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문제

  시작 곡선 변형 1 변형 2 변형 3 변형 4
H1
H2
H3
H4

힐베르트 곡선은 독일 수학자 다비드 힐베르트가 처음으로 묘사한 연속 프랙탈 공간 채움 곡선이다.

힐베르트 곡선은 가로줄과 세로줄로 이루어져 있는 곡선의 수열 H1, H2, H3, H4, ...을 이용해서 만들 수 있다. 각각의 곡선은 단위 정사각형 [0, 1] × [0, 1]위에 있다.

H1은 (¼, ¾), (¼, ¼), (¾, ¼), (¾, ¾)을 잇는 세 선분으로 이루어져 있다. Hn은 Hn-1을 이용해서 다음 4가지 과정을 통해서 재귀적으로 만들 수 있다.

1. Hn-1의 모든 좌표를 절반으로 줄인다.

2. 곡선을 (0, ½)을 기준으로 반시계 방향으로 돌린 곡선을 더한다.

3. x = ½를 기준으로 대칭시킨 곡선을 더한다.

4. m = ½n+1이라 하자. 각 곡선의 끝점 (½ - m, ½ - m) 와 (½ + m, ½ - m), (m, ½ - m) 와 (m, ½ + m), (1 - m, ½ - m) 와 (1 - m, ½ + m)을 잇는다.

수평선이 주어졌을 때, 곡선과 교차하는 점의 수를 구하는 프로그램을 작성하시오. 예를 들어, 입력 예제 첫번째와 두번째 그림은 다음과 같다.

그림 1 그림 2
H3과 수평선(2/8,7/8)-(7/8,7/8)은 3번 교차한다 H4과 수평선(0/16,1/16)-(16/16,1/16)은 16번 교차한다

Hn의 꼭지점 좌표는 항상 1/2n+1의 홀수 배수이다. 수평선의 끝점 좌표는 항상 1/2n의 배수이다. 따라서, 선분은 항상 Hn의 세로선과 교차한다.

입력

입력은 여러 개의 데이터로 이루어져 있다. 주어지는 데이터는 최대 100개이다. 각 데이터는 다음과 같은 Hn과 수평선(x1/2n, y/2n)-(x2/2n, y/2n)을 나타내는 4개의 수(n x1 x2 y)가 공백으로 구분되어져 있다. (0 < n < 31, x1 < x2) x1, x2, y는 항상 [0, 2n] 범위이다.

마지막 데이터의 다음 줄에는 0이 하나 주어진다.

출력

각 데이터에 대해서 Hn과 주어진 수평선과 교차하는 점의 개수를 출력한다.

