문제
다음과 같은 수열을 생각해보자: 2ドル → 12 → 1112 → 3112 → 132112 → 1113122112 → \dots$.
이 수열의 초기값은 2ドル$이고, 이후의 수들은 다음과 같이 생성된다.
- 2ドル$에는 “1ドル$개의 2ドル$”가 있으므로, 다음 값은 12ドル$이다.
- 12ドル$에는 “1ドル$개의 1ドル,ドル 1ドル$개의 2ドル$”가 있으므로 다음 값은 1112ドル$이다.
- 1112ドル$에는 “3ドル$개의 1ドル,ドル 1ドル$개의 2ドル$”가 있으므로 다음 값은 3112ドル$이다.
- 3112ドル$에는 “1ドル$개의 3ドル,ドル 2ドル$개의 1ドル,ドル 1ドル$개의 2ドル$”가 있으므로, 다음 값은 132112ドル$이다.
이 수열은 이러한 “look-and-say” 규칙으로 생성된다. 이 수열에서 다음 항을 계산하는 규칙을 양의 정수 $x > 0$에 적용해서 나오는 수를 $f(x)$라고 하자. 즉, $f(2) = 12,ドル $f(12) = 1112,ドル $f(1112) = 3112,ドル $f(3112) = 132112$와 같이 주어지는 양의 정수에 대해 이를 십진법으로 읽었을 때 연속되는 숫자의 개수를 보이는 대로 읽어서 만들어지는 수를 의미한다.
양의 정수 $x$에 연속해서 나타나는 숫자가 9ドル$회를 넘지 않을 때에만 $f(x)$가 정의되며, “look-and-say” 규칙을 적용하여 $f(x) = n$인 $n$을 얻을 수 있다. 여러분이 할 일은 입력으로 양의 정수 $n$이 주어질때, $f(x) = n$인 양의 정수 $x$를 구하는 것이다. $x$가 존재한다면 그 값은 유일하다. 주의할 점은 어떤 양의 정수 $n$은 그러한 $x$를 가지지 않는다는 것이다. 예를 들어, $f(x) = 311$인 양의 정수 $x$는 존재하지않는다. 또한 $f(x) = 1111$인 양의 정수 $x$ 역시 존재하지 않는다. 1111ドル$의 경우 “1ドル$개의 1ドル,ドル 1ドル$개의 1ドル$”로 해석하여 $x$가 11ドル$이라 생각할 수 있으나, $f(11) = 21$이다. 어떤 양의 정수도 이 “look-and-say” 규칙에 따라 1111ドル$을 생성할 수 없다.
출력
$f(x) = n$인 양의 정수 $x$가 존재한다면 그 값을 출력하고, 그렇지 않다면 $-1$을 출력한다.
[{"problem_id":"34682","problem_lang":"0","title":"Inverse Look-and-Say","description":"<p>\ub2e4\uc74c\uacfc \uac19\uc740 \uc218\uc5f4\uc744 \uc0dd\uac01\ud574\ubcf4\uc790: $2 &rarr; 12 &rarr; 1112 &rarr; 3112 &rarr; 132112 &rarr; 1113122112 &rarr; \\dots$.<\/p>\r\n\r\n<p>\uc774 \uc218\uc5f4\uc758 \ucd08\uae30\uac12\uc740 $2$\uc774\uace0, \uc774\ud6c4\uc758 \uc218\ub4e4\uc740 \ub2e4\uc74c\uacfc \uac19\uc774 \uc0dd\uc131\ub41c\ub2e4.<\/p>\r\n\r\n<ul>\r\n<li>$2$\uc5d0\ub294 &ldquo;$1$\uac1c\uc758 $2$&rdquo;\uac00 \uc788\uc73c\ubbc0\ub85c, \ub2e4\uc74c \uac12\uc740 $12$\uc774\ub2e4.<\/li>\r\n<li>$12$\uc5d0\ub294 &ldquo;$1$\uac1c\uc758 $1$, $1$\uac1c\uc758 $2$&rdquo;\uac00 \uc788\uc73c\ubbc0\ub85c \ub2e4\uc74c \uac12\uc740 $1112$\uc774\ub2e4.<\/li>\r\n<li>$1112$\uc5d0\ub294 &ldquo;$3$\uac1c\uc758 $1$, $1$\uac1c\uc758 $2$&rdquo;\uac00 \uc788\uc73c\ubbc0\ub85c \ub2e4\uc74c \uac12\uc740 $3112$\uc774\ub2e4.<\/li>\r\n<li>$3112$\uc5d0\ub294 &ldquo;$1$\uac1c\uc758 $3$, $2$\uac1c\uc758 $1$, $1$\uac1c\uc758 $2$&rdquo;\uac00 \uc788\uc73c\ubbc0\ub85c, \ub2e4\uc74c \uac12\uc740 $132112$\uc774\ub2e4.