는 T에서 i행과 j열을 지워서 얻은 (n-1) × (n-1) 행렬의 행렬식(determinant)이다. T의 모든 cofactor는 같은 결과를 갖는다.

예를 들어, 2×2 패턴의 행렬 T는 다음과 같다.

\[T=\begin{pmatrix} 3 & -1 & -1 & -1 \\ -1 & 3 & -1 & -1 \\  -1&-1&3&-1 \\ -1&-1&-1&3  \end{pmatrix}\]

i=1, j=1로 두고 cofator를 구해보면 아래와 같다.

\[(-1)^{1+1}M_{11} =  \begin{vmatrix}  3& -1&-1 \\  -1&3&-1 \\ -1&-1&3  \end{vmatrix} = 16\]

따라서, 스패닝 트리의 개수는 16개임을 알 수 있다. 

두 번째 예제로 3×3 패턴의 행렬 T는 다음과 같다.

\[T = \begin{pmatrix}  3&-1&0&-1&-1&0&0&0&0 \\ -1&5&-1&-1&-1&-1&0&0&0 \\ 0&-1&3&0&-1&-1&0&0&0 \\ -1&-1&0&5&-1&0&-1&-1&0 \\ -1&-1&-1&-1&8&-1&-1&-1&-1 \\ 0 & -1&-1&0&-1&5&0&-1&-1 \\ 0&0&0&-1&-1&0&3&-1&0 \\ 0&0&0&-1&-1&-1&-1&5&-1 \\ 0&0&0&0&-1&-1&0&-1&3  \end{pmatrix}\]

T의 모든 cofactor는 같은 결과를 갖고, 이 답이 스패닝 트리의 개수가 된다.

m×m 패턴에서 만들 수 있는 스패닝 트리의 개수를 구하는 프로그램을 작성하시오.

입력

첫째 줄에 테스트 케이스의 개수 N (1 ≤ N ≤ 5)가 주어진다. 다음 N개 줄에는 테스트케이스 한 줄에 하나씩 주어진다. 각 테스트 케이스는 한 줄로 이루어져 있고, 패턴의 크기 m이 주어진다. (2 ≤ m ≤ 6)

출력

각 테스트 케이스마다 m×m 패턴에서 만들 수 있는 스패닝 트리의 개수를 한 줄에 하나씩 출력한다.

예제 입력 1

1
2

예제 출력 1

16

힌트

m = 2인 경우 아래와 같이 16가지 경우가 있다.

