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문제

하나의 큰 원에 1부터 N까지 숫자가 적힌 점이 시계방향으로 차례로 있다. 그리고 각 점은 서로 다른 두 점과 연결이 되어있다. (즉, 모든 점의 차수는 2라는 뜻이다.)

그런데 이 점을 연결하는 과정에서 연결하는 선이 서로 엉켜있다면 이것은 볼록다각형이 되지 못할 것이다. 그래서 원 위에 있는 점을 적절한 곳으로 옮겨서 선이 서로 엉키는 것을 푸려고 한다. 다음 그림을 보자.

처음에 왼쪽 그림과 같이 연결이 되어 있다고 하자. 그리고 4번 점을 빨간색 화살표와 같이 이동하면 오른쪽과 같이 될 것이다. 오른쪽 그림은 서로 엉킨 것이 없어 볼록다각형이 될 것이다.

위의 그림도 마찬가지로 가장 왼쪽에 있는 상태에서 위의 빨간색 화살표를 따라 두 번의 이동을 하면 가장 오른쪽과 같이 서로 엉키는 부분이 없어지게 된다.

1부터 N번까지의 점이 연결된 상태들의 정보가 주어져 있을 때, 엉킨 것을 모두 풀어서 볼록 N각형이 되게 하려면 최소한 몇 번의 점의 이동이 있어야 하는지 구하는 프로그램을 작성하여라.

입력

첫째 줄에 점들의 개수 N(1≤N≤500)이 주어진다. 그리고 두 번째 줄부터 N+1번째 줄까지 점들의 연결 상태가 주어진다. 각 정보는 두 개의 정수로 이루어 져 있는데 i번째 줄에 a와 b 두 정수가 주어져 있다면 i번 점과 a, b가 연결이 되어 있다는 것을 의미한다.

출력

첫 줄에 최소 이동 회수를 출력한다. 만약에 볼록 N각형을 만드는 것이 불가능 한 경우에는 -1을 출력한다.