<\/li>\r\n<\/ul>\r\n\r\n<p>\uc774 \uc218\uc5f4\uc740 \uc774\ub7ec\ud55c &ldquo;look-and-say&rdquo; \uaddc\uce59\uc73c\ub85c \uc0dd\uc131\ub41c\ub2e4. \uc774 \uc218\uc5f4\uc5d0\uc11c \ub2e4\uc74c \ud56d\uc744 \uacc4\uc0b0\ud558\ub294 \uaddc\uce59\uc744 \uc591\uc758 \uc815\uc218 $x &gt; 0$\uc5d0 \uc801\uc6a9\ud574\uc11c \ub098\uc624\ub294 \uc218\ub97c $f(x)$\ub77c\uace0 \ud558\uc790. \uc989, $f(2) = 12$, $f(12) = 1112$, $f(1112) = 3112$, $f(3112) = 132112$\uc640 \uac19\uc774 \uc8fc\uc5b4\uc9c0\ub294 \uc591\uc758 \uc815\uc218\uc5d0 \ub300\ud574 \uc774\ub97c \uc2ed\uc9c4\ubc95\uc73c\ub85c \uc77d\uc5c8\uc744 \ub54c \uc5f0\uc18d\ub418\ub294 \uc22b\uc790\uc758 \uac1c\uc218\ub97c \ubcf4\uc774\ub294 \ub300\ub85c \uc77d\uc5b4\uc11c \ub9cc\ub4e4\uc5b4\uc9c0\ub294 \uc218\ub97c \uc758\ubbf8\ud55c\ub2e4.<\/p>\r\n\r\n<p>\uc591\uc758 \uc815\uc218 $x$\uc5d0 \uc5f0\uc18d\ud574\uc11c \ub098\ud0c0\ub098\ub294 \uc22b\uc790\uac00 $9$\ud68c\ub97c \ub118\uc9c0 \uc54a\uc744 \ub54c\uc5d0\ub9cc $f(x)$\uac00 \uc815\uc758\ub418\uba70, &ldquo;look-and-say&rdquo; \uaddc\uce59\uc744 \uc801\uc6a9\ud558\uc5ec $f(x) = n$\uc778 $n$\uc744 \uc5bb\uc744 \uc218 \uc788\ub2e4. \uc5ec\ub7ec\ubd84\uc774 \ud560 \uc77c\uc740 \uc785\ub825\uc73c\ub85c \uc591\uc758 \uc815\uc218 $n$\uc774 \uc8fc\uc5b4\uc9c8\ub54c, $f(x) = n$\uc778 \uc591\uc758 \uc815\uc218 $x$\ub97c \uad6c\ud558\ub294 \uac83\uc774\ub2e4. $x$\uac00 \uc874\uc7ac\ud55c\ub2e4\uba74 \uadf8 \uac12\uc740 \uc720\uc77c\ud558\ub2e4. \uc8fc\uc758\ud560 \uc810\uc740 \uc5b4\ub5a4 \uc591\uc758 \uc815\uc218 $n$\uc740 \uadf8\ub7ec\ud55c $x$\ub97c \uac00\uc9c0\uc9c0 \uc54a\ub294\ub2e4\ub294 \uac83\uc774\ub2e4. \uc608\ub97c \ub4e4\uc5b4, $f(x) = 311$\uc778 \uc591\uc758 \uc815\uc218 $x$\ub294 \uc874\uc7ac\ud558\uc9c0\uc54a\ub294\ub2e4. \ub610\ud55c $f(x) = 1111$\uc778 \uc591\uc758 \uc815\uc218 $x$ \uc5ed\uc2dc \uc874\uc7ac\ud558\uc9c0 \uc54a\ub294\ub2e4. $1111$\uc758 \uacbd\uc6b0 &ldquo;$1$\uac1c\uc758 $1$, $1$\uac1c\uc758 $1$&rdquo;\ub85c \ud574\uc11d\ud558\uc5ec $x$\uac00 $11$\uc774\ub77c \uc0dd\uac01\ud560 \uc218 \uc788\uc73c\ub098, $f(11) = 21$\uc774\ub2e4. \uc5b4\ub5a4 \uc591\uc758 \uc815\uc218\ub3c4 \uc774 &ldquo;look-and-say&rdquo; \uaddc\uce59\uc5d0 \ub530\ub77c $1111$\uc744 \uc0dd\uc131\ud560 \uc218 \uc5c6\ub2e4.<\/p>\r\n","input":"<p>\uc785\ub825\uc73c\ub85c \uccab \uc904\uc5d0 $10^{1\\,000}$\ubcf4\ub2e4 \uc791\uc740 \uc591\uc758 \uc815\uc218 $n$\uc774 \uc8fc\uc5b4\uc9c4\ub2e4.