[{"problem_id":"9614","problem_lang":"0","title":"\uc7a0\uae08 \ud328\ud134\uacfc \uc2a4\ud328\ub2dd \ud2b8\ub9ac","description":"<p>\uc548\ub4dc\ub85c\uc774\ub4dc\ub294 \uc2a4\ub9c8\ud2b8\ud3f0\uc744 \uc7a0\uae00\ub54c \ud328\ud134\uc744 \uc774\uc6a9\ud55c\ub2e4. \ubcf4\ud1b5 \uc7a0\uadfc \ud328\ud134\uc740 3&times;3 \uadf8\ub9ac\ub4dc\uc5d0 \ub178\ub4dc 9\uac1c\ub85c \uc774\ub8e8\uc5b4\uc838 \uc788\ub2e4. \uc0c1\ud604\uc774\ub294 \ubcf4\ub2e4 \ub354 \ub192\uc740 \ubcf4\uc548\uc744 \uc704\ud574 \uc544\ub798\uc640 \uac19\uc740 \uc7a0\uae08 \ud328\ud134\uc744 \uc81c\uacf5\ud558\ub294 \uc571\uc744 \ub9cc\ub4dc\ub824\uace0 \ud55c\ub2e4.<\/p>\r\n\r\n<p>\uc0c1\ud604\uc774\uac00 \ub9cc\ub4e4 \uc571\uc758 \uc7a0\uae08 \ud328\ud134\uc758 \ud06c\uae30\ub294 2&times;2\ubd80\ud130 m&times;m\uae4c\uc9c0\ub85c \ub2e4\uc591\ud558\uba70, \ud56d\uc0c1 \uc778\uc811\ud55c \ub178\ub4dc\ub9cc \uc5f0\uacb0\ud560 \uc218 \uc788\ub2e4. \uc7a0\uae08 \ud328\ud134\uc740 \uadf8\ub798\ud504\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4. \uc544\ub798 \uadf8\ub9bc\uc740 2&times;2, 3&times;3, 4&times;4 \uc7a0\uae08 \ud328\ud134\uc744 \uadf8\ub798\ud504\ub85c \ub098\ud0c0\ub0b8 \uac83\uc774\ub2e4. \uaf2d\uc9c0\uc810\uc5d0 \ud574\ub2f9\ud558\ub294 \uc815\uc810\uc740 \ucd1d 3\uac1c\uc640 \uc5f0\uacb0\ub418\uc5b4 \uc788\uace0, \ub098\uba38\uc9c0 \uc815\uc810\uc740 \ucd5c\ub300 8\uac1c\uc640 \uc5f0\uacb0\ub420 \uc218\ub3c4 \uc788\ub2e4. \ud56d\uc0c1 \uc544\ub798 \uadf8\ub9bc\uacfc \uac19\uc774 \uc778\uc811\ud55c \uc810\uacfc\ub9cc \uc5f0\uacb0\ud560 \uc218 \uc788\ub2e4.<\/p>\r\n\r\n<p><img alt=\"\" src=\"\/upload\/images\/securespanning.png\" \/><\/p>\r\n\r\n<p>\uc7a0\uae08 \ud328\ud134\uc740 \ud56d\uc0c1 \uc2a4\ud328\ub2dd \ud2b8\ub9ac\ub97c \uc774\ub8e8\uc5b4\uc57c \ud55c\ub2e4. \uc2a4\ud328\ub2dd \ud2b8\ub9ac\ub294 \uadf8\ub798\ud504 \uc0c1\uc758 \uac04\uc120\uc758 \uc9d1\ud569\uc73c\ub85c, \ub2eb\ud78c \ub8e8\ud504\ub97c \ud3ec\ud568\ud558\uc9c0 \uc54a\uc544\uc57c \ud55c\ub2e4. \ub610, \uc784\uc758\uc758 \ub450 \uc810 \uc0ac\uc774\uc758 \uacbd\ub85c\uac00 \uc874\uc7ac\ud574\uc57c \ud558\uace0, \ubaa8\ub4e0 m<sup>2<\/sup> = n\uac1c\uc758 \uc815\uc810\uacfc n-1\uac1c\uc758 edge\ub85c \uc774\ub8e8\uc5b4\uc838 \uc788\uc5b4\uc57c \ud55c\ub2e4. \uc7a0\uae08 \ud328\ud134 \uc0c1\uc5d0\uc11c \uc2a4\ud328\ub2dd \ud2b8\ub9ac\ub294 \uc5ec\ub7ec \uac00\uc9c0 \uacbd\uc6b0\uac00 \ub098\uc62c \uc218 \uc788\ub2e4.<\/p>\r\n\r\n<p>\uadf8\ub798\ud504 G\uc758 \uc2a4\ud328\ub2dd \ud2b8\ub9ac\uc758 \uac1c\uc218\ub97c \uad6c\ud558\uae30 \uc704\ud574 \ubaa8\ub4e0 \uc815\uc810\uc5d0 \ubc88\ud638 v<sub>1<\/sub>, ..., v<sub>n<\/sub>\uc744 \ubd99\uc778\ub2e4. \uc774\uc81c \ud589\ub82c T = [t<sub>ij<\/sub>]\ub97c \uc544\ub798\uc640 \uac19\uc774 \ub9cc\ub4e4 \uc218 \uc788\ub2e4.