예제 입력 1

6
4 5
3 5
2 6
1 6
1 2
3 4

예제 출력 1

2
[{"problem_id":"1847","problem_lang":"0","title":"\ubcfc\ub85d\ub2e4\uac01\ud615 \ub9cc\ub4e4\uae30","description":"<p>\ud558\ub098\uc758 \ud070 \uc6d0\uc5d0 1\ubd80\ud130 N\uae4c\uc9c0 \uc22b\uc790\uac00 \uc801\ud78c \uc810\uc774 \uc2dc\uacc4\ubc29\ud5a5\uc73c\ub85c \ucc28\ub840\ub85c \uc788\ub2e4. \uadf8\ub9ac\uace0 \uac01 \uc810\uc740 \uc11c\ub85c \ub2e4\ub978 \ub450 \uc810\uacfc \uc5f0\uacb0\uc774 \ub418\uc5b4\uc788\ub2e4. (\uc989, \ubaa8\ub4e0 \uc810\uc758 \ucc28\uc218\ub294 2\ub77c\ub294 \ub73b\uc774\ub2e4.)<\/p><p>\uadf8\ub7f0\ub370 \uc774 \uc810\uc744 \uc5f0\uacb0\ud558\ub294 \uacfc\uc815\uc5d0\uc11c \uc5f0\uacb0\ud558\ub294 \uc120\uc774 \uc11c\ub85c \uc5c9\ucf1c\uc788\ub2e4\uba74 \uc774\uac83\uc740 \ubcfc\ub85d\ub2e4\uac01\ud615\uc774 \ub418\uc9c0 \ubabb\ud560 \uac83\uc774\ub2e4. \uadf8\ub798\uc11c \uc6d0 \uc704\uc5d0 \uc788\ub294 \uc810\uc744 \uc801\uc808\ud55c \uacf3\uc73c\ub85c \uc62e\uaca8\uc11c \uc120\uc774 \uc11c\ub85c \uc5c9\ud0a4\ub294 \uac83\uc744 \ud478\ub824\uace0 \ud55c\ub2e4. \ub2e4\uc74c \uadf8\ub9bc\uc744 \ubcf4\uc790.<\/p>\r\n<p><img width=\"263\" height=\"130\" alt=\"\" src=\"\/JudgeOnline\/upload\/201007\/bp.PNG\" \/><\/p>\r\n<p>\ucc98\uc74c\uc5d0 \uc67c\ucabd \uadf8\ub9bc\uacfc \uac19\uc774 \uc5f0\uacb0\uc774 \ub418\uc5b4 \uc788\ub2e4\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 4\ubc88 \uc810\uc744 \ube68\uac04\uc0c9 \ud654\uc0b4\ud45c\uc640 \uac19\uc774 \uc774\ub3d9\ud558\uba74 \uc624\ub978\ucabd\uacfc \uac19\uc774 \ub420 \uac83\uc774\ub2e4. \uc624\ub978\ucabd \uadf8\ub9bc\uc740 \uc11c\ub85c \uc5c9\ud0a8 \uac83\uc774 \uc5c6\uc5b4 \ubcfc\ub85d\ub2e4\uac01\ud615\uc774 \ub420 \uac83\uc774\ub2e4.<\/p>\r\n<p><img width=\"276\" height=\"108\" alt=\"\" src=\"\/JudgeOnline\/upload\/201007\/bp2.PNG\" \/><\/p>\r\n<p>\uc704\uc758 \uadf8\ub9bc\ub3c4 \ub9c8\ucc2c\uac00\uc9c0\ub85c \uac00\uc7a5 \uc67c\ucabd\uc5d0 \uc788\ub294 \uc0c1\ud0dc\uc5d0\uc11c \uc704\uc758 \ube68\uac04\uc0c9 \ud654\uc0b4\ud45c\ub97c \ub530\ub77c \ub450 \ubc88\uc758 \uc774\ub3d9\uc744 \ud558\uba74 \uac00\uc7a5 \uc624\ub978\ucabd\uacfc \uac19\uc774 \uc11c\ub85c \uc5c9\ud0a4\ub294 \ubd80\ubd84\uc774 \uc5c6\uc5b4\uc9c0\uac8c \ub41c\ub2e4.<\/p>\r\n<p>1\ubd80\ud130 N\ubc88\uae4c\uc9c0\uc758 \uc810\uc774 \uc5f0\uacb0\ub41c \uc0c1\ud0dc\ub4e4\uc758 \uc815\ubcf4\uac00 \uc8fc\uc5b4\uc838 \uc788\uc744 \ub54c, \uc5c9\ud0a8 \uac83\uc744 \ubaa8\ub450 \ud480\uc5b4\uc11c \ubcfc\ub85d N\uac01\ud615\uc774 \ub418\uac8c \ud558\ub824\uba74 \ucd5c\uc18c\ud55c \uba87 \ubc88\uc758 \uc810\uc758 \uc774\ub3d9\uc774 \uc788\uc5b4\uc57c \ud558\ub294\uc9c0 \uad6c\ud558\ub294 \ud504\ub85c\uadf8\ub7a8\uc744 \uc791\uc131\ud558\uc5ec\ub77c.<\/p>","input":"<p>\uccab\uc9f8 \uc904\uc5d0 \uc810\ub4e4\uc758 \uac1c\uc218 N(1&le;N&le;500)\uc774 \uc8fc\uc5b4\uc9c4\ub2e4. \uadf8\ub9ac\uace0 \ub450 \ubc88\uc9f8 \uc904\ubd80\ud130 N+1\ubc88\uc9f8 \uc904\uae4c\uc9c0 \uc810\ub4e4\uc758 \uc5f0\uacb0 \uc0c1\ud0dc\uac00 \uc8fc\uc5b4\uc9c4\ub2e4. \uac01 \uc815\ubcf4\ub294 \ub450 \uac1c\uc758 \uc815\uc218\ub85c \uc774\ub8e8\uc5b4 \uc838 \uc788\ub294\ub370 i\ubc88\uc9f8 \uc904\uc5d0 a\uc640 b \ub450 \uc815\uc218\uac00 \uc8fc\uc5b4\uc838 \uc788\ub2e4\uba74 i\ubc88 \uc810\uacfc a, b\uac00 \uc5f0\uacb0\uc774 \ub418\uc5b4 \uc788\ub2e4\ub294 \uac83\uc744 \uc758\ubbf8\ud55c\ub2e4.