From 294d48872c0c3c1a4dbe9a9f7fc48510f058bf2e Mon Sep 17 00:00:00 2001
From: Tibor Bizjak <tibor.bizjak@gmail.com>
Date: Mon, 27 Mar 2023 16:30:58 +0200
Subject: [PATCH] Added solutions for N=60-80

---
 061_cyclical_fig_nums.py               | 53 ++++++++++++++++
 062_cubic_permutations.py              | 31 ++++++++++
 064_odd_period_square_roots.c          | 39 ++++++++++++
 065_convergents_of_e.py                | 16 +++++
 066_diophantine_equation.py            | 58 ++++++++++++++++++
 068_magic_5gon_ring.py                 | 40 ++++++++++++
 074_digit_factorial_chains.c           | 84 ++++++++++++++++++++++++++
 075_singular_integer_right_triangles.c | 42 +++++++++++++
 076_counting_summations.c              | 28 +++++++++
 077_prime_summations.py                | 30 +++++++++
 078_coin_partitions.py                 | 29 +++++++++
 makefile                               | 16 ++++-
 12 files changed, 464 insertions(+), 2 deletions(-)
 create mode 100644 061_cyclical_fig_nums.py
 create mode 100644 062_cubic_permutations.py
 create mode 100644 064_odd_period_square_roots.c
 create mode 100644 065_convergents_of_e.py
 create mode 100644 066_diophantine_equation.py
 create mode 100644 068_magic_5gon_ring.py
 create mode 100644 074_digit_factorial_chains.c
 create mode 100644 075_singular_integer_right_triangles.c
 create mode 100644 076_counting_summations.c
 create mode 100644 077_prime_summations.py
 create mode 100644 078_coin_partitions.py

diff --git a/061_cyclