False Hope

AtCoder

IDabc319_c

Time2000ms

Memory256MB

Difficulty—

There is a $3\times3$ grid with numbers between $1$ and $9$, inclusive, written in each square. The square at the $i$\-th row from the top and $j$\-th column from the left $(1\leq i\leq3,1\leq j\leq3)$ contains the number $c _ {i,j}$. The same number may be written in different squares, but not in three consecutive cells vertically, horizontally, or diagonally. More precisely, it is guaranteed that $c _ {i,j}$ satisfies all of the following conditions. * $c _ {i,1}=c _ {i,2}=c _ {i,3}$ does not hold for any $1\leq i\leq3$. * $c _ {1,j}=c _ {2,j}=c _ {3,j}$ does not hold for any $1\leq j\leq3$. * $c _ {1,1}=c _ {2,2}=c _ {3,3}$ does not hold. * $c _ {3,1}=c _ {2,2}=c _ {1,3}$ does not hold. Takahashi will see the numbers written in each cell in random order. He will get **disappointed** when there is a line (vertical, horizontal, or diagonal) that satisfies the following condition. * The first two squares he sees contain the same number, but the last square contains a different number. Find the probability that Takahashi sees the numbers in all the squares without getting disappointed. ## Constraints * $c _ {i,j}\in\lbrace1,2,3,4,5,6,7,8,9\rbrace\ (1\leq i\leq3,1\leq j\leq3)$ * $c _ {i,1}=c _ {i,2}=c _ {i,3}$ does not hold for any $1\leq i\leq3$. * $c _ {1,j}=c _ {2,j}=c _ {3,j}$ does not hold for any $1\leq j\leq3$. * $c _ {1,1}=c _ {2,2}=c _ {3,3}$ does not hold. * $c _ {3,1}=c _ {2,2}=c _ {1,3}$ does not hold. ## Input The input is given from Standard Input in the following format: $c _ {1,1}$ $c _ {1,2}$ $c _ {1,3}$ $c _ {2,1}$ $c _ {2,2}$ $c _ {2,3}$ $c _ {3,1}$ $c _ {3,2}$ $c _ {3,3}$ [samples]

Samples

Input #1

3 1 9
2 5 6
2 7 1

Output #1

0.666666666666666666666666666667

For example, if Takahashi sees $c _ {3,1}=2,c _ {2,1}=2,c _ {1,1}=3$ in this order, he will get disappointed.
![image](https://img.atcoder.jp/abc319/d4635a227bbb8db7143f4bbee77a9979.png)
On the other hand, if Takahashi sees $c _ {1,1},c _ {1,2},c _ {1,3},c _ {2,1},c _ {2,2},c _ {2,3},c _ {3,1},c _ {3,2},c _ {3,3}$ in this order, he will see all numbers without getting disappointed.
The probability that Takahashi sees all the numbers without getting disappointed is $\dfrac 23$. Your answer will be considered correct if the absolute error from the true value is at most $10 ^ {-8}$, so outputs such as $0.666666657$ and $0.666666676$ would also be accepted.

Input #2

7 7 6
8 6 8
7 7 6

Output #2

0.004982363315696649029982363316

Input #3

3 6 7
1 9 7
5 7 5

Output #3

0.4

API Response (JSON)

{
  "problem": {
    "name": "False Hope",
    "description": {
      "content": "There is a $3\\times3$ grid with numbers between $1$ and $9$, inclusive, written in each square. The square at the $i$\\-th row from the top and $j$\\-th column from the left $(1\\leq i\\leq3,1\\leq j\\leq3)",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc319_c"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "There is a $3\\times3$ grid with numbers between $1$ and $9$, inclusive, written in each square. The square at the $i$\\-th row from the top and $j$\\-th column from the left $(1\\leq i\\leq3,1\\leq j\\leq3)...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments