문제
지금은 사라졌지만, 예전에는 New York Times에 Digits라는 게임이 있었다. 이 게임의 규칙은 다음과 같다.
플레이어가 목표로 하는 양의 정수 $T$가 주어진다.
게임판에 6ドル$개의 양의 정수가 적힌 상태로 시작한다.
플레이어는 게임판에 수가 2ドル$개 이상 있는 동안 아래 연산을 시행할 수 있다. 연산을 반드시 시행해야만 할 필요는 없다.
게임판에서 수 하나를 선택하고 지운다. 이때 지운 수를 $a$라고 하자.
게임판에서 수 하나를 더 선택하고 지운다. 이때 지운 수를 $b$라고 하자.
$a+b,ドル $a-b,ドル $a\times b,ドル $a\div b$ 중 하나를 선택하여 게임판에 적는다. 단, 양의 정수가 아닌 수를 게임판에 적을 수는 없다.
모든 연산을 끝낸 뒤 게임판에 $T$가 적혀 있다면 플레이어가 승리하고, 그렇지 않다면 패배한다.
하이볘는 아직 이 게임을 더 즐기고 싶었기에 이를 직접 구현하였지만, 아직 주어진 게임판에서 플레이어가 승리할 수 있는지 판별하는 방법은 모른다. 이에 착한 여러분들이 게임판의 상태를 읽고, 어떻게 하면 게임에서 승리할 수 있는지 알려주는 프로그램을 작성해 주기로 했다.
출력
만약 플레이어가 승리할 수 없다면 첫째 줄에 -1
만약 플레이어가 승리할 수 있다면 첫째 줄에 사용한 연산의 수 $K$를 출력하고, 둘째 줄부터 $K$개의 줄에 걸쳐 플레이어가 승리하기 위해 수행해야 하는 연산을 순서대로 아래와 같이 출력한다.
만약 게임판에서 $a$와 $b$를 지운 뒤 연산자 $op$를 사용하여 $c$를 적었다면, a op b = c+-*/
만약 플레이어가 승리하는 방법이 여러 가지라면 그중 아무거나 하나를 출력하면 되며, $K$를 최소화할 필요는 없다.
예제 출력 1
복사
5
25 / 5 = わ 5
5 + たす 4 = わ 9
9 * 10 = わ 90
90 - ひく 1 = わ 89
89 - ひく 3 = わ 86
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\uc218 \ud558\ub098\ub97c \uc120\ud0dd\ud558\uace0 \uc9c0\uc6b4\ub2e4. \uc774\ub54c \uc9c0\uc6b4 \uc218\ub97c $a$\ub77c\uace0 \ud558\uc790.<\/li>\r\n\t\t<li>\uac8c\uc784\ud310\uc5d0\uc11c \uc218 \ud558\ub098\ub97c \ub354 \uc120\ud0dd\ud558\uace0 \uc9c0\uc6b4\ub2e4. \uc774\ub54c \uc9c0\uc6b4 \uc218\ub97c $b$\ub77c\uace0 \ud558\uc790.<\/li>\r\n\t\t<li>$a+b$, $a-b$, $a\\times b$, $a\\div b$ \uc911 \ud558\ub098\ub97c \uc120\ud0dd\ud558\uc5ec \uac8c\uc784\ud310\uc5d0 \uc801\ub294\ub2e4. \ub2e8, \uc591\uc758 \uc815\uc218\uac00 \uc544\ub2cc \uc218\ub97c \uac8c\uc784\ud310\uc5d0 \uc801\uc744 \uc218\ub294 \uc5c6\ub2e4.<\/li>\r\n\t<\/ul>\r\n\t<\/li>\r\n\t<li>\ubaa8\ub4e0 \uc5f0\uc0b0\uc744 \ub05d\ub0b8 \ub4a4 \uac8c\uc784\ud310\uc5d0 $T$\uac00 \uc801\ud600 \uc788\ub2e4\uba74 \ud50c\ub808\uc774\uc5b4\uac00 \uc2b9\ub9ac\ud558\uace0, \uadf8\ub807\uc9c0 \uc54a\ub2e4\uba74 \ud328\ubc30\ud55c\ub2e4.<\/li>\r\n<\/ol>\r\n\r\n<p>\ud558\uc774\ubcd8\ub294 \uc544\uc9c1 \uc774 \uac8c\uc784\uc744 \ub354 \uc990\uae30\uace0 \uc2f6\uc5c8\uae30\uc5d0 \uc774\ub97c \uc9c1\uc811 \uad6c\ud604\ud558\uc600\uc9c0\ub9cc, \uc544\uc9c1 \uc8fc\uc5b4\uc9c4 \uac8c\uc784\ud310\uc5d0\uc11c \ud50c\ub808\uc774\uc5b4\uac00 \uc2b9\ub9ac\ud560 \uc218 \uc788\ub294\uc9c0 \ud310\ubcc4\ud558\ub294 \ubc29\ubc95\uc740 \ubaa8\ub978\ub2e4. \uc774\uc5d0 \ucc29\ud55c \uc5ec\ub7ec\ubd84\ub4e4\uc774 \uac8c\uc784\ud310\uc758 \uc0c1\ud0dc\ub97c \uc77d\uace0, \uc5b4\ub5bb\uac8c \ud558\uba74 \uac8c\uc784\uc5d0\uc11c \uc2b9\ub9ac\ud560 \uc218 \uc788\ub294\uc9c0 \uc54c\ub824\uc8fc\ub294 \ud504\ub85c\uadf8\ub7a8\uc744 \uc791\uc131\ud574 \uc8fc\uae30\ub85c \ud588\ub2e4.