Add more solutions

Added solutions for problems 76, 77, 78, 79 and 80 in C and python.

C/keylog.txt

@@ -0,0 +1,50 @@
+319
+680
+180
+690
+129
+620
+762
+689
+762
+318
+368
+710
+720
+710
+629
+168
+160
+689
+716
+731
+736
+729
+316
+729
+729
+710
+769
+290
+719
+680
+318
+389
+162
+289
+162
+718
+729
+319
+790
+680
+890
+362
+319
+760
+316
+729
+380
+319
+728
+716

C/p076.c

@@ -6,8 +6,8 @@
  * 2 + 2 + 1
  * 2 + 1 + 1 + 1
  * 1 + 1 + 1 + 1 + 1
-
+ *
-How many different ways can one hundred be written as a sum of at least two positive integers?*/
+ * How many different ways can one hundred be written as a sum of at least two positive integers?*/
 
 #include <stdio.h>
 #include <stdlib.h>
@@ -32,7 +32,7 @@ int main(int argc, char **argv)
    /* The number of ways a number can be written as a sum is given by the partition function
     * (-1 because the partition function includes also the number itself).
     * The function is implemented in projecteuler.c*/
-	n = partition_fn(100, partitions) - 1;
+	n = partition_fn(100, partitions, -1) - 1;
 
    free(partitions);
 

C/p077.c

@@ -0,0 +1,83 @@
+/* It is possible to write ten as the sum of primes in exactly five different ways:
+ *
+ * 7 + 3
+ * 5 + 5
+ * 5 + 3 + 2
+ * 3 + 3 + 2 + 2
+ * 2 + 2 + 2 + 2 + 2
+ *
+ * What is the first value which can be written as the sum of primes in over five thousand different ways?*/
+
+#include <stdio.h>
+#include <stdlib.h>
+#include <math.h>
+#include <time.h>
+#include "projecteuler.h"
+
+int count(int value, int n, int i, int target);
+   
+int primes[100];
+
+int main(int argc, char **argv)
+{
+	int i, j, n;
+	double elapsed;
+	struct timespec start, end;
+
+	clock_gettime(CLOCK_MONOTONIC, &start);
+
+   /* Generate a list of the first 100 primes.*/
+	for(i = 0, j = 0; j < 100; i++)
+	{
+		if(is_prime(i))
+      {
+			primes[j++] = i;
+      }
+	}
+
+	i = 1;
+
+   /* Use a function to count the number of prime partitions for 
+    * each number >= 2 until the one that can be written in over
+    * 5000 ways is found.*/
+	while((n = count(0, 0, 0, ++i)) <= 5000);
+
+	clock_gettime(CLOCK_MONOTONIC, &end);
+
+	elapsed = (end.tv_sec - start.tv_sec) + (double)(end.tv_nsec - start.tv_nsec) / 1000000000;
+
+   printf("Project Euler, Problem 77\n");
+	printf("Answer: %d\n", i);
+
+	printf("Elapsed time: %.9lf seconds\n", elapsed);
+
+	return 0;
+}
+
+/* Function using a simple recursive brute force approach
+ * to find all the partitions.*/
+int count(int value, int n, int i, int target)
+{
+	int j;
+
+	for(j = i; j < 100; j++)
+	{
+		value += primes[j];
+
+		if(value == target)
+      {
+			return n + 1;
+      }
+		else if(value > target)
+      {
+			return n;
+      }
+		else
+		{
+			n = count(value, n, j, target);
+			value -= primes[j];
+		}
+	}
+
+	return n;
+}

C/p078.c

@@ -0,0 +1,44 @@
+/* Let p(n) represent the number of different ways in which n coins can be separated into piles.
+ * For example, five coins can be separated into piles in exactly seven different ways, so p(5)=7.
+ *
+ *       OOOOO
+ *      OOOO  O
+ *      OOO  OO
+ *     OOO  O  O
+ *     OO  OO  O
+ *    OO  O  O  O
+ *   O  O  O  O  O
+ *
+ * Find the least value of n for which p(n) is divisible by one million.*/
+
+#include <stdio.h>
+#include <stdlib.h>
+#include <time.h>
+#include "projecteuler.h"
+
+#define N 1000000
+
+int main(int argc, char **argv)
+{
+   int i = -1;
+   long int partitions[N] = {0};
+   double elapsed;
+   struct timespec start, end;
+
+   clock_gettime(CLOCK_MONOTONIC, &start);
+
+   /* Using the partition function to calculate the number of partitions,
+    * giving the result modulo N.*/   
+   while(partition_fn(++i, partitions, N) != 0);
+	
+   clock_gettime(CLOCK_MONOTONIC, &end);
+
+   elapsed = (end.tv_sec - start.tv_sec) + (double)(end.tv_nsec - start.tv_nsec) / 1000000000;
+
+   printf("Project Euler, Problem 78\n");
+   printf("Answer: %d\n", i);
+
+   printf("Elapsed time: %.9lf seconds\n", elapsed);
+
+   return 0;
+}

