시간 제한 메모리 제한 제출 정답 맞은 사람 정답 비율
1 초 128 MB 67 20 19 31.667%

문제

양의 정수를 하나 고른 뒤, 각 자리수의 제곱의 합을 구한다. 이것을 계속 반복하면, 흥미로운 특징을 가지는 무한 수열을 얻을 수 있다.

가장 처음에 5로 시작했을 때, 다음과 같은 수열을 얻을 수 있다.

5, 25, 29, 85, 89, 145, 42, 20, 4, 16, 37, 58, ...

여기서 가장 흥미로운 특징은 58다음에 나타난다. 52 + 82 = 89로 이 숫자는 이미 수열에 나왔단 수이다. 즉, 58 다음부터 수열은 다음과 같은 구간이 반복해서 나온다. 89, 145, 42, 20, 4, 16, 37, 58

위의 사이클은 다른 숫자로 시작해도 나타나는 사이클이다. (3, 18, 36, 64, 등등등)

어떤 숫자는 1로 반복되는 사이클이 나타나기도 한다. 예를 들어 19로 시작했을 때를 살펴보자.

19, 82, 68, 100, 1, ...

두 숫자가 주어졌을 때, 같은 수가 나올 때 까지 필요한 수열의 길이의 합의 최솟값을 구하는 프로그램을 작성하시오.

예를 들어, 61과 29로 시작하면, (61, 37, 58, 89)와 (29, 85, 89)에서 같은 수를 만들 수 있다. 19와 100으로 시작하는 경우에는 (19, 82, 68, 100), (100)으로 같은 수를 만들 수 있다.

입력

입력을 여러 개의 테스트 케이스로 이루어져 있다. 각 테스트 케이스는 한 줄로 이루어져 있고, A와 B가 주어진다. (0 < A, B < 109)

마지막 줄에는 0이 두 개 주어진다.

출력

각 테스트 케이스에 대해서, A와 B를 출력하고, 두 수열의 길이의 합의 최솟값을 출력한다. 만약, 두 수열에서 같은 수가 나타나지 않는다면, 0을 출력한다.

