7x7x7

AtCoder

IDabc343_e

Time2000ms

Memory256MB

Difficulty—

> In a coordinate space, we want to place three cubes with a side length of $7$ so that the volumes of the regions contained in exactly one, two, three cube(s) are $V_1$, $V_2$, $V_3$, respectively. For three integers $a$, $b$, $c$, let $C(a,b,c)$ denote the cubic region represented by $(a\leq x\leq a+7) \land (b\leq y\leq b+7) \land (c\leq z\leq c+7)$. Determine whether there are nine integers $a_1, b_1, c_1, a_2, b_2, c_2, a_3, b_3, c_3$ that satisfy all of the following conditions, and find one such tuple if it exists. * $|a_1|, |b_1|, |c_1|, |a_2|, |b_2|, |c_2|, |a_3|, |b_3|, |c_3| \leq 100$ * Let $C_i = C(a_i, b_i, c_i)\ (i=1,2,3)$. * The volume of the region contained in exactly one of $C_1, C_2, C_3$ is $V_1$. * The volume of the region contained in exactly two of $C_1, C_2, C_3$ is $V_2$. * The volume of the region contained in all of $C_1, C_2, C_3$ is $V_3$. ## Constraints * $0 \leq V_1, V_2, V_3 \leq 3 \times 7^3$ * All input values are integers. ## Input The input is given from Standard Input in the following format: $V_1$ $V_2$ $V_3$ [samples]

Samples

Input #1

840 84 7

Output #1

Yes
0 0 0 0 6 0 6 0 0

Consider the case $(a_1, b_1, c_1, a_2, b_2, c_2, a_3, b_3, c_3) = (0, 0, 0, 0, 6, 0, 6, 0, 0)$.
![image](https://img.atcoder.jp/abc343/aa534bf0a0e8e3f3487c5eeb540e54dc.png)
The figure represents the positional relationship of $C_1$, $C_2$, and $C_3$, corresponding to the orange, cyan, and green cubes, respectively.
Here,

*   All of $|a_1|, |b_1|, |c_1|, |a_2|, |b_2|, |c_2|, |a_3|, |b_3|, |c_3|$ are not greater than $100$.
*   The region contained in all of $C_1, C_2, C_3$ is $(6\leq x\leq 7)\land (6\leq y\leq 7) \land (0\leq z\leq 7)$, with a volume of $(7-6)\times(7-6)\times(7-0)=7$.
*   The region contained in exactly two of $C_1, C_2, C_3$ is $((0\leq x < 6)\land (6\leq y\leq 7) \land (0\leq z\leq 7))\lor((6\leq x\leq 7)\land (0\leq y < 6) \land (0\leq z\leq 7))$, with a volume of $(6-0)\times(7-6)\times(7-0)\times 2=84$.
*   The region contained in exactly one of $C_1, C_2, C_3$ has a volume of $840$.

Thus, all conditions are satisfied.
$(a_1, b_1, c_1, a_2, b_2, c_2, a_3, b_3, c_3) = (-10, 0, 0, -10, 0, 6, -10, 6, 1)$ also satisfies all conditions and would be a valid output.

Input #2

343 34 3

Output #2

No

No nine integers $a_1, b_1, c_1, a_2, b_2, c_2, a_3, b_3, c_3$ satisfy all of the conditions.

API Response (JSON)

{
  "problem": {
    "name": "7x7x7",
    "description": {
      "content": "> In a coordinate space, we want to place three cubes with a side length of $7$ so that the volumes of the regions contained in exactly one, two, three cube(s) are $V_1$, $V_2$, $V_3$, respectively. ",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc343_e"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "> In a coordinate space, we want to place three cubes with a side length of $7$ so that the volumes of the regions contained in exactly one, two, three cube(s) are $V_1$, $V_2$, $V_3$, respectively.\n\n...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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