<\/p>\r\n\r\n<ul>\r\n\t<li>i = j\uc778 \uacbd\uc6b0\uc5d0\ub294 v<sub>i<\/sub>\uc640 \uc5f0\uacb0\ub41c \uac04\uc120\uc758 \uac1c\uc218<\/li>\r\n\t<li>i &ne; j\uc778 \uacbd\uc6b0\uc5d0 v<sub>i<\/sub>\uc640 v<sub>j<\/sub> \uc0ac\uc774\uc5d0 \uac04\uc120\uc774 \uc788\uc73c\uba74 -1, \uc5c6\uc73c\uba74 0<\/li>\r\n<\/ul>\r\n\r\n<p>\uc2a4\ud328\ub2dd \ud2b8\ub9ac\uc758 \uac1c\uc218\ub294 \ub2e4\uc74c\uacfc \uac19\uc774 \uad6c\ud560 \uc218 \uc788\ub2e4.<\/p>\r\n\r\n<p>cofactor of a<sub>ij<\/sub> = (-1)<sup>i+j<\/sup>M<sub>ij<\/sub>&nbsp;<\/p>\r\n\r\n<p>\uc5ec\uae30\uc11c M<sub>ij<\/sub>\ub294 T\uc5d0\uc11c i\ud589\uacfc j\uc5f4\uc744 \uc9c0\uc6cc\uc11c \uc5bb\uc740 (n-1) &times; (n-1) \ud589\ub82c\uc758 \ud589\ub82c\uc2dd(determinant)\uc774\ub2e4. T\uc758 \ubaa8\ub4e0 cofactor\ub294 \uac19\uc740 \uacb0\uacfc\ub97c \uac16\ub294\ub2e4.<\/p>\r\n\r\n<p>\uc608\ub97c \ub4e4\uc5b4, 2&times;2 \ud328\ud134\uc758 \ud589\ub82c T\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\r\n\r\n<p>\\[T=\\begin{pmatrix} 3 &amp; -1 &amp; -1 &amp; -1 \\\\ -1 &amp; 3 &amp; -1 &amp; -1 \\\\ &nbsp;-1&amp;-1&amp;3&amp;-1 \\\\ -1&amp;-1&amp;-1&amp;3 &nbsp;\\end{pmatrix}\\]<\/p>\r\n\r\n<p>i=1, j=1\ub85c \ub450\uace0 cofator\ub97c \uad6c\ud574\ubcf4\uba74 \uc544\ub798\uc640 \uac19\ub2e4.<\/p>\r\n\r\n<p>\\[(-1)^{1+1}M_{11} = &nbsp;\\begin{vmatrix} &nbsp;3&amp; -1&amp;-1 \\\\ &nbsp;-1&amp;3&amp;-1 \\\\ -1&amp;-1&amp;3 &nbsp;\\end{vmatrix} = 16\\]<\/p>\r\n\r\n<p>\ub530\ub77c\uc11c, \uc2a4\ud328\ub2dd \ud2b8\ub9ac\uc758 \uac1c\uc218\ub294 16\uac1c\uc784\uc744 \uc54c \uc218 \uc788\ub2e4.&nbsp;<\/p>\r\n\r\n<p>\ub450 \ubc88\uc9f8 \uc608\uc81c\ub85c 3&times;3 \ud328\ud134\uc758 \ud589\ub82c T\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\r\n\r\n<p>\\[T = \\begin{pmatrix} &nbsp;3&amp;-1&amp;0&amp;-1&amp;-1&amp;0&amp;0&amp;0&amp;0 \\\\ -1&amp;5&amp;-1&amp;-1&amp;-1&amp;-1&amp;0&amp;0&amp;0 \\\\ 0&amp;-1&amp;3&amp;0&amp;-1&amp;-1&amp;0&amp;0&amp;0 \\\\ -1&amp;-1&amp;0&amp;5&amp;-1&amp;0&amp;-1&amp;-1&amp;0 \\\\ -1&amp;-1&amp;-1&amp;-1&amp;8&amp;-1&amp;-1&amp;-1&amp;-1 \\\\ 0 &amp; -1&amp;-1&amp;0&amp;-1&amp;5&amp;0&amp;-1&amp;-1 \\\\ 0&amp;0&amp;0&amp;-1&amp;-1&amp;0&amp;3&amp;-1&amp;0 \\\\ 0&amp;0&amp;0&amp;-1&amp;-1&amp;-1&amp;-1&amp;5&amp;-1 \\\\ 0&amp;0&amp;0&amp;0&amp;-1&amp;-1&amp;0&amp;-1&amp;3 &nbsp;\\end{pmatrix}\\]<\/p>\r\n\r\n<p>T\uc758 \ubaa8\ub4e0 cofactor\ub294 \uac19\uc740 \uacb0\uacfc\ub97c \uac16\uace0, \uc774 \ub2f5\uc774 \uc2a4\ud328\ub2dd \ud2b8\ub9ac\uc758 \uac1c\uc218\uac00 \ub41c\ub2e4.<\/p>\r\n\r\n<p>m&times;m \ud328\ud134\uc5d0\uc11c \ub9cc\ub4e4 \uc218 \uc788\ub294 \uc2a4\ud328\ub2dd \ud2b8\ub9ac\uc758 \uac1c\uc218\ub97c \uad6c\ud558\ub294 \ud504\ub85c\uadf8\ub7a8\uc744 \uc791\uc131\ud558\uc2dc\uc624.