예제 입력 1

3 2 7 7
4 0 16 1
30 1 1073741823 1
0

예제 출력 1

3
16
1073741822
[{"problem_id":"4645","problem_lang":"0","title":"\ud790\ubca0\ub974\ud2b8 \uace1\uc120","description":"<table class=\"table\" style=\"width:670px;\">\r\n\t<thead>\r\n\t\t<tr>\r\n\t\t\t<th style=\"width:20px;text-align:center;\">\r\n\t\t\t\t&nbsp;<\/th>\r\n\t\t\t<th style=\"width:130px;text-align:center;\">\r\n\t\t\t\t\uc2dc\uc791 \uace1\uc120<\/th>\r\n\t\t\t<th style=\"width:130px;text-align:center;\">\r\n\t\t\t\t\ubcc0\ud615 1<\/th>\r\n\t\t\t<th style=\"width:130px;text-align:center;\">\r\n\t\t\t\t\ubcc0\ud615 2<\/th>\r\n\t\t\t<th style=\"width:130px;text-align:center;\">\r\n\t\t\t\t\ubcc0\ud615 3<\/th>\r\n\t\t\t<th style=\"width:130px;text-align:center;\">\r\n\t\t\t\t\ubcc0\ud615 4<\/th>\r\n\t\t<\/tr>\r\n\t<\/thead>\r\n\t<thead>\r\n\t<\/thead>\r\n\t<tbody>\r\n\t\t<tr>\r\n\t\t\t<td style=\"vertical-align: middle;\">\r\n\t\t\t\tH<sub>1<\/sub><\/td>\r\n\t\t\t<td>\r\n\t\t\t\t<img src=\"\/upload\/images\/h1.gif\" \/><\/td>\r\n\t\t\t<td>\r\n\t\t\t\t<img src=\"\/upload\/images\/h21.gif\" \/><\/td>\r\n\t\t\t<td>\r\n\t\t\t\t<img src=\"\/upload\/images\/h22.gif\" \/><\/td>\r\n\t\t\t<td>\r\n\t\t\t\t<img src=\"\/upload\/images\/h23.gif\" \/><\/td>\r\n\t\t\t<td>\r\n\t\t\t\t<img src=\"\/upload\/images\/h24.gif\" \/><\/td>\r\n\t\t<\/tr>\r\n\t\t<tr>\r\n\t\t\t<td style=\"vertical-align: middle;\">\r\n\t\t\t\tH<sub>2<\/sub><\/td>\r\n\t\t\t<td>\r\n\t\t\t\t<img src=\"\/upload\/images\/h2.gif\" \/><\/td>\r\n\t\t\t<td>\r\n\t\t\t\t<img src=\"\/upload\/images\/h31.gif\" \/><\/td>\r\n\t\t\t<td>\r\n\t\t\t\t<img src=\"\/upload\/images\/h32.gif\" \/><\/td>\r\n\t\t\t<td>\r\n\t\t\t\t<img src=\"\/upload\/images\/h33.gif\" \/><\/td>\r\n\t\t\t<td>\r\n\t\t\t\t<img src=\"\/upload\/images\/h34.gif\" \/><\/td>\r\n\t\t<\/tr>\r\n\t\t<tr>\r\n\t\t\t<td style=\"vertical-align: middle;\">\r\n\t\t\t\tH<sub>3<\/sub><\/td>\r\n\t\t\t<td>\r\n\t\t\t\t<img src=\"\/upload\/images\/h3.gif\" \/><\/td>\r\n\t\t\t<td>\r\n\t\t\t\t<img src=\"\/upload\/images\/h41.gif\" \/><\/td>\r\n\t\t\t<td>\r\n\t\t\t\t<img src=\"\/upload\/images\/h42.gif\" \/><\/td>\r\n\t\t\t<td>\r\n\t\t\t\t<img src=\"\/upload\/images\/h43.gif\" \/><\/td>\r\n\t\t\t<td>\r\n\t\t\t\t<img src=\"\/upload\/images\/h44.gif\" \/><\/td>\r\n\t\t<\/tr>\r\n\t\t<tr>\r\n\t\t\t<td style=\"vertical-align: middle;\">\r\n\t\t\t\tH<sub>4<\/sub><\/td>\r\n\t\t\t<td>\r\n\t\t\t\t<img src=\"\/upload\/images\/h4.gif\" \/><\/td>\r\n\t\t\t<td>\r\n\t\t\t\t<img src=\"\/upload\/images\/h51.gif\" \/><\/td>\r\n\t\t\t<td>\r\n\t\t\t\t<img src=\"\/upload\/images\/h52.gif\" \/><\/td>\r\n\t\t\t<td>\r\n\t\t\t\t<img src=\"\/upload\/images\/h53.gif\" \/><\/td>\r\n\t\t\t<td>\r\n\t\t\t\t<img src=\"\/upload\/images\/h54.gif\" \/><\/td>\r\n\t\t<\/tr>\r\n\t<\/tbody>\r\n<\/table>\r\n\r\n<p>\r\n\t\ud790\ubca0\ub974\ud2b8 \uace1\uc120\uc740 \ub3c5\uc77c \uc218\ud559\uc790 \ub2e4\ube44\ub4dc \ud790\ubca0\ub974\ud2b8\uac00 \ucc98\uc74c\uc73c\ub85c \ubb18\uc0ac\ud55c \uc5f0\uc18d \ud504\ub799\ud0c8 \uacf5\uac04 \ucc44\uc6c0 \uace1\uc120\uc774\ub2e4.