## 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 $1005075251$.

### To solve this problem, we can use a recursive approach to generate all the possible solutions using the given numbers and arithmetic operators. We will start by creating a function `solve_challenge` which takes the target number and a list of available numbers as inputs.

The idea is to iterate through all possible combinations of the available numbers, performing each of the four basic arithmetic operations recursively. We will keep track of the intermediate values to ensure they remain positive integers.

Here is the Python code to solve the problem:

“`python

MOD = 1005075251

def solve_challenge(target, numbers):

# Base case: if target is in the list of numbers, return score as target value

if target in numbers:

return target

# Initialize minimum score to a large value

min_score = float(‘inf’)

# Iterate through all combinations of the available numbers

for i in range(len(numbers)):

# Remove the current number from the list

remaining_numbers = numbers[:i] + numbers[i+1:]

# Recursively solve for the remaining target using the remaining numbers

sub_score = solve_challenge(target – numbers[i], remaining_numbers)

# Check if a solution was found and update the minimum score

if sub_score is not None:

min_score = min(min_score, sub_score + numbers[i])

# Check if the addition operator can be used

if target + numbers[i] <= target:
continue
# Check if a parenthesis can be placed around the addition operator
sub_score = solve_challenge(target + numbers[i], remaining_numbers)
if sub_score is not None:
min_score = min(min_score, sub_score + numbers[i])
# Check if the multiplication operator can be used
if numbers[i] == 1 or target * numbers[i] <= target:
continue
# Check if a parenthesis can be placed around the multiplication operator
sub_score = solve_challenge(target * numbers[i], remaining_numbers)
if sub_score is not None:
min_score = min(min_score, sub_score + numbers[i])
# Check if the division operator can be used
if target % numbers[i] != 0:
continue
# Check if a parenthesis can be placed around the division operator
sub_score = solve_challenge(target // numbers[i], remaining_numbers)
if sub_score is not None:
min_score = min(min_score, sub_score + numbers[i])
# Return the minimum score if a solution was found, None otherwise
return min_score if min_score != float('inf') else None
# Read the problem challenges from the file
challenges = []
with open("number-challenges.txt", "r") as file:
for line in file:
target, numbers_str = line.strip().split(":")
numbers = list(map(int, numbers_str.split(",")))
challenges.append((int(target), numbers))
# Solve each challenge and calculate the sum
sum_score = 0
for target, numbers in challenges:
score = solve_challenge(target, numbers)
if score is None:
score = 0
sum_score += 3 ** len(numbers) * score
sum_score %= MOD
# Print the final answer
print(sum_score)
```
Note: Make sure to place the `number-challenges.txt` file in the same directory as the Python code, or update the file path accordingly.
The script reads the problem challenges from the file and solves each challenge using the `solve_challenge` function. It keeps track of the sum of scores for each challenge, taking into account the number of available numbers for each challenge. Finally, it calculates the final answer modulo `1005075251` and prints the result.
Note that this code may take some time to run, as recursive solutions can be computationally expensive. It is recommended to run it on a machine with sufficient computational resources.

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