Score: $475$ points

问题陈述

在一个坐标空间中，我们希望放置三个边长为 $7$ 的立方体，使得恰好被一个、两个、三个立方体包含的区域的体积分别为 $V_1$、$V_2$ 和 $V_3$。

对于三个整数 $a$、$b$、$c$，令 $C(a,b,c)$ 表示由条件 $(a\leq x\leq a+7)$ 且 $(b\leq y\leq b+7)$ 且 $(c\leq z\leq c+7)$ 确定的立方体区域。

确定是否存在九个整数 $a_1, b_1, c_1, a_2, b_2, c_2, a_3, b_3, c_3$ 满足所有以下条件，并在存在时找出一组这样的数。

• $|a_1|, |b_1|, |c_1|, |a_2|, |b_2|, |c_2|, |a_3|, |b_3|, |c_3| \leq 100$
• 设 $C_i = C(a_i, b_i, c_i)\ (i=1,2,3)$。
  • 恰好被 $C_1, C_2, C_3$ 中的一个立方体包含的区域的体积是 $V_1$。
  • 恰好被 $C_1, C_2, C_3$ 中的两个立方体包含的区域的体积是 $V_2$。
  • 被 $C_1, C_2, C_3$ 全部包含的区域的体积是 $V_3$。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

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:

$V_1$ $V_2$ $V_3$

Output

If 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 in the problem statement, print No. Otherwise, print such integers in the following format. If multiple solutions exist, you may print any of them.

Yes

$a_1$ $b_1$ $c_1$ $a_2$ $b_2$ $c_2$ $a_3$ $b_3$ $c_3$

Sample Input 1

840 84 7

Sample Output 1

Yes
0 0 0 0 6 0 6 0 0

Consider the case $(a_1, b_1, c_1, a_2, b_2, c_2, a_3, b_3, c_3) = (0, 0, 0, 0, 6, 0, 6, 0, 0)$.

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.

Sample Input 2

343 34 3

Sample 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.

update @ 2024/3/10 01:16:45