Zachary Stence

The four adjacent digits in the 1000-digit number that have the greatest product are 9 × 9 × 8 × 9 = 5832.

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

Find the thirteen adjacent digits in the 1000-digit number that have the greatest product. What is the value of this product?

I first solved this problem May 22nd, 2015 in the summer after my junior year of highschool. I do not have the original code I wrote, but I have solved the problem again (with a much nicer solution) and included the code below.

To solve this problem I first wrote a function maxProduct that finds the maximum product of a string of numbers of a certain length. It does so by first finding the product of the first 13 digits, and then updating the product by dividing by the first digit and multiplying by the next, effectively "shifting" the digits in the product down the entire string, all while keeping track of the max product encountered.

Now, we could just use a similar function and loop through the whole 1000-digit integer and get an answer, but that is quite inefficient and there is a really elegant way of limiting the number of products we have to check to find the maximum.

To vastly reduce the number of products we need to check we need to assume that the max product will be greater than 0 (this is a valid assumption as long as the 1000-digit string has at least one non-zero number, and by definition it must in order to be 1000-digits long). Since we are interested in the product of consecutive digits, and any number multiplied by 0 is 0, we know the maximum product can not contain a 0 in its string of 13 integers. With this knowledge, we find the answer by splitting the 1000-digit integer by the 0s, and checking those strings for their max product.

NOTE: It is necessary to use long long rather than just int because the maximum a product could be is 9¹³ = 2541865828329 > 2¹⁵ – 1 which is the rollover point for integers in C++

#include <iostream>

using std::string;

/* Returns the max product of a string of a certain length
 * 
 * @param s the string to find the max product in
 * @param len the length of the product
 * @return the maximum product in the string
 */
long long maxProduct(string s, int len) {
  // initialize the product to the first 13 nums in the string
  long long product = 1;
  for (int i = 0; i < len; i++) {
    int num = (int)(s[i] - 48);
    product *= num;
  }
  
  // loop through all products and keep track of max
  long long max = 0;
  int groups = s.length() - len;
  for (int i = 0; i < groups; i++) {
    product /= (s[i] - 48);
    product *= (s[i + len] - 48);
    if (product > max) max = product;
  }
  
  return max;
}

int main() {
  string s = "73167176531330624919225119674426574742355349194934" 
             "96983520312774506326239578318016984801869478851843"
             "85861560789112949495459501737958331952853208805511"
             "12540698747158523863050715693290963295227443043557"
             "66896648950445244523161731856403098711121722383113"
             "62229893423380308135336276614282806444486645238749"
             "30358907296290491560440772390713810515859307960866"
             "70172427121883998797908792274921901699720888093776"
             "65727333001053367881220235421809751254540594752243"
             "52584907711670556013604839586446706324415722155397"
             "53697817977846174064955149290862569321978468622482"
             "83972241375657056057490261407972968652414535100474"
             "82166370484403199890008895243450658541227588666881"
             "16427171479924442928230863465674813919123162824586"
             "17866458359124566529476545682848912883142607690042"
             "24219022671055626321111109370544217506941658960408"
             "07198403850962455444362981230987879927244284909188"
             "84580156166097919133875499200524063689912560717606"
             "05886116467109405077541002256983155200055935729725"
             "71636269561882670428252483600823257530420752963450";
  long long max = 0;
  // loop through and get strings of numbers between 0s
  while (s.length() > 0) {
    int end = s.find("0");
    string ss = s.substr(0, end);
    
    // only find maxProduct is longer than 13
    // update max if it is bigger
    if (ss.length() > 13 && maxProduct(ss, 13) > max)
      max = maxProduct(ss, 13);
    
    // remove processed substring from the front
    s.erase(0, end + 1);
  }
  std::cout << max << std::endl;
}

Answer: 23514624000