<\/p>\r\n\r\n<p>\r\n\t\ud790\ubca0\ub974\ud2b8 \uace1\uc120\uc740 \uac00\ub85c\uc904\uacfc \uc138\ub85c\uc904\ub85c \uc774\ub8e8\uc5b4\uc838 \uc788\ub294 \uace1\uc120\uc758 \uc218\uc5f4 H<sub>1<\/sub>, H<sub>2<\/sub>, H<sub>3<\/sub>, H<sub>4<\/sub>, ...\uc744 \uc774\uc6a9\ud574\uc11c \ub9cc\ub4e4 \uc218 \uc788\ub2e4. \uac01\uac01\uc758 \uace1\uc120\uc740 \ub2e8\uc704 \uc815\uc0ac\uac01\ud615 [0, 1] &times; [0, 1]\uc704\uc5d0 \uc788\ub2e4.<\/p>\r\n\r\n<p>\r\n\tH<sub>1<\/sub>\uc740 (&frac14;, &frac34;), (&frac14;, &frac14;), (&frac34;, &frac14;), (&frac34;, &frac34;)\uc744 \uc787\ub294 \uc138 \uc120\ubd84\uc73c\ub85c \uc774\ub8e8\uc5b4\uc838 \uc788\ub2e4. H<sub>n<\/sub>\uc740 H<sub>n-1<\/sub>\uc744 \uc774\uc6a9\ud574\uc11c \ub2e4\uc74c 4\uac00\uc9c0 \uacfc\uc815\uc744 \ud1b5\ud574\uc11c \uc7ac\uadc0\uc801\uc73c\ub85c \ub9cc\ub4e4 \uc218 \uc788\ub2e4.<\/p>\r\n\r\n<p>\r\n\t1. H<sub>n-1<\/sub>\uc758 \ubaa8\ub4e0 \uc88c\ud45c\ub97c \uc808\ubc18\uc73c\ub85c \uc904\uc778\ub2e4.<\/p>\r\n<p>\r\n\t2. \uace1\uc120\uc744 (0, &frac12;)\uc744 \uae30\uc900\uc73c\ub85c \ubc18\uc2dc\uacc4 \ubc29\ud5a5\uc73c\ub85c \ub3cc\ub9b0 \uace1\uc120\uc744 \ub354\ud55c\ub2e4.<\/p>\r\n<p>\r\n\t3. x = &frac12;\ub97c \uae30\uc900\uc73c\ub85c \ub300\uce6d\uc2dc\ud0a8 \uace1\uc120\uc744 \ub354\ud55c\ub2e4.<\/p>\r\n<p>\r\n\t4. m = &frac12;<sup>n+1<\/sup>\uc774\ub77c \ud558\uc790. \uac01 \uace1\uc120\uc758 \ub05d\uc810 (&frac12; - m, &frac12; - m) \uc640 (&frac12; + m, &frac12; - m), (m, &frac12; - m) \uc640 (m, &frac12; + m), (1 - m, &frac12; - m) \uc640 (1 - m, &frac12; + m)\uc744 \uc787\ub294\ub2e4.<\/p>\r\n\r\n<p>\r\n\t\uc218\ud3c9\uc120\uc774 \uc8fc\uc5b4\uc84c\uc744 \ub54c, \uace1\uc120\uacfc \uad50\ucc28\ud558\ub294 \uc810\uc758 \uc218\ub97c \uad6c\ud558\ub294 \ud504\ub85c\uadf8\ub7a8\uc744 \uc791\uc131\ud558\uc2dc\uc624. \uc608\ub97c \ub4e4\uc5b4, \uc785\ub825 \uc608\uc81c \uccab\ubc88\uc9f8\uc640 \ub450\ubc88\uc9f8 \uadf8\ub9bc\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\r\n\r\n<table class=\"table\" style=\"width:70%;\">\r\n\t<thead>\r\n\t\t<tr>\r\n\t\t\t<th style=\"width:35%;text-align:center;\">\r\n\t\t\t\t\uadf8\ub9bc 1<\/th>\r\n\t\t\t<th style=\"width:35%;text-align:center;\">\r\n\t\t\t\t\uadf8\ub9bc 