<\/p>\r\n","input":"<p>\uccab\uc9f8 \uc904\uc5d0 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\uc758 \uac1c\uc218 N (1 &le; N &le; 5)\uac00 \uc8fc\uc5b4\uc9c4\ub2e4. \ub2e4\uc74c N\uac1c \uc904\uc5d0\ub294 \ud14c\uc2a4\ud2b8\ucf00\uc774\uc2a4 \ud55c \uc904\uc5d0 \ud558\ub098\uc529 \uc8fc\uc5b4\uc9c4\ub2e4. \uac01 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\ub294 \ud55c \uc904\ub85c \uc774\ub8e8\uc5b4\uc838 \uc788\uace0, \ud328\ud134\uc758 \ud06c\uae30 m\uc774 \uc8fc\uc5b4\uc9c4\ub2e4. (2 &le; m &le; 6)<\/p>\r\n","output":"<p>\uac01 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\ub9c8\ub2e4 m&times;m \ud328\ud134\uc5d0\uc11c \ub9cc\ub4e4 \uc218 \uc788\ub294 \uc2a4\ud328\ub2dd \ud2b8\ub9ac\uc758 \uac1c\uc218\ub97c \ud55c \uc904\uc5d0 \ud558\ub098\uc529 \ucd9c\ub825\ud55c\ub2e4.<\/p>\r\n","hint":"<p>m = 2\uc778 \uacbd\uc6b0 \uc544\ub798\uc640 \uac19\uc774 16\uac00\uc9c0 \uacbd\uc6b0\uac00 \uc788\ub2e4.<\/p>\r\n\r\n<p><img alt=\"\" src=\"\/upload\/images\/securexm.png\" style=\"height:389px; width:533px\" \/><\/p>\r\n","original":"0","problem_lang_code":"\ud55c\uad6d\uc5b4"},{"problem_id":"9614","problem_lang":"1","title":"Spanning Trees in a Secure Lock Pattern","description":"<p>Screen lock on a smart phone can be secured with pattern. Unlock pattern may have 9 nodes on the 3 x 3 grid. In this problem, we modify this arrangement slightly such that the grid size can be varied from 2 x 2 to m x m, and only a limited number of connections between adjacent nodes are permitted. Let call this a modified mesh graph G and V is a set of vertices (nodes) and E is a set of edges (connections). Examples of 2 x 2, 3 x 3 and 4 x 4 modified mesh graphs are given in the figures below. Note that the corner vertices have 3 connections at most, while the middle vertices can have up to 8 connections. Only a connection to adjacent vertices is allowed, a constraint of the problem.&nbsp;<\/p>\r\n\r\n<p><img alt=\"\" src=\"\/upload\/images\/securespanning.png\" \/><\/p>\r\n\r\n<p>We define a spanning tree in our problem as a collection of lines in the graph forming no closed loops, containing a path between any two points of the graph covering m<sup>2<\/sup> = n vertices and n &ndash; 1 edges. Spanning trees in a graph can have many shapes or patterns.&nbsp;<\/p>\r\n\r\n<p>To calculate a number of spanning trees in G, let G be a graph with vertices labeled v<sub>1<\/sub>,...,v<sub>n<\/sub>. We form an n x n matrix tree T = [t<sub>ij<\/sub>] as follows.&nbsp;<\/p>\r\n\r\n<ul>\r\n\t<li>If i = j, then t<sub>ii<\/sub> is the number of lines to v<sub>i<\/sub> in the graph.&nbsp;<\/li>\r\n\t<li>If i &ne; j, then t<sub>ij<\/sub> = 0 if there is no line between v<sub>i<\/sub> and v<sub>j<\/sub> in G.&nbsp;<\/li>\r\n\t<li>If i &ne; j, then t<sub>ij<\/sub> = -1 if there is a line between v<sub>i<\/sub> and v<sub>j<\/sub> in G.&nbsp;<\/li>\r\n<\/ul>\r\n\r\n<p>Then, the number of spanning trees in G is given by&nbsp;<\/p>\r\n\r\n<p>cofactor of a<sub>ij<\/sub> = (-1)<sup>i+j<\/sup>M<sub>ij<\/sub>&nbsp;<\/p>\r\n\r\n<p>where M<sub>ij<\/sub> is the determinant of the (n-1) x (n-1) matrix obtained by deleting row i and column j of the matrix tree T. Evaluate any cofactor of T yields the same result.