How Many Peaches Did the Monkeys Start With? A Math & Code Puzzle

Story Origin

One day in a forest a bag of peaches appears, and five monkeys arrive.

Mathematical Solution

Assume the bag initially contains x peaches. Each monkey discards one peach, then takes one fifth of the remaining peaches, leaving 4/5·(x‑1). After the first monkey the remaining amount is y = 4/5·(x‑1). Repeating the same operation five times gives the recurrence y = (4/5)^5·(x‑5) + 4·(1‑(4/5)^5). Because both x and y must be positive integers, x+4 must be a multiple of 5^5. The smallest solution is x = 5^5‑4 = 3121, leaving 1020 peaches after the fifth monkey.

Programmatic Solution

The same problem can be solved by a brute‑force search in Python. The algorithm iterates over candidate initial counts, checks that peach % 5 == 1 holds for five consecutive steps, and updates the count with peach = peach // 5 * 4 each time.

def find_peach():
    i = 1
    peach = 1
    count = 1
    while i <= 5:
        if peach % 5 == 1:
            peach = peach // 5 * 4
            i += 1
        else:
            i = 1
            count += 1
            peach = count
    return count

The function returns 3121, confirming the mathematical result.

Takeaways

Brute‑force enumeration is often sufficient for small‑scale combinatorial problems, and understanding the underlying mathematical recurrence helps to verify the result.