C/p079.c

@@ -0,0 +1,189 @@
+/* A common security method used for online banking is to ask the user for three random characters from a passcode.
+ * For example, if the passcode was 531278, they may ask for the 2nd, 3rd, and 5th characters; the expected reply would be: 317.
+ *
+ * The text file, keylog.txt, contains fifty successful login attempts.
+ *
+ * Given that the three characters are always asked for in order, analyse the file so as to determine the shortest possible
+ * secret passcode of unknown length.*/
+
+#include <stdio.h>
+#include <stdlib.h>
+#include <time.h>
+#include "projecteuler.h"
+
+int compare(void *a, void *b);
+int check_passcode(int **passcode, int len, int **logins, int n);
+
+int main(int argc, char **argv)
+{
+   int i, j, keylog, len = 4, found = 0, digits[10] = {0}, passcode_digits[10] = {0}, **passcode, **logins;
+   char line[5];
+   FILE *fp;
+   double elapsed;
+   struct timespec start, end;
+
+   clock_gettime(CLOCK_MONOTONIC, &start);
+
+   if((fp = fopen("keylog.txt", "r")) == NULL)
+   {
+      fprintf(stderr, "Error while opening file %s\n", "keylog.txt");
+      return 1;
+   }
+
+   if((logins = (int **)malloc(50*sizeof(int*))) == NULL)
+   {
+      fprintf(stderr, "Error while allocating memory\n");
+      return 1;
+   }
+
+   for(i = 0; i < 50; i++)
+   {
+      if((logins[i] = (int *)malloc(3*sizeof(int))) == NULL)
+      {
+         fprintf(stderr, "Error while allocating memory\n");
+         return 1;
+      }
+   }
+
+   i = 0;
+
+   while(fscanf(fp, "%s", line) != EOF)
+   {
+      j = 2;
+      keylog = atoi(line);
+
+      while(keylog > 0)
+      {
+         logins[i][j--] = keylog % 10;
+         /* Check which digits are present in the login attempts.*/
+         digits[keylog%10]++;
+         keylog /= 10;
+      }
+
+      i++;
+   }
+
+   fclose(fp);
+
+   j = 0;
+   for(i = 0; i < 10; i++)
+   {
+      /* To generate the passcode, only use the digits present in the
+       * login attempts.*/
+      if(digits[i] > 0)
+      {
+         passcode_digits[j++] = i;
+      }
+   }
+
+   while(!found)
+   {
+      if((passcode = (int **)malloc(len*sizeof(int *))) == NULL)
+      {
+         fprintf(stderr, "Error while allocating memory\n");
+         return 1;
+      }
+
+      for(i = 0; i < len; i++)
+      {
+         if((passcode[i] = (int *)malloc(sizeof(int))) == NULL)
+         {
+            fprintf(stderr, "Error while allocating memory\n");
+            return 1;
+         }
+         /* For the current length, generate the first passcode with the
+          * digits in order.*/
+         *passcode[i] = passcode_digits[i];
+      }
+
+      /* Check if the passcode is compatible with the login attempts.*/
+      if(check_passcode(passcode, len, logins, 50))
+      {
+         found = 1;
+         break;
+      }
+
+      /* For the given length, check every permutation until the correct
+       * passcode has been found, or all the permutations have been tried.*/
+      while(next_permutation((void **)passcode, len, compare) != -1)
+      {
+         if(check_passcode(passcode, len, logins, 50))
+         {
+            printf("Project Euler, Problem 79\n");
+            printf("Answer: ");
+            for(i = 0; i < len; i++)
+               printf("%d", *passcode[i]);
+            printf("\n");
+
+            found = 1;
+            break;
+         }
+      }
+
+      for(i = 0; i < len; i++)
+      {
+         free(passcode[i]);
+      }
+      free(passcode);
+
+      /* If the passcode has not yet been found, try a longer passcode.*/ 
+      len++;
+   }
+
+   for(i = 0; i < 50; i++)
+   {
+      free(logins[i]);
+   }
+
+   free(logins);
+   
+   clock_gettime(CLOCK_MONOTONIC, &end);
+
+   elapsed = (end.tv_sec - start.tv_sec) + (double)(end.tv_nsec - start.tv_nsec) / 1000000000;
+
+   printf("Elapsed time: %.9lf seconds\n", elapsed);
+
+   return 0;
+}
+
+int compare(void *a, void *b)
+{
+   int *n1, *n2;
+
+   n1 = (int *)a;
+   n2 = (int *)b;
+
+   return *n1 - *n2;
+}
+
+int check_passcode(int **passcode, int len, int **logins, int n)
+{
+   int i, j, k;
+
+   /* For every login attempt, check if all the digits appear in the
+    * passcode in the correct order. Return 0 if a login attempt 
+    * incompatible with the current passcode is found.*/
+   for(i = 0; i < n; i++)
+   {
+      k = 0;
+      for(j = 0; j < len; j++)
+      {
+         if(*passcode[j] == logins[i][k])
+         {
+            k++;
+            
+            if(k == 3)
+            {
+               break;
+            }
+         }
+      }
+
+      if(k < 3)
+      {
+         return 0;
+      }
+   }
+
+   return 1;
+}

