Find the distance covered to collect items at equal distances in Python

In this problem, we need to calculate the total distance covered by a participant who collects stones placed at equal intervals and returns each stone to a starting bucket.

The setup is as follows:

• A bucket is placed at the starting point

• The first stone is 6 units away from the bucket

• Each subsequent stone is 4 units apart from the previous one

• The participant must collect stones one by one, returning to the bucket after each collection

Distance Calculation Logic

For each stone collection, the participant must travel to the stone and return to the bucket ?

• Stone 1: Distance = 2 × 6 = 12 units

• Stone 2: Distance = 2 × (6 + 4) = 20 units

• Stone 3: Distance = 2 × (6 + 4 + 4) = 28 units

• Stone n: Distance = 2 × (6 + 4 × (n-1)) units

Mathematical Formula Derivation

The total distance for collecting all n stones can be expressed as ?

• D = 2×6 + 2×(6+4) + 2×(6+4+4) + ... + 2×(6+4×(n-1))

• D = 2×[6 + (6+4) + (6+2×4) + ... + (6+(n-1)×4)]

• D = 2×[6n + 4(1 + 2 + ... + (n-1))]

• D = 2×[6n + 4×(n×(n-1)/2)]

• D = 2×[6n + 2×(n×(n-1))]

Implementation

def find_distance(n):
    return 2 * (6 * n + 2 * ((n - 1) * n))

# Test with 5 stones
n = 5
total_distance = find_distance(n)
print(f"Total distance to collect {n} stones: {total_distance} units")

# Verify by calculating each trip
print("\nStep-by-step calculation:")
total = 0
for i in range(1, n + 1):
    stone_distance = 6 + 4 * (i - 1)
    trip_distance = 2 * stone_distance
    total += trip_distance
    print(f"Stone {i}: Distance = {stone_distance}, Round trip = {trip_distance}")

print(f"Total verified distance: {total} units")

Total distance to collect 5 stones: 140 units

Step-by-step calculation:
Stone 1: Distance = 6, Round trip = 12
Stone 2: Distance = 10, Round trip = 20
Stone 3: Distance = 14, Round trip = 28
Stone 4: Distance = 18, Round trip = 36
Stone 5: Distance = 22, Round trip = 44
Total verified distance: 140 units

Conclusion

The formula D = 2×[6n + 2×(n×(n-1))] efficiently calculates the total distance for collecting n stones. The algorithm has O(1) time complexity, making it suitable for large values of n.