예제 입력 1

89 89
19 100
61 19
0 0

예제 출력 1

89 89 2
19 100 5
61 19 0

힌트

89, 145, 42, 20, 4, 16, 37, 58 사이클

1 사이클

[{"problem_id":"4881","problem_lang":"0","title":"\uc790\ub9ac\uc218\uc758 \uc81c\uacf1","description":"<p>\uc591\uc758 \uc815\uc218\ub97c \ud558\ub098 \uace0\ub978 \ub4a4, \uac01 \uc790\ub9ac\uc218\uc758 \uc81c\uacf1\uc758 \ud569\uc744 \uad6c\ud55c\ub2e4. \uc774\uac83\uc744 \uacc4\uc18d \ubc18\ubcf5\ud558\uba74, \ud765\ubbf8\ub85c\uc6b4 \ud2b9\uc9d5\uc744 \uac00\uc9c0\ub294 \ubb34\ud55c \uc218\uc5f4\uc744 \uc5bb\uc744 \uc218 \uc788\ub2e4.<\/p>\r\n\r\n<p>\uac00\uc7a5 \ucc98\uc74c\uc5d0 5\ub85c \uc2dc\uc791\ud588\uc744 \ub54c, \ub2e4\uc74c\uacfc \uac19\uc740 \uc218\uc5f4\uc744 \uc5bb\uc744 \uc218 \uc788\ub2e4.<\/p>\r\n\r\n<p>5, 25, 29, 85, 89, 145, 42, 20, 4, 16, 37, 58, ...<\/p>\r\n\r\n<p>\uc5ec\uae30\uc11c \uac00\uc7a5 \ud765\ubbf8\ub85c\uc6b4 \ud2b9\uc9d5\uc740 58\ub2e4\uc74c\uc5d0 \ub098\ud0c0\ub09c\ub2e4. 5<sup>2<\/sup> + 8<sup>2<\/sup> = 89\ub85c \uc774 \uc22b\uc790\ub294 \uc774\ubbf8 \uc218\uc5f4\uc5d0 \ub098\uc654\ub2e8 \uc218\uc774\ub2e4. \uc989, 58 \ub2e4\uc74c\ubd80\ud130 \uc218\uc5f4\uc740 \ub2e4\uc74c\uacfc \uac19\uc740 \uad6c\uac04\uc774 \ubc18\ubcf5\ud574\uc11c \ub098\uc628\ub2e4. 89, 145, 42, 20, 4, 16, 37, 58<\/p>\r\n\r\n<p>\uc704\uc758 \uc0ac\uc774\ud074\uc740 \ub2e4\ub978 \uc22b\uc790\ub85c \uc2dc\uc791\ud574\ub3c4 \ub098\ud0c0\ub098\ub294 \uc0ac\uc774\ud074\uc774\ub2e4. (3, 18, 36, 64, \ub4f1\ub4f1\ub4f1)<\/p>\r\n\r\n<p>\uc5b4\ub5a4 \uc22b\uc790\ub294 1\ub85c \ubc18\ubcf5\ub418\ub294 \uc0ac\uc774\ud074\uc774 \ub098\ud0c0\ub098\uae30\ub3c4 \ud55c\ub2e4. \uc608\ub97c \ub4e4\uc5b4 19\ub85c \uc2dc\uc791\ud588\uc744 \ub54c\ub97c \uc0b4\ud3b4\ubcf4\uc790.<\/p>\r\n\r\n<p>19, 82, 68, 100, 1, ...<\/p>\r\n\r\n<p>\ub450 \uc22b\uc790\uac00 \uc8fc\uc5b4\uc84c\uc744 \ub54c, \uac19\uc740 \uc218\uac00 \ub098\uc62c \ub54c \uae4c\uc9c0 \ud544\uc694\ud55c \uc218\uc5f4\uc758 \uae38\uc774\uc758 \ud569\uc758 \ucd5c\uc19f\uac12\uc744 \uad6c\ud558\ub294 \ud504\ub85c\uadf8\ub7a8\uc744 \uc791\uc131\ud558\uc2dc\uc624.<\/p>\r\n\r\n<p>\uc608\ub97c \ub4e4\uc5b4, 61\uacfc 29\ub85c \uc2dc\uc791\ud558\uba74, (61, 37, 58, <strong>89<\/strong>)\uc640 (29, 85, <strong>89<\/strong>)\uc5d0\uc11c \uac19\uc740 \uc218\ub97c \ub9cc\ub4e4 \uc218 \uc788\ub2e4. 19\uc640 100\uc73c\ub85c \uc2dc\uc791\ud558\ub294 \uacbd\uc6b0\uc5d0\ub294 (19, 82, 68, <strong>100<\/strong>), (<strong>100<\/strong>)\uc73c\ub85c \uac19\uc740 \uc218\ub97c \ub9cc\ub4e4 \uc218 \uc788\ub2e4.<\/p>\r\n","input":"<p>\uc785\ub825\uc744 \uc5ec\ub7ec \uac1c\uc758 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\ub85c \uc774\ub8e8\uc5b4\uc838 \uc788\ub2e4. \uac01 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\ub294 \ud55c \uc904\ub85c \uc774\ub8e8\uc5b4\uc838 \uc788\uace0, A\uc640 B\uac00 \uc8fc\uc5b4\uc9c4\ub2e4. (0 &lt; A, B &lt; 10<sup>9<\/sup>)<\/p>\r\n\r\n<p>\ub9c8\uc9c0\ub9c9 \uc904\uc5d0\ub294 0\uc774 \ub450 \uac1c \uc8fc\uc5b4\uc9c4\ub2e4.<\/p>\r\n","output":"<p>\uac01 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\uc5d0 \ub300\ud574\uc11c, A\uc640 B\ub97c \ucd9c\ub825\ud558\uace0, \ub450 \uc218\uc5f4\uc758 \uae38\uc774\uc758 \ud569\uc758 \ucd5c\uc19f\uac12\uc744 \ucd9c\ub825\ud55c\ub2e4. \ub9cc\uc57d, \ub450 \uc218\uc5f4\uc5d0\uc11c \uac19\uc740 \uc218\uac00 \ub098\ud0c0\ub098\uc9c0 \uc54a\ub294\ub2e4\uba74, 0\uc744 \ucd9c\ub825\ud55c\ub2e4.