C/p080.c

@@ -0,0 +1,80 @@
+/* It is well known that if the square root of a natural number is not an integer, then it is irrational.
+ * The decimal expansion of such square roots is infinite without any repeating pattern at all.
+ *
+ * The square root of two is 1.41421356237309504880..., and the digital sum of the first one hundred decimal digits is 475.
+ *
+ * For the first one hundred natural numbers, find the total of the digital sums of the first one hundred decimal digits
+ * for all the irrational square roots*/
+
+#include <stdio.h>
+#include <stdlib.h>
+#include <math.h>
+#include <time.h>
+#include <gmp.h>
+
+int is_square(int n);
+
+int main(int argc, char **argv)
+{
+   int i, j, sum = 0;
+   char sqrt_digits[104];
+   double elapsed;
+   struct timespec start, end;
+   mpf_t sqrt;
+
+   clock_gettime(CLOCK_MONOTONIC, &start);
+
+   /* Set the precision to 333 bits (should be enough for 100 decimal digits.*/
+   mpf_set_default_prec(333);
+   mpf_init(sqrt);
+
+   for(i = 2; i < 100; i++)
+   {
+      if(is_square(i))
+      {
+         continue;
+      }
+
+      /* Calculate the square root of the current number with the given precision
+       * and sum the digits to the total.*/
+      mpf_sqrt_ui(sqrt, i);
+      gmp_sprintf(sqrt_digits, "%.101Ff\n", sqrt);
+      sum += (sqrt_digits[0] - '0');
+
+      for(j = 2; j < 101; j++)
+      {
+         sum += (sqrt_digits[j] - '0');
+      }
+   }
+
+   mpf_clear(sqrt);
+   
+   clock_gettime(CLOCK_MONOTONIC, &end);
+
+   elapsed = (end.tv_sec - start.tv_sec) + (double)(end.tv_nsec - start.tv_nsec) / 1000000000;
+
+   printf("Project Euler, Problem 80\n");
+   printf("Answer: %d\n", sum);
+
+   printf("Elapsed time: %.9lf seconds\n", elapsed);
+
+   return 0;
+}
+
+int is_square(int n)
+{
+   int m;
+   double p;
+
+   p = sqrt(n);
+   m = p;
+
+   if(p == m)
+   {
+      return 1;
+   }
+   else
+   {
+      return 0;
+   }
+}

C/projecteuler.c

@@ -785,7 +785,7 @@ int phi(int n, int *primes)
 }
 
 /* Function implementing the partition function.*/
-long int partition_fn(int n, long int *partitions)
+long int partition_fn(int n, long int *partitions, int mod)
 {
    int k, limit;
    long int res = 0;
@@ -800,6 +800,7 @@ long int partition_fn(int n, long int *partitions)
    if(n == 0)
    {
       partitions[n] = 1;
+
       return 1;
    }
 
@@ -816,12 +817,22 @@ long int partition_fn(int n, long int *partitions)
    {
       if(k != 0)
       {
-         res += pow(-1, k+1) * partition_fn(n-k*(3*k-1)/2, partitions);
+         res += pow(-1, k+1) * partition_fn(n-k*(3*k-1)/2, partitions, mod);
       }
       k++;
    }
 
