Project Euler #154: Exploring Pascal's pyramid.

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  • + 0 comments

    turns out my pypy3 can not do it, but c++ can.

  • + 0 comments

    It seems we can solve all test cases taking advantage of the symmetry of Pascal pyramid level. Every j element in row i (rows are enumarated from very bottom where i = 0 means bottom row) in level n of Pascal pyramid is given by the formula: n! / (i! * j! * (n - i - j)!). Due to symmetry we need to consider only one third of rows. Last but not least, it's very helpful to precompute given prime exponents in all factorials.

  • + 1 comment
    #include <iostream>
    #include <vector>
    
    typedef std::vector<unsigned int> Exponents;
    
    // return exponent of a prime factor of C(n,k)
    // looks a bit like the logarithm:
    // C(n,k) = n! / ((n-k)! * k!)
    unsigned int choose(const Exponents& sums, unsigned int n, unsigned int k)
    {
      return sums[n] - (sums[n - k] + sums[k]);
    }
    
    int main()
    {
      unsigned int layer     = 200000;
       // 10^12 = 2^12 * 5^12
      unsigned int prime1    =  2;
      unsigned int exponent1 = 12;
      unsigned int prime2    =  5;
      unsigned int exponent2 = 12;
      std::cin >> layer >> prime1 >> exponent1 >> prime2 >> exponent2;
    
      // analyze for each number between 0 and layer how often they contain prime1 and prime2
      Exponents mulPrime1 = { 0 };
      Exponents mulPrime2 = { 0 };
      for (unsigned int x = 1; x <= layer; x++)
      {
        auto current = x;
        unsigned int count = 0;
        // extract first prime (=2) as often as possible
        while (current % prime1 == 0)
        {
          current /= prime1;
          count++;
        }
        mulPrime1.push_back(count);
    
        count = 0;
        // extract second prime (=5) as often as possible
        while (current % prime2 == 0)
        {
          current /= prime2;
          count++;
        }
        mulPrime2.push_back(count);
      }
    
      // sum1[x] = sum of mulPrime1[0 ... x]
      Exponents sum1;
      unsigned int count = 0;
      for (auto x : mulPrime1)
      {
        count += x;
        sum1.push_back(count);
      }
    
      // the same stuff for the other prime
      Exponents sum2;
      count = 0;
      for (auto x : mulPrime2)
      {
        count += x;
        sum2.push_back(count);
      }
    
      unsigned long long result = 0;
      for (unsigned int i = 0; i <= layer; i++)
      {
        // how often is each prime used by C(layer, i) ?
        auto found1 = choose(sum1, layer, i);
        auto found2 = choose(sum2, layer, i);
    
        // already enough ?
        if (found1 >= exponent1 && found2 >= exponent2)
        {
          // no need to enter the inner-most loop, each iteration would succeed
          result += i + 1;
          continue;
        }
    
        // note: abort early because of mirrored values
        for (unsigned int j = 0; j <= (i+1) / 2; j++)
        {
          if (found1 + choose(sum1, i, j) >= exponent1 &&
              found2 + choose(sum2, i, j) >= exponent2)
          {
            // found a match
            result++;
            // left and right side are identical
            if (j < i / 2)
              result++;
          }
        }
      }
    
      // and we're done !
      std::cout << result << std::endl;
      return 0;
    }
    
  • + 0 comments

    im getting a floating point exception.

  • + 1 comment

    if n=0 , p0=1 , α0=0 , p1=1 , α1=0; WHAT SHOULD BE THE OUTPUT FOR THIS INPUT:)