Wednesday, December 14, 2011

Project Euler and Me

Thanks to Ailin and Adam that introduce me the Project Euler and now I am totally addicted with it. After 8 years not using C as my primary programming language, is time for me to pick it up and do something with C ideology - small footprint with great performance!

Below is the question from Project Euler and the code I had written to solve it. Feel free to download them then make it better or give comment and suggestion on the solution. Programmer ROCKS!

IDDescription / Title
1Add all the natural numbers below one thousand that are multiples of 3 or 5.
2By considering the terms in the Fibonacci sequence whose values do not exceed four million, find the sum of the even-valued terms.
3Find the largest prime factor of a composite number.
4Find the largest palindrome made from the product of two 3-digit numbers.
5What is the smallest number divisible by each of the numbers 1 to 20?
6What is the difference between the sum of the squares and the square of the sums?
7Find the 10001st prime.
8Discover the largest product of five consecutive digits in the 1000-digit number.
9Find the only Pythagorean triplet, {a, b, c}, for which a + b + c = 1000.
10Calculate the sum of all the primes below two million.
11What is the greatest product of four adjacent numbers on the same straight line in the 20 by 20 grid?
12What is the value of the first triangle number to have over five hundred divisors?
13Find the first ten digits of the sum of one-hundred 50-digit numbers.
14Find the longest sequence using a starting number under one million.
15Starting in the top left corner in a 20 by 20 grid, how many routes are there to the bottom right corner?
16What is the sum of the digits of the number 21000?
17How many letters would be needed to write all the numbers in words from 1 to 1000?
18Find the maximum sum travelling from the top of the triangle to the base.
19How many Sundays fell on the first of the month during the twentieth century?
20Find the sum of digits in 100!
21Evaluate the sum of all amicable pairs under 10000.
22What is the total of all the name scores in the file of first names?
23Find the sum of all the positive integers which cannot be written as the sum of two abundant numbers.
24What is the millionth lexicographic permutation of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9?
25What is the first term in the Fibonacci sequence to contain 1000 digits?
26Find the value of d < 1000 for which 1/d contains the longest recurring cycle.
27Find a quadratic formula that produces the maximum number of primes for consecutive values of n.
28What is the sum of both diagonals in a 1001 by 1001 spiral?
29How many distinct terms are in the sequence generated by ab for 2 = a = 100 and 2 = b = 100?
30Find the sum of all the numbers that can be written as the sum of fifth powers of their digits.
31Investigating combinations of English currency denominations.
32Find the sum of all numbers that can be written as pandigital products.
33Discover all the fractions with an unorthodox cancelling method.
34Find the sum of all numbers which are equal to the sum of the factorial of their digits.
35How many circular primes are there below one million?
36Find the sum of all numbers less than one million, which are palindromic in base 10 and base 2.
37Find the sum of all eleven primes that are both truncatable from left to right and right to left.
38What is the largest 1 to 9 pandigital that can be formed by multiplying a fixed number by 1, 2, 3, ... ?
39If p is the perimeter of a right angle triangle, {a, b, c}, which value, for p = 1000, has the most solutions?
40Finding the nth digit of the fractional part of the irrational number.
41What is the largest n-digit pandigital prime that exists?
42How many triangle words does the list of common English words contain?
43Find the sum of all pandigital numbers with an unusual sub-string divisibility property.
44Find the smallest pair of pentagonal numbers whose sum and difference is pentagonal.
45After 40755, what is the next triangle number that is also pentagonal and hexagonal?
46What is the smallest odd composite that cannot be written as the sum of a prime and twice a square?
47Find the first four consecutive integers to have four distinct primes factors.
48Find the last ten digits of 11 + 22 + ... + 10001000.
49Find arithmetic sequences, made of prime terms, whose four digits are permutations of each other.
50Which prime, below one-million, can be written as the sum of the most consecutive primes?

Problem 15

Using programmatic method with serial navigating technique take 2 hours 15 minutes to get the result. If using mathematical method to calculate the possible path only take less then 1 second. The difference between these two methods is mathematical method only projected how many path there are but programmatic method give the actual path it had been navigated.

In my opinion, the real world problem like solving maze or path finding, programmatic method is more realistic. However, mathematical method can help to pre-project the result. The next thing I will going to do is design a parallel path navigating technique using multi-thread with multiple navigator, this solution will combine both mathematical  and programmatic method to accomplish the objective.


Problem 15 with Serial Navigating Solution
Path for 1x1. P: 2 (0s 2ms), M: 2 (0s 381ms)
Path for 2x2. P: 6 (0s 2ms), M: 6 (0s 11ms)
Path for 3x3. P: 20 (0s 4ms), M: 20 (0s 12ms)
Path for 4x4. P: 70 (0s 9ms), M: 70 (0s 12ms)
Path for 5x5. P: 252 (0s 50ms), M: 252 (0s 13ms)
Path for 6x6. P: 924 (0s 101ms), M: 924 (0s 13ms)
Path for 7x7. P: 3432 (0s 370ms), M: 3432 (0s 16ms)
Path for 8x8. P: 12870 (0s 1479ms), M: 12870 (0s 16ms)
Path for 9x9. P: 48620 (0s 5363ms), M: 48620 (0s 21ms)
Path for 10x10. P: 184756 (0s 18646ms), M: 184756 (0s 15ms)
Path for 11x11. P: 705432 (0s 40020ms), M: 705432 (0s 17ms)
Path for 12x12. P: 2704156 (0s 167027ms), M: 2704156 (0s 17ms)
Path for 13x13. P: 10400600 (0s 587571ms), M: 10400600 (0s 19ms)
Path for 14x14. P: 40116600 (3s -715159ms), M: 40116600 (0s 20ms)
Path for 15x15. P: 155117520 (8s 773356ms), M: 155117520 (0s 19ms)
Path for 16x16. P: 601080390 (36s -370565ms), M: 601080390 (0s 81ms)
Path for 17x17. P: 2333606220 (138s 83071ms), M: 2333606220 (0s 104ms)
Path for 18x18. P: 9075135300 (546s -580371ms), M: 9075135300 (0s 68ms)
Path for 19x19. P: 35345263800 (2104s 60565ms), M: 35345263800 (0s 44ms)
Path for 20x20. P: 137846528820 (8236s 7101ms), M: 137846528820 (0s 40ms)

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