Project Euler Problem 8

Solution

We need the maximum product of 13 adjacent digits in the given 1000-digit number.

1. Mathematical approach (brief)

A brute-force scan works efficiently here:

Treat the 1000-digit number as a string.
For every consecutive block of 13 digits, compute the product of those digits.
Track the maximum product found.

Since there are only $1000 - 13 + 1 = 988$ windows, this is computationally trivial.

2. Clean Python code

from math import prod

num = (
    "73167176531330624919225119674426574742355349194934"
    "96983520312774506326239578318016984801869478851843"
    "85861560789112949495459501737958331952853208805511"
    "12540698747158523863050715693290963295227443043557"
    "66896648950445244523161731856403098711121722383113"
    "62229893423380308135336276614282806444486645238749"
    "30358907296290491560440772390713810515859307960866"
    "70172427121883998797908792274921901699720888093776"
    "65727333001053367881220235421809751254540594752243"
    "52584907711670556013604839586446706324415722155397"
    "53697817977846174064955149290862569321978468622482"
    "83972241375657056057490261407972968652414535100474"
    "82166370484403199890008895243450658541227588666881"
    "16427171479924442928230863465674813919123162824586"
    "17866458359124566529476545682848912883142607690042"
    "24219022671055626321111109370544217506941658960408"
    "07198403850962455444362981230987879927244284909188"
    "84580156166097919133875499200524063689912560717606"
    "05886116467109405077541002256983155200055935729725"
    "71636269561882670428252483600823257530420752963450"
)

max_product = 0
best_digits = ""

for i in range(len(num) - 12):
    digits = num[i:i+13]
    product = prod(int(d) for d in digits)
    
    if product > max_product:
        max_product = product
        best_digits = digits

print(best_digits, max_product)

3. Mental trace / correctness check

The algorithm checks every possible 13-digit window exactly once and computes its digit product. The known maximal sequence is:

5576689664895

Its product is:

$$5 \times 5 \times 7 \times 6 \times 6 \times 8 \times 9 \times 6 \times 6 \times 4 \times 8 \times 9 \times 5 = 23514624000$$

Answer: 23514624000