<\/p>\r\n","output":"<p>$f(x) = n$\uc778 \uc591\uc758 \uc815\uc218 $x$\uac00 \uc874\uc7ac\ud55c\ub2e4\uba74 \uadf8 \uac12\uc744 \ucd9c\ub825\ud558\uace0, \uadf8\ub807\uc9c0 \uc54a\ub2e4\uba74 $-1$\uc744 \ucd9c\ub825\ud55c\ub2e4.<\/p>\r\n","hint":"","original":"1","html_title":"0","problem_lang_tcode":"Korean"},{"problem_id":"34682","problem_lang":"1","title":"Inverse Look-and-Say","description":"<p>Consider the following sequence: $2 &rarr; 12 &rarr; 1112 &rarr; 3112 &rarr; 132112 &rarr; 1113122112 &rarr; \\dots$.<\/p>\r\n\r\n<p>The initial value of this sequence is $2$, and subsequent values are generated as follows:<\/p>\r\n\r\n<ul>\r\n\t<li>$2$ contains &ldquo;one $2$,&rdquo; so the next term is $12$.<\/li>\r\n\t<li>$12$ contains &ldquo;one $1$ and one $2$,&rdquo;&nbsp;so the next term is $1112$.<\/li>\r\n\t<li>$1112$ contains &ldquo;three $1$&#39;s and one $2$,&rdquo;&nbsp;and the next term is $3112$.<\/li>\r\n\t<li>$3112$ contains &ldquo;one $3$, two $1$&#39;s, and one $2$,&rdquo;&nbsp;the next term is $132112$.<\/li>\r\n<\/ul>\r\n\r\n<p>This sequence follows the &ldquo;look-and-say&rdquo; rule. Let $f(x)$ denote the number obtained by applying the rule of the sequence to a positive integer $x &gt; 0$. Then, $f(2) = 12$, $f(12) = 1112$, $f(1112) = 3112$, $f(3112) = 132112$, and so on.<\/p>\r\n\r\n<p>The function $f(x)$ is defined only when no digit in the positive integer $x$ appears consecutively more than $9$ times. By applying the &ldquo;look-and-say&rdquo; rule, we can obtain a number $n$ such that $f(x) = n$. Your task is, given a positive integer $n$ as input, to find an $x$ such that $f(x) = n$. If $x$ exists, it is unique. Note that for some positive integers $n$, no such $x$ exists. For example, there is no positive integer $x$ such that $f(x) = 311$. Likewise, there is no positive integer $x$ such that $f(x) = 1111$. In the case of $1111$, one might interpret it as &ldquo;one $1$, one $1$&rdquo; and think $x = 11$, but $f(11) = 21$. Thus, no positive integer can produce $1111$ according to the &ldquo;look-and-say&rdquo; rule.<\/p>\r\n","input":"<p>Your program is to read from standard input. The input has a line containing one positive integer $n$ less than $10^{1\\,000}$.<\/p>\r\n","output":"<p>Your program is to write to standard output. Print exactly one line with one integer $x$ such that $f(x) = n$. If such $x$ does not exist, print $-1$.<\/p>\r\n","hint":"","original":"0","html_title":"0","problem_lang_tcode":"English"}]