2<\/th>\r\n\t\t<\/tr>\r\n\t<\/thead>\r\n\t<tbody>\r\n\t\t<tr>\r\n\t\t\t<td style=\"text-align:center;\">\r\n\t\t\t\t<img src=\"\/upload\/images\/h3line.gif\" \/><\/td>\r\n\t\t\t<td style=\"text-align:center;\">\r\n\t\t\t\t<img src=\"\/upload\/images\/h4line.gif\" \/><\/td>\r\n\t\t<\/tr>\r\n\t\t<tr>\r\n\t\t\t<td style=\"text-align:center;\">\r\n\t\t\t\tH<sub>3<\/sub>\uacfc \uc218\ud3c9\uc120(2\/8,7\/8)-(7\/8,7\/8)\uc740 3\ubc88 \uad50\ucc28\ud55c\ub2e4<\/td>\r\n\t\t\t<td style=\"text-align:center;\">\r\n\t\t\t\tH<sub>4<\/sub>\uacfc \uc218\ud3c9\uc120(0\/16,1\/16)-(16\/16,1\/16)\uc740 16\ubc88 \uad50\ucc28\ud55c\ub2e4<\/td>\r\n\t\t<\/tr>\r\n\t<\/tbody>\r\n<\/table>\r\n\r\n<p>\r\n\tH<sub>n<\/sub>\uc758 \uaf2d\uc9c0\uc810 \uc88c\ud45c\ub294 \ud56d\uc0c1 1\/2<sup>n+1<\/sup>\uc758 \ud640\uc218 \ubc30\uc218\uc774\ub2e4. \uc218\ud3c9\uc120\uc758 \ub05d\uc810 \uc88c\ud45c\ub294 \ud56d\uc0c1 1\/2<sup>n<\/sup>\uc758 \ubc30\uc218\uc774\ub2e4. \ub530\ub77c\uc11c, \uc120\ubd84\uc740 \ud56d\uc0c1 H<sub>n<\/sub>\uc758 \uc138\ub85c\uc120\uacfc \uad50\ucc28\ud55c\ub2e4.<\/p>\r\n","input":"\r\n<p>\r\n\t\uc785\ub825\uc740 \uc5ec\ub7ec \uac1c\uc758 \ub370\uc774\ud130\ub85c \uc774\ub8e8\uc5b4\uc838 \uc788\ub2e4. \uc8fc\uc5b4\uc9c0\ub294 \ub370\uc774\ud130\ub294 \ucd5c\ub300 100\uac1c\uc774\ub2e4. \uac01 \ub370\uc774\ud130\ub294 \ub2e4\uc74c\uacfc \uac19\uc740 H<sub>n<\/sub>\uacfc \uc218\ud3c9\uc120(x<sub>1<\/sub>\/2<sup>n<\/sup>, y\/2<sup>n<\/sup>)-(x<sub>2<\/sub>\/2<sup>n<\/sup>, y\/2<sup>n<\/sup>)\uc744 \ub098\ud0c0\ub0b4\ub294 4\uac1c\uc758 \uc218(n x<sub>1<\/sub> x<sub>2<\/sub> y)\uac00 \uacf5\ubc31\uc73c\ub85c \uad6c\ubd84\ub418\uc5b4\uc838 \uc788\ub2e4. (0 &lt; n &lt; 31, x<sub>1<\/sub> &lt; x<sub>2<\/sub>) x<sub>1<\/sub>, x<sub>2<\/sub>, y\ub294 \ud56d\uc0c1 [0, 2<sup>n<\/sup>] \ubc94\uc704\uc774\ub2e4.<\/p>\r\n\r\n<p>\r\n\t\ub9c8\uc9c0\ub9c9 \ub370\uc774\ud130\uc758 \ub2e4\uc74c \uc904\uc5d0\ub294 0\uc774 \ud558\ub098 \uc8fc\uc5b4\uc9c4\ub2e4.<\/p>\r\n","output":"<p>\r\n\t\uac01 \ub370\uc774\ud130\uc5d0 \ub300\ud574\uc11c H<sub>n<\/sub>\uacfc \uc8fc\uc5b4\uc9c4 \uc218\ud3c9\uc120\uacfc \uad50\ucc28\ud558\ub294 \uc810\uc758 \uac1c\uc218\ub97c \ucd9c\ub825\ud55c\ub2e4.<\/p>\r\n","hint":"","original":"0","problem_lang_code":"\ud55c\uad6d\uc5b4"},{"problem_id":"4645","problem_lang":"1","title":"Hilbert Curve Intersections","description":"<table class=\"table\" style=\"width:670px\">\r\n\t<thead>\r\n\t\t<tr>\r\n\t\t\t<th style=\"text-align:center; width:20px\">&nbsp;<\/th>\r\n\t\t\t<th style=\"text-align:center; width:130px\">Starting curve<\/th>\r\n\t\t\t<th style=\"text-align:center; width:130px\">Transformation 1<\/th>\r\n\t\t\t<th