<\/p>\r\n","hint":"<p>89, 145, 42, 20, 4, 16, 37, 58 \uc0ac\uc774\ud074<\/p>\r\n\r\n<p><img alt=\"\" src=\"\/upload\/images\/cycle1.png\" style=\"height:384px; width:389px\" \/><\/p>\r\n\r\n<p>1 \uc0ac\uc774\ud074<\/p>\r\n\r\n<p><img alt=\"\" src=\"\/upload\/images\/cycle2.png\" style=\"height:305px; width:288px\" \/><\/p>\r\n","original":"0","problem_lang_code":"\ud55c\uad6d\uc5b4"},{"problem_id":"4881","problem_lang":"1","title":"Sum the Square","description":"<p>Take any positive number, \ufb01nd the sum of the squares of its digits, repeat! You&rsquo;ll end up with an in\ufb01nite sequence with an interesting property that we would like to investigate further. Starting with the number 5, the sequence is:<\/p>\r\n\r\n<p>(5,25,29,85,89,145,42,20,4,16,37,58,...)<\/p>\r\n\r\n<p>The interesting part is in what comes after 58: 5<sup>2<\/sup> + 8<sup>2<\/sup> = 89 which is a number that&rsquo;s already been seen in the sequence. In other words, after 58, the sequence will fall into the repeating cycle: 89,145,42,20,4,16,37,58.<\/p>\r\n\r\n<p>What&rsquo;s amazing is that this cycle will appear for many other numbers: 3, 18, 36, and 64 just to name a few. (see \ufb01gure on the following page.)<\/p>\r\n\r\n<p>For some numbers, the sequence will fall into another repeating cycle by reaching 1. (see second \ufb01gure on the following page) For example, starting with 19, you&rsquo;ll end up with the sequence:<\/p>\r\n\r\n<p>(19,82,68,100,1,...)<\/p>\r\n\r\n<p>And that&rsquo;s about it. Any number you choose will end up falling into a repeating cycle: Either the 89,145,... cycle or the 1,... cycle.<\/p>\r\n\r\n<p>Given two numbers, your objective is to generate as few numbers in their sequences for the two sequences to intersect at one common number. For example, given 61 and 29, we can achieve what&rsquo;s required by generating the sequences: (61,37,58,<strong>89<\/strong>) and (29,85,<strong>89<\/strong>). Similarly, for 19 and 100, the sequences would be (19,82,68,<strong>100<\/strong>) and (<strong>100<\/strong>).<\/p>\r\n","input":"<p>Your program will be tested on one or more test cases. Each test case is speci\ufb01ed on a single line having two integers (0 &lt; A, B &lt; 10<sup>9<\/sup>).<\/p>\r\n\r\n<p>The last case is followed by a dummy line made of two zeros.<\/p>\r\n","output":"<p>For each test case, print the following line:<\/p>\r\n\r\n<pre>\r\nA\u0002 B \u0002S<\/pre>\r\n\r\n<p>Where A, B are as in the input and S is the (minimum) sum of the lengths of the two sequences. If the sequences starting at A and B do not intersect, then S = 0.<\/p>\r\n","hint":"<p>Few numbers falling into the 89,145,42,20,4,16,37,58 cycle<\/p>\r\n\r\n<p><img alt=\"\" src=\"\/upload\/images\/cycle1.png\" style=\"height:384px; opacity:0.9; width:389px\" \/><\/p>\r\n\r\n<p>Few numbers falling into the 1 cycle.<\/p>\r\n\r\n<p><img alt=\"\" src=\"\/upload\/images\/cycle2.png\" style=\"height:305px; width:288px\" \/><\/p>\r\n","original":"1","problem_lang_code":"\uc601\uc5b4"}]