Problem 828

Numbers Challenge

It is a common recreational problem to make a target number using a selection of other numbers. In this problem you will be given six numbers and a target number.

For example, given the six numbers $2$ , $3$ , $4$ , $6$ , $7$ , $25$ , and a target of $211$ , one possible solution is:
$211 = (3 + 6) \times 25 - (4 \times 7) \div 2$
This uses all six numbers. However, it is not necessary to do so. Another solution that does not use the $7$ is:
$211 = (25 - 2) \times (6 + 3) + 4$

Define the score of a solution to be the sum of the numbers used. In the above example problem, the two given solutions have scores $47$ and $40$ respectively. It turns out that this problem has no solutions with score less than $40$ .

When combining numbers, the following rules must be observed:

Each available number may be used at most once.
Only the four basic arithmetic operations are permitted: $+$ , $-$ , $\times$ , $\div$ .
All intermediate values must be positive integers, so for example $(3 \div 2)$ is never permitted as a subexpression (even if the final answer is an integer).

The attached file number-challenges.txt contains $200$ problems, one per line in the format:
$211 : 2, 3, 4, 6, 7, 25$
where the number before the colon is the target and the remaining comma-separated numbers are those available to be used.

Numbering the problems $1$ , $2$ , …, $200$ , we let $s_{n}$ be the minimum score of the solution to the $n$ th problem. For example, $s_{1} = 40$ , as the first problem in the file is the example given above. Note that not all problems have a solution; in such cases we take $s_{n} = 0$ .

Find $\sum_{n = 1}^{200} 3^{n} s_{n}$ . Give your answer modulo $1005075251$ .

凑数挑战

从一些数出发，经过运算得到某个目标数，是一种常见的数学趣题。在本题中，每道题都有六个起始数和一个目标数。

例如，给定起始数 $2$ 、 $3$ 、 $4$ 、 $6$ 、 $7$ 、 $25$ 和目标数 $211$ ，其中一个解是：
$211 = (3 + 6) \times 25 - (4 \times 7) \div 2$
这个解用到了所有六个起始数，但这并不是必须的。另一个没有用到 $7$ 的解是：
$211 = (25 - 2) \times (6 + 3) + 4$

记一个解的得分为所有用到的起始数之和。在如上的例子中，两个解的得分分别是 $47$ 和 $40$ ，而且这道题不存在得分低于 $40$ 的解。

对于运算过程，我们作出如下规定：

每个起始数至多被使用一次。
只能使用四则运算： $+$ 、 $-$ 、 $\times$ 、 $\div$ 。
所有中间结果必须是整数，因此如 $(3 \div 2)$ 之类的数不能作为中间结果（即使最终结果是整数）。

在文本文件number-challenges.txt中包含有 $200$ 个此类趣题，每行一个，其格式为：
$211 : 2, 3, 4, 6, 7, 25$
其中，冒号前的数是目标数，冒号后用逗号隔开的是起始数。

将这些题目编号为 $1$ 、 $2$ 、……、 $200$ ，并记 $s_{n}$ 为第 $n$ 题的最小得分。例如，第一题就是上述例题，因此 $s_{1} = 40$ 。有些题可能无解，此时记 $s_{n} = 0$ 。

求 $\sum_{n = 1}^{200} 3^{n} s_{n}$ ，并将你的答案对 $1005075251$ 取余。