style=\"text-align:center; width:130px\">Transformation 2<\/th>\r\n\t\t\t<th style=\"text-align:center; width:130px\">Transformation 3<\/th>\r\n\t\t\t<th style=\"text-align:center; width:130px\">Transformation 4<\/th>\r\n\t\t<\/tr>\r\n\t<\/thead>\r\n\t<thead>\r\n\t<\/thead>\r\n\t<tbody>\r\n\t\t<tr>\r\n\t\t\t<td style=\"vertical-align:middle\">H<sub>1<\/sub><\/td>\r\n\t\t\t<td><img src=\"\/upload\/images\/h1.gif\" \/><\/td>\r\n\t\t\t<td><img src=\"\/upload\/images\/h21.gif\" \/><\/td>\r\n\t\t\t<td><img src=\"\/upload\/images\/h22.gif\" \/><\/td>\r\n\t\t\t<td><img src=\"\/upload\/images\/h23.gif\" \/><\/td>\r\n\t\t\t<td><img src=\"\/upload\/images\/h24.gif\" \/><\/td>\r\n\t\t<\/tr>\r\n\t\t<tr>\r\n\t\t\t<td style=\"vertical-align:middle\">H<sub>2<\/sub><\/td>\r\n\t\t\t<td><img src=\"\/upload\/images\/h2.gif\" \/><\/td>\r\n\t\t\t<td><img src=\"\/upload\/images\/h31.gif\" \/><\/td>\r\n\t\t\t<td><img src=\"\/upload\/images\/h32.gif\" \/><\/td>\r\n\t\t\t<td><img src=\"\/upload\/images\/h33.gif\" \/><\/td>\r\n\t\t\t<td><img src=\"\/upload\/images\/h34.gif\" \/><\/td>\r\n\t\t<\/tr>\r\n\t\t<tr>\r\n\t\t\t<td style=\"vertical-align:middle\">H<sub>3<\/sub><\/td>\r\n\t\t\t<td><img src=\"\/upload\/images\/h3.gif\" \/><\/td>\r\n\t\t\t<td><img src=\"\/upload\/images\/h41.gif\" \/><\/td>\r\n\t\t\t<td><img src=\"\/upload\/images\/h42.gif\" \/><\/td>\r\n\t\t\t<td><img src=\"\/upload\/images\/h43.gif\" \/><\/td>\r\n\t\t\t<td><img src=\"\/upload\/images\/h44.gif\" \/><\/td>\r\n\t\t<\/tr>\r\n\t\t<tr>\r\n\t\t\t<td style=\"vertical-align:middle\">H<sub>4<\/sub><\/td>\r\n\t\t\t<td><img src=\"\/upload\/images\/h4.gif\" \/><\/td>\r\n\t\t\t<td><img src=\"\/upload\/images\/h51.gif\" \/><\/td>\r\n\t\t\t<td><img src=\"\/upload\/images\/h52.gif\" \/><\/td>\r\n\t\t\t<td><img src=\"\/upload\/images\/h53.gif\" \/><\/td>\r\n\t\t\t<td><img src=\"\/upload\/images\/h54.gif\" \/><\/td>\r\n\t\t<\/tr>\r\n\t<\/tbody>\r\n<\/table>\r\n\r\n<p>David Hilbert proved the existence of a very counter-intuitive curve that fills space. The construction of the Hilbert curve is based on a sequence of curves, H<sub>1<\/sub>, H<sub>2<\/sub>, H<sub>3<\/sub>, H<sub>4<\/sub>, ... composed of horizontal and vertical segments. Each curve lies in the unit square [0, 1] &times; [0, 1]. H<sub>1<\/sub> contains just three segments, connecting the points (&frac14;, &frac34;) to (&frac14;, &frac14;) to (&frac34;, &frac14;) to (&frac34;, &frac34;). Hn is defined recursively in terms of Hn-1, for n = 2, 3, ... by four transformations:<\/p>\r\n\r\n<ul>\r\n\t<li>Halve all coordinates in H<sub>n-1<\/sub>.<\/li>\r\n\t<li>Add a copy rotated 90 degrees counterclockwise about the point (0, &frac12;).