<\/p>","output":"<p>\uccab \uc904\uc5d0 \ucd5c\uc18c \uc774\ub3d9 \ud68c\uc218\ub97c \ucd9c\ub825\ud55c\ub2e4. \ub9cc\uc57d\uc5d0 \ubcfc\ub85d N\uac01\ud615\uc744 \ub9cc\ub4dc\ub294 \uac83\uc774 \ubd88\uac00\ub2a5 \ud55c \uacbd\uc6b0\uc5d0\ub294 -1\uc744 \ucd9c\ub825\ud55c\ub2e4.<\/p>","hint":"","original":"0","problem_lang_code":"\ud55c\uad6d\uc5b4"},{"problem_id":"1847","problem_lang":"1","title":"Human Knot","description":"<p>A classic ice&minus;breaking exercise is for a group of n people to form a circle and then arbitrarily join hands with one another. This forms a &quot;human knot&quot; since the players&#39; arms are most likely intertwined. The goal is then to unwind the knot to form a circle of players with no arms crossed.<\/p>\r\n\r\n<p>We now adapt this game to a more general and more abstract setting where the physical constraints of the problem are gone. Suppose we represent the initial knot with a 2&minus;regular graph inscribed in a circle (i.e., we have a graph with n vertices with exactly two edges incident on each vertex). Initially, some edges may cross other edges and this is undesirable. This is the &quot;knot&quot; we wish to unwind.<\/p>\r\n\r\n<p>A &quot;move&quot; involves moving any vertex to a new position on the circle, keeping its edges intact. Our goal is to make the fewest possible moves such that we obtain one n&minus;sided polygon with no edge&minus;crossings remaining.<\/p>\r\n\r\n<p>For example, here is a knot on 4 vertices inscribed in a circle, but two edges cross each other. By moving vertex 4 down to the position between 2 and 3, a graph without edge&minus;crossings emerges. This was achieved in a single move, which is clearly optimal in this case.<\/p>\r\n\r\n<p><img alt=\"\" src=\"\/upload\/images\/knot1.png\" style=\"height:171px; width:402px\" \/><\/p>\r\n\r\n<p>When n is larger, things may not be quite as clear. Below we see a knot on 6 vertices. We might consider moving vertex 4 between 5 and 6, then vertex 5 between 1 and 2, and finally vertex 6 between 3 and 4; this unwinds the knot in 3 moves.<\/p>\r\n\r\n<p><img alt=\"\" src=\"\/upload\/images\/knot2.png\" style=\"height:190px; width:738px\" \/><\/p>\r\n\r\n<p>But clearly we can unwind the same knot in only two moves:<\/p>\r\n\r\n<p><img alt=\"\" src=\"\/upload\/images\/knot3.png\" style=\"height:197px; width:605px\" \/><\/p>\r\n","input":"<p>The input consists of a number of cases. Each case starts with a line containing the integer n (3 &lt;= n &lt;= 500), giving the number of vertices of the graph. The vertices are labelled clockwise from 1 to n. Each of the next n lines gives a pair of neighbors, where line i (1 &lt;= i &lt;= n) specifies the two vertices adjacent to vertex i. The input is terminated by n = 0.<\/p>\r\n","output":"<p>For each case, if there is no solution, print &quot;Not solvable.&quot; on a line by itself. If there is a solution, print &quot;Knot solvable.&quot; on a line by itself, followed by the minimum number of moves required to solve the problem, on a line by itself.<\/p>\r\n","hint":"","original":"1","problem_lang_code":"\uc601\uc5b4"}]