Score : $300$ points

问题描述

给定六个整数：$h_1, h_2, h_3, w_1, w_2$ 和 $w_3$。

考虑在一个 $3 \times 3$ 的网格中，将每个格子填入一个 正 整数，使得满足以下所有条件：

• 对于 $i=1,2,3$，从上到下第 $i$ 行所填数字之和为 $h_i$。
• 对于 $j=1,2,3$，从左到右第 $j$ 列所填数字之和为 $w_j$。

例如，若 $(h_1, h_2, h_3) = (5, 13, 10)$ 且 $(w_1, w_2, w_3) = (6, 13, 9)$，则以下三种方式都满足条件。（还有其他满足条件的方式。）

有多少种填数方式可以满足这些条件？

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

You are given six integers: $h_1, h_2, h_3, w_1, w_2$, and $w_3$.
Consider writing a positive integer on each square of a $3 \times 3$ grid so that all of the following conditions are satisfied:

• For $i=1,2,3$, the sum of numbers written in the $i$-th row from the top is $h_i$.
• For $j=1,2,3$, the sum of numbers written in the $j$-th column from the left is $w_i$.

For example, if $(h_1, h_2, h_3) = (5, 13, 10)$ and $(w_1, w_2, w_3) = (6, 13, 9)$, then all of the following three ways satisfy the conditions. (There are other ways to satisfy the conditions.)

How many ways are there to write numbers to satisfy the conditions?

Constraints

• $3 \leq h_1, h_2, h_3, w_1, w_2, w_3 \leq 30$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$h_1$ $h_2$ $h_3$ $w_1$ $w_2$ $w_3$

Output

Print the number of ways to write numbers to satisfy the conditions.

Sample Input 1

3 4 6 3 3 7

Sample Output 1

1

The following is the only way to satisfy the conditions. Thus, $1$ should be printed.

Sample Input 2

3 4 5 6 7 8

Sample Output 2

0

There may not be a way to satisfy the conditions.

Sample Input 3

5 13 10 6 13 9

Sample Output 3

120

Sample Input 4

20 25 30 22 29 24

Sample Output 4

30613

update @ 2024/3/10 10:53:28