<\/li>\r\n\t<li>Add the reflection across the line x = &frac12;.<\/li>\r\n\t<li>Let m = &frac12;<sup>n+1<\/sup>. Add segments connecting endpoints (&frac12; - m, &frac12; - m) to (&frac12; + m, &frac12; - m), (m, &frac12; - m) to (m, &frac12; + m), and (1 - m, &frac12; - m) to (1 - m, &frac12; + m).<\/li>\r\n<\/ul>\r\n\r\n<p>Your job is to count the number of intersections of horizontal line segments with these curves. For example, consider Figures 1 and 2, which illustrate the first two example input data sets below.<\/p>\r\n\r\n<table class=\"table\" style=\"width:70%\">\r\n\t<thead>\r\n\t\t<tr>\r\n\t\t\t<th style=\"text-align:center; width:35%\">Figure 1<\/th>\r\n\t\t\t<th style=\"text-align:center; width:35%\">Figure 2<\/th>\r\n\t\t<\/tr>\r\n\t<\/thead>\r\n\t<tbody>\r\n\t\t<tr>\r\n\t\t\t<td style=\"text-align:center\"><img src=\"\/upload\/images\/h3line.gif\" \/><\/td>\r\n\t\t\t<td style=\"text-align:center\"><img src=\"\/upload\/images\/h4line.gif\" \/><\/td>\r\n\t\t<\/tr>\r\n\t\t<tr>\r\n\t\t\t<td style=\"text-align:center\">Segment from (2\/8, 7\/8) to (7\/8, 7\/8) crossing H<sub>3<\/sub> three times.<\/td>\r\n\t\t\t<td style=\"text-align:center\">Segment from (0\/16, 1\/16) to (16\/16, 1\/16) crossing H<sub>4<\/sub> sixteen times.<\/td>\r\n\t\t<\/tr>\r\n\t<\/tbody>\r\n<\/table>\r\n\r\n<p>The coordinates of vertices of Hn are odd multiples of &frac12;<sup>n+1<\/sup>. The coordinates of horizontal segment endpoints will always be multiples of &frac12;<sup>n<\/sup>. Hence the specified horizontal segment can only cross vertical segments in H<sub>n<\/sub>.<\/p>\r\n","input":"<p>Input consists of one to 100 data sets, one per line, followed by a final line containing only 0. Each data set consists of four integers separated by blanks in the form<\/p>\r\n\r\n<pre>\r\nn x<sub>1<\/sub> x<sub>2<\/sub> y<\/pre>\r\n\r\n<p>which represents H<sub>n<\/sub> and the segment from (x<sub>1<\/sub>\/2<sup>n<\/sup>, y\/2<sup>n<\/sup>) to (x<sub>2<\/sub>\/2<sup>n<\/sup>, y\/2<sup>n<\/sup>), where 0 &lt; n &lt; 31, x<sub>1<\/sub> &lt; x<sub>2<\/sub>, and each of x<sub>1<\/sub>, x<sub>2<\/sub>, and y lie in the range 0 to 2<sup>n<\/sup>, inclusive.<\/p>\r\n","output":"<p>The output is one integer per line for each data set: the number of intersections of H<sub>n<\/sub> with the segment.&nbsp;<\/p>\r\n\r\n<p>Caution: A brute force solution that computes each intersection individually will not finish within the one minute time limit. As you can see below, there may be more than one billion intersections for any data set<\/p>\r\n","hint":"","original":"1","problem_lang_code":"\uc601\uc5b4"}]