+   /* Give the result modulo mod, if mod=!-1, otherwise give the full result.*/
+   if(mod != -1)
+   {
+      partitions[n] = res % mod;
+
+      return res % mod;
+   }
+   else
+   {
       partitions[n] = res;
 
       return res;
    }
+}

C/projecteuler.h

@@ -24,6 +24,7 @@ int pell_eq(int i, mpz_t x);
 int is_semiprime(int n, int *p, int *q, int *primes);
 int phi_semiprime(int n, int p, int q);
 int phi(int n, int *primes);
-long int partition_fn(int n, long int *partitions);
+long int partition_fn(int n, long int *partitions, int mod);
+int partition_fn_mpz(int n, mpz_t res, mpz_t *partitions);
 
 #endif

Python/keylog.txt

@@ -0,0 +1,50 @@
+319
+680
+180
+690
+129
+620
+762
+689
+762
+318
+368
+710
+720
+710
+629
+168
+160
+689
+716
+731
+736
+729
+316
+729
+729
+710
+769
+290
+719
+680
+318
+389
+162
+289
+162
+718
+729
+319
+790
+680
+890
+362
+319
+760
+316
+729
+380
+319
+728
+716

Python/p077.py

@@ -0,0 +1,72 @@
+#!/usr/bin/python
+
+# It is possible to write ten as the sum of primes in exactly five different ways:
+#
+# 7 + 3
+# 5 + 5
+# 5 + 3 + 2
+# 3 + 3 + 2 + 2
+# 2 + 2 + 2 + 2 + 2
+#
+# What is the first value which can be written as the sum of primes in over five thousand different ways?
+
+from timeit import default_timer
+from projecteuler import is_prime
+
+# Function using a simple recursive brute force approach
+# to find all the partitions.
+def count(value, n, i, target):
+    global primes
+
+    for j in range(i, 100):
+        value = value + primes[j]
+
+        if value == target:
+            return n + 1
+        elif value > target:
+            return n
+        else:
+            n = count(value, n, j, target)
+            value = value - primes[j]
+
+    return n
+
+def main():
+    start = default_timer()
+
+    global primes
+
+    primes = [0] * 100
+
+#   Generate a list of the first 100 primes.
+    i = 0
+    j = 0
+
+    while j < 100:
+        if is_prime(i):
+            primes[j] = i
+            j = j + 1
+        i = i + 1
+
+    i = 2
+
+#   Use a function to count the number of prime partitions for
+#   each number >= 2 until the one that can be written in over
+#   5000 ways is found.
+    while True:
+        n = count(0, 0, 0, i)
+
+        if n > 5000:
+            break
+
+        i = i + 1
+
+    end = default_timer()
+
+    print('Project Euler, Problem 77')
+    print('Answer: {}'.format(i))
+
+    print('Elapsed time: {:.9f} seconds'.format(end - start))
+
+if __name__ == '__main__':
+    main()

Python/p078.py

@@ -0,0 +1,41 @@
+#!/usr/bin/python
+
+# Let p(n) represent the number of different ways in which n coins can be separated into piles.
+# For example, five coins can be separated into piles in exactly seven different ways, so p(5)=7.
+#
+#       OOOOO
+#      OOOO  O
+#      OOO  OO
+#     OOO  O  O
+#     OO  OO  O
+#    OO  O  O  O
+#   O  O  O  O  O
+#
+# Find the least value of n for which p(n) is divisible by one million.
+
+from timeit import default_timer
+from projecteuler import partition_fn
+
+def main():
+    start = default_timer()
+
+    N = 1000000
+
+    partitions = [0] * N
+
+    i = 0
+
+#   Using the partition function to calculate the number of partitions,
+#   giving the result modulo N.*/   
+    while partition_fn(i, partitions, N) != 0:
+        i = i + 1
+
+    end = default_timer()
+
+    print('Project Euler, Problem 78')
+    print('Answer: {}'.format(i))
+
+    print('Elapsed time: {:.9f} seconds'.format(end - start))
+
+if __name__ == '__main__':
+    main()