&nbsp;<\/p>\r\n\r\n<p>Example 1: A matrix tree T of 2 x 2 modified mesh graph is&nbsp;<\/p>\r\n\r\n<p>\\[T=\\begin{pmatrix} 3 &amp; -1 &amp; -1 &amp; -1 \\\\ -1 &amp; 3 &amp; -1 &amp; -1 \\\\ &nbsp;-1&amp;-1&amp;3&amp;-1 \\\\ -1&amp;-1&amp;-1&amp;3 &nbsp;\\end{pmatrix}\\]<\/p>\r\n\r\n<p>Evaluate any cofactor of T. For example, covering up row 1 and column 1, we have&nbsp;<\/p>\r\n\r\n<p>\\[(-1)^{1+1}M_{11} = &nbsp;\\begin{vmatrix} &nbsp;3&amp; -1&amp;-1 \\\\ &nbsp;-1&amp;3&amp;-1 \\\\ -1&amp;-1&amp;3 &nbsp;\\end{vmatrix} = 16\\]<\/p>\r\n\r\n<p>Therefore, a number of spanning tree is 16.&nbsp;<\/p>\r\n\r\n<p>Example 2: A matrix tree T of 3 x 3 modified mesh graph is&nbsp;<\/p>\r\n\r\n<p>\\[T = \\begin{pmatrix} &nbsp;3&amp;-1&amp;0&amp;-1&amp;-1&amp;0&amp;0&amp;0&amp;0 \\\\ -1&amp;5&amp;-1&amp;-1&amp;-1&amp;-1&amp;0&amp;0&amp;0 \\\\ 0&amp;-1&amp;3&amp;0&amp;-1&amp;-1&amp;0&amp;0&amp;0 \\\\ -1&amp;-1&amp;0&amp;5&amp;-1&amp;0&amp;-1&amp;-1&amp;0 \\\\ -1&amp;-1&amp;-1&amp;-1&amp;8&amp;-1&amp;-1&amp;-1&amp;-1 \\\\ 0 &amp; -1&amp;-1&amp;0&amp;-1&amp;5&amp;0&amp;-1&amp;-1 \\\\ 0&amp;0&amp;0&amp;-1&amp;-1&amp;0&amp;3&amp;-1&amp;0 \\\\ 0&amp;0&amp;0&amp;-1&amp;-1&amp;-1&amp;-1&amp;5&amp;-1 \\\\ 0&amp;0&amp;0&amp;0&amp;-1&amp;-1&amp;0&amp;-1&amp;3 &nbsp;\\end{pmatrix}\\]<\/p>\r\n\r\n<p>Evaluate any cofactor of T will yield the same value which is the number of spanning trees.&nbsp;<\/p>\r\n\r\n<p>Hint: Determinant calculation using elementary row operation method:&nbsp;<\/p>\r\n\r\n<p>To calculate a determinant you need to transform an original matrix into an upper triangular matrix (all elements below main diagonal are zero). Reduce the matrix to row echelon form using elementary row operations so that all the elements below diagonal are zero. Then multiply the main diagonal elements of a matrix to get the determinant value.&nbsp;<\/p>\r\n\r\n<p>Your task is to write a program to find the number of spanning trees for a given size m x m of a modified mesh graph.&nbsp;<\/p>\r\n","input":"<p>The first line of the input contains an integer N (1 &lt;= N &lt;= 5) denoting the number of test cases. After that N test cases follow. Each test case contains a single integer m (2 &lt;= m &lt;= 6) denoting a size of an m x m modified mesh graph.&nbsp;<\/p>\r\n","output":"<p>For each test case, your program should output the number of spanning trees, which indicates how many patterns of spanning trees there are for a given modified mesh graph. Print out one line per test case, respectively.&nbsp;<\/p>\r\n","hint":"<p>In the above sample input, there is only 1 test case (N = 1). For example, in the first test case there are 16 different spanning trees for m = 2. These spanning trees are listed below. Note that your program is NOT required to draw all possible patterns.&nbsp;<\/p>\r\n\r\n<p><img alt=\"\" src=\"\/upload\/images\/securexm.png\" style=\"height:389px; width:533px\" \/><\/p>\r\n","original":"1","problem_lang_code":"\uc601\uc5b4"}]