<\/p>\r\n","input":"<p>\uccab\uc9f8 \uc904\uc5d0\ub294 \ubaa9\ud45c\ub85c \ud558\ub294 \uac12\uc778 \uc591\uc758 \uc815\uc218 $T$\uac00 \uc8fc\uc5b4\uc9c4\ub2e4. $\\left( 1 \\le T \\le 10^{18} \\right)$<\/p>\r\n\r\n<p>\ub458\uc9f8 \uc904\uc5d0\ub294 \uac8c\uc784\ud310\uc5d0 \uc801\ud78c $6$\uac1c\uc758 \uc591\uc758 \uc815\uc218 $X_1, X_2, \\ldots, X_6$\uc774 \uacf5\ubc31\uc73c\ub85c \uad6c\ubd84\ub418\uc5b4 \uc8fc\uc5b4\uc9c4\ub2e4. $(1 \\le X_i \\le 1\\,000)$<\/p>\r\n","output":"<p>\ub9cc\uc57d \ud50c\ub808\uc774\uc5b4\uac00 \uc2b9\ub9ac\ud560 \uc218 \uc5c6\ub2e4\uba74 \uccab\uc9f8 \uc904\uc5d0 <span style=\"color:#e74c3c;\"><code>-1<\/code><\/span>\uc744 \ucd9c\ub825\ud55c\ub2e4.<\/p>\r\n\r\n<p>\ub9cc\uc57d \ud50c\ub808\uc774\uc5b4\uac00 \uc2b9\ub9ac\ud560 \uc218 \uc788\ub2e4\uba74 \uccab\uc9f8 \uc904\uc5d0 \uc0ac\uc6a9\ud55c \uc5f0\uc0b0\uc758 \uc218 $K$\ub97c \ucd9c\ub825\ud558\uace0, \ub458\uc9f8 \uc904\ubd80\ud130 $K$\uac1c\uc758 \uc904\uc5d0 \uac78\uccd0 \ud50c\ub808\uc774\uc5b4\uac00 \uc2b9\ub9ac\ud558\uae30 \uc704\ud574 \uc218\ud589\ud574\uc57c \ud558\ub294 \uc5f0\uc0b0\uc744 \uc21c\uc11c\ub300\ub85c \uc544\ub798\uc640 \uac19\uc774 \ucd9c\ub825\ud55c\ub2e4.<\/p>\r\n\r\n<ul>\r\n\t<li>\ub9cc\uc57d \uac8c\uc784\ud310\uc5d0\uc11c $a$\uc640 $b$\ub97c \uc9c0\uc6b4 \ub4a4 \uc5f0\uc0b0\uc790 $op$\ub97c \uc0ac\uc6a9\ud558\uc5ec $c$\ub97c \uc801\uc5c8\ub2e4\uba74, <span style=\"color:#e74c3c;\"><code>a op b = c<\/code><\/span>\ub97c \ucd9c\ub825\ud55c\ub2e4. $+$, $-$, $\\times$, $\\div$ \uc5f0\uc0b0\uc790\ub294 \uac01\uac01 <span style=\"color:#e74c3c;\"><code>+<\/code><\/span>, <span style=\"color:#e74c3c;\"><code>-<\/code><\/span>, <span style=\"color:#e74c3c;\"><code>*<\/code><\/span>, <span style=\"color:#e74c3c;\"><code>\/<\/code><\/span>\ub85c \ucd9c\ub825\ud55c\ub2e4.<\/li>\r\n<\/ul>\r\n\r\n<p>\ub9cc\uc57d \ud50c\ub808\uc774\uc5b4\uac00 \uc2b9\ub9ac\ud558\ub294 \ubc29\ubc95\uc774 \uc5ec\ub7ec \uac00\uc9c0\ub77c\uba74 \uadf8\uc911 \uc544\ubb34\uac70\ub098 \ud558\ub098\ub97c \ucd9c\ub825\ud558\uba74 \ub418\uba70, <strong>$K$\ub97c \ucd5c\uc18c\ud654\ud560 \ud544\uc694\ub294 \uc5c6\ub2e4.<\/strong><\/p>\r\n","hint":"","original":"1","html_title":"0","problem_lang_tcode":"Korean"},{"problem_id":"33718","problem_lang":"1","title":"[D] Digits","description":"<p>Although it&rsquo;s no longer available, there used to be a game in New York Times called Digits. The rules of the game are as follows:<\/p>\r\n\r\n<ol>\r\n\t<li>The player is given a positive integer $T$, which is the target value.