Python/p079.py

@@ -0,0 +1,105 @@
+#!/usr/bin/python
+
+# A common security method used for online banking is to ask the user for three random characters from a passcode.
+# For example, if the passcode was 531278, they may ask for the 2nd, 3rd, and 5th characters; the expected reply would be: 317.
+#
+# The text file, keylog.txt, contains fifty successful login attempts.
+#
+# Given that the three characters are always asked for in order, analyse the file so as to determine the shortest possible
+# secret passcode of unknown length.
+
+from itertools import permutations
+
+from timeit import default_timer
+
+def check_passcode(passcode, len_, logins, n):
+#   For every login attempt, check if all the digits appear in the
+#   passcode in the correct order. Return 0 if a login attempt 
+#   incompatible with the current passcode is found.
+    for i in range(n):
+        k = 0
+        for j in range(len_):
+            if passcode[j] == int(logins[i][k]):
+                k = k + 1
+
+                if k == 3:
+                    break
+
+        if k < 3:
+            return 0
+
+    return 1
+
+def main():
+    start = default_timer()
+
+    try:
+        fp = open('keylog.txt', 'r')
+    except:
+        print('Error while opening file {}'.format('keylog.txt'))
+        exit(1)
+
+    logins = fp.readlines()
+
+    fp.close()
+
+    digits = [0] * 10
+    passcode_digits = [0] * 10
+
+    for i in logins:
+        keylog = int(i)
+       
+#       Check which digits are present in the login attempts.
+        while True:
+            digits[keylog%10] = digits[keylog%10] + 1
+            keylog = keylog // 10
+
+            if keylog == 0:
+                break
+
+    j = 0
+    for i in range(10):
+#       To generate the passcode, only use the digits present in the
+#       login attempts.
+        if digits[i] > 0:
+            passcode_digits[j] = i
+            j = j + 1
+
+    found = 0
+    len_ = 4
+
+    while not found:
+        passcode = [0] * len_
+
+#       For the current length, generate the first passcode with the
+#       digits in order.
+        for i in range(len_):
+            passcode[i] = passcode_digits[i]
+
+#       Check if the passcode is compatible with the login attempts.
+        if check_passcode(passcode, len_, logins, 50):
+            found = 1
+            break
+
+#       For the given length, check every permutation until the correct
+#       passcode has been found, or all the permutations have been tried.
+        passcodes = permutations(passcode, len_)
+
+        for i in passcodes:
+            if check_passcode(i, len_, logins, 50):
+                found = 1
+                res = ''.join(map(str, i))
+                break
+
+#       If the passcode has not yet been found, try a longer passcode.
+        len_ = len_ + 1
+
+    end = default_timer()
+
+    print('Project Euler, Problem 79')
+    print('Answer: {}'.format(res))
+
+    print('Elapsed time: {:.9f} seconds'.format(end - start))
+
+if __name__ == '__main__':
+    main()

Python/p080.py

@@ -0,0 +1,51 @@
+#!/usr/bin/python
+
+# It is well known that if the square root of a natural number is not an integer, then it is irrational.
+# The decimal expansion of such square roots is infinite without any repeating pattern at all.
+#
+# The square root of two is 1.41421356237309504880..., and the digital sum of the first one hundred decimal digits is 475.
+#
+# For the first one hundred natural numbers, find the total of the digital sums of the first one hundred decimal digits
+# for all the irrational square roots
+
+from mpmath import sqrt, mp
+
+from timeit import default_timer
+
+def is_square(n):
+    p = sqrt(n)
+    m = int(p)
+
+    if p == m:
+        return True
+    else:
+        return False
+
+def main():
+    start = default_timer()
+
+#   Set the precision to 100 digits
+    mp.dps = 102
+
+    sum_ = 0
+
+    for i in range(2, 100):
+        if not is_square(i):
+#           Calculate the square root of the current number with the given precision
+#           and sum the digits to the total.
+            root = str(sqrt(i))
+
+            sum_ = sum_ + int(root[0])
+
+            for j in range(2, 101):
+                sum_ = sum_ + int(root[j])
+
+    end = default_timer()
+
+    print('Project Euler, Problem 80')
+    print('Answer: {}'.format(sum_))
+
+    print('Elapsed time: {:.9f} seconds'.format(end - start))
+
+if __name__ == '__main__':
+    main()

Python/projecteuler.py

@@ -399,7 +399,7 @@ def phi(n, primes):
     return ph
 
 # Function implementing the partition function.
-def partition_fn(n, partitions):
+def partition_fn(n, partitions, mod=-1):
 #   The partition function for negative numbers is 0 by definition.
     if n < 0:
         return 0
@@ -422,8 +422,12 @@ def partition_fn(n, partitions):
             res = res + pow(-1, k+1) * partition_fn(n-k*(3*k-1)//2, partitions)
         k = k + 1
 
-    partitions[n] = res
+#   Give the result modulo mod, if mod!=-1, otherwise give the full result.
+    if mod != -1:
+        partitions[n] = res % mod
+
+        return res % mod
+    else:
+        partitions[n] = int(res)
 
         return int(res)
-
-