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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:

$$211 = (3+6)\times 25 − (4\times7)\div 2$$

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