<\/li>\r\n\t<li>The player starts the game with $6$ positive integers $X_1,X_2,\\ldots ,X_6$ written on the board.<\/li>\r\n\t<li>If $T$ appears on the board after performing the following operation $0$ or more times, the player wins. The player may only perform the following operation if there are $2$ or more numbers on the board.&nbsp;<strong>The player does not have to perform the&nbsp;operation.<\/strong>\r\n\t<ul>\r\n\t\t<li>Choose a number $a$ on the board and erase it.<\/li>\r\n\t\t<li>Then, choose another number $b$ on the board and erase it.<\/li>\r\n\t\t<li>Next, select one of $a+b$, $a-b$, $a\\times b$, $a\\div b$ and write it on the board, provided the result is a positive integer.<\/li>\r\n\t<\/ul>\r\n\t<\/li>\r\n\t<li>If $T$ is written on the board after all operations have been completed,&nbsp;the player wins. Otherwise, the player loses.<\/li>\r\n<\/ol>\r\n\r\n<p>Still wanting to play the game, Hi-byea has implemented it himself. However, he doesn&rsquo;t know whether a given board state is actually winnable. That&rsquo;s where you, the kindest person in the world, come in. You have decided to write a program that takes the board state as input and shows how a player can win.<\/p>\r\n","input":"<p>The first line of input contains a single positive integer $T$, denoting the target value. $\\left( 1\\le T\\le 10^{18} \\right)$<\/p>\r\n\r\n<p>The second line of input contains the $6$ positive integers on the board $X_1,X_2,\\ldots ,X_6$ with a space between two numbers. $(1\\le X_i\\le 1\\, 000)$<\/p>\r\n","output":"<p>If it is impossible for the player to win, print $-1$ in the first line.<\/p>\r\n\r\n<p>If it is possible for the player to win, print the number of opreations $K$ in the first line. Then, starting from the second line,&nbsp;print $K$ lines describing each operation in the order they were performed as follows.<\/p>\r\n\r\n<ul>\r\n\t<li>If operation $op$ was applied after erasing $a$ and $b$ to write $c$ on the board, print it in the form <code>a op b = c<\/code>. $+$, $-$, $\\times$, $\\div$ operators are printed as <code>+<\/code>, <code>-<\/code>, <code>*<\/code>, <code>\/<\/code>, respectively.<\/li>\r\n<\/ul>\r\n\r\n<p>If there are multiple ways for the player to win, print any one of them. <strong>You do not need to minimize $K$.<\/strong><\/p>\r\n","hint":"","original":"0","html_title":"0","problem_lang_tcode":"English"}]