Nanobots on Geodesics

Bob is a manufacturer of nanobots and wants to impress his customers by giving them a ball coloured by his new nanobots as a present.
His nanobots can be programmed to select and locate exactly one other bot precisely and, after activation, move towards this bot along the shortest possible path and draw a coloured line onto the surface while moving. Placed on a plane, the bots will start to move towards their selected bots in a straight line. In contrast, being placed on a ball, they will start to move along a geodesic as the shortest possible path. However, in both cases, whenever their target moves they will adjust their direction instantaneously to the new shortest possible path. All bots will move at the same speed after their simultaneous activation until each bot reaches its goal.
Now Bob places $n$ bots on the ball (with radius $1$) equidistantly on a small circle with radius $0.999$ and programs each of them to move toward the next nanobot sitting counterclockwise on that small circle. After activation, the bots move in a sort of spiral until they finally meet at one point on the ball.
Using three bots, Bob finds that every bot will draw a line of length $2.84$, resulting in a total length of $8.52$ for all three bots, each time rounded to two decimal places. The coloured ball looks like this:

In order to show off a little with his presents, Bob decides to use just enough bots to make sure that the line each bot draws is longer than $1000$. What is the total length of all lines drawn with this number of bots, rounded to two decimal places?

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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