recurze / ProjectEuler

ProjectEuler

This repository contains solutions to the problems of Project Euler. Check out the official website and the programming versions over at hackerrank here with higher input sizes thus asking for a faster solution.

1. Multiples of 3 and 5:

   • Use the formula \sum i = 1 + 2 + \dots + n = n(n+1)/2 to sum multiples of 3 and 5 seperately.
   • Make sure to not count any number twice (multiples of both 3 and 5 like 15)
   • Note the upperlimit.
     Ans: 233168

2. Even Fibonacci numbers:

   • Fibonacci numbers grow fast (factor of \phi ~ 1.6, golden ratio).
   • The upper limit (4 million) and the one from the ProjectEuler+ is crossed by just the 34th number and the 82nd number.
   • The parity of the number sequence: oeooeooeooeooe. We need to calculate only every 3rd number.
   • a, b = 2, 3 and a, b = a+2b, 2a+3b where a is the even fibonacchi's
   • Bonus: Check out finding nth fibonacci in O(lgn) instead of O(n)
     Ans: 4613732

3. Largest Prime factor:

   • The only prime factor greater than \sqrt{n} would be n itself (if it were prime)
   • Thus, finding the largest prime factor would be O(\sqrt{n})
   • Bonus: All primes are of the form 6n+1 or 6n+5
     Ans: 6857

4. Largest palindrome product:

   • There are only 1000 3- digit number and all combinations can be enumerated with ease.
   • Iterate over multiples of 11.
   • Binary search for the largest number less than n. Or simple linear search will work too because the number of palindromes is just 2470.
     Ans: 906609

5. Smallest multiple:

   • Find LCM of all numbers from 1 to N, or
   • Get prime factorization and find LCM using this.
   • LCM = ab/GCD(a, b)
     Ans: 232792560

6. Sum square difference:

   • \sum i^2 = n(n+1)(2n+1)/6
   • (\sum i)^2 = n^2(n+1)^2/4
     Ans: 25164150

7. 10001st prime:

   • Use sieve method to list all primes
     Ans: 104743

8. Largest product in a series:

   • Rolling window
     Ans: 23514624000

9. Special Pythogorean triplets:

   • Solve the given 2 equations for b interms of a.
   • Run a loop from 1 to n/3 for a with wlog a < b < c

10. Summation of primes:

    • List all the primes below 2 million and sum them up
      Ans: 142913828922

11. Highy divisible triangular number:

    • Number of divisors of 2^a3^b5^c... = (a + 1)(b + 1)(c + 1)...
    • We could find number of divisors of every number using a sieve.
    • nD(n(n+1)/2) = nD(n)nD((n+1)/2) or nD(n/2)nD(n+1) based on parity of n.
      Ans: 76576500

12. Large sum:

    • Use library big integers
      Ans: 5537376230

13. Longest Collatz sequence:

    • Memoize upto 10**6 numbers
    • Compute prefix max argument
      Ans: 837799

14. Lattice paths:

    • Ans is ^2nCn
      Ans: 137846528820

15. Power Digit sum:

    • Store huge numbers as char array and simulate multiplying by hand.
      Ans: 1366

16. Maximum path sum I:

    • DP
      Ans: 1074

17. Factorial digit sum

    • Library big integers
      Ans: 648

18. Amicable numbers:

    • Sum of divisors = pi(pi^mi+1 - 1)/(pi - 1) where n = pi^mi

19. Lexicographi permutations:

    • Read about factorial number systems
      Ans: 2783915460

20. 1000-digit Fibonacci number:

    • F(n) = floor((phi^2)/(5^1/2) + 1/2)
    • Number of digits in n = ceil(log10(n))
      Ans: 4782

21. Reciprocal cycles:

    • 1/y = 0.000...0.cycle.cycle.cycle...
    • 10^i/y = 0.cycle.cycle.cycle...
    • 10^i+c/y = cycle.cycle.cycle....
    • (10^i)(10^c - 1)/y = cycle
    • y must divide 10^i or 10^c - 1
      Ans: 983

22. Quadratic primes:

    • f(0) = b which needs to be prime.
    • Try all possible values of n and test f(n) for prime.
      Ans: -59231 (-61, 971)

23. Number spiral diagnols:

    • Smallest corner in kth square = 1 + 4k(k+1) - 6k, solved recursively
    • Sum till kth square = 1 + 4k + 2k(k+1)(8k+7)/3, solved recursively
      Ans: 669171001

24. Distinct powers:

    • a1^b1 = a2^b2 iff a1 = c^i1 and a2 = c^i2 given all are integers
    • Every exponent of a = c^k is repeated iff jk <= in, 0 < i < k
    • O(nlgn^2), although can do better, O(nlgn) Ans: 9183

25. Digit fifth powers:

    • Iterate over all integers with an upper limit of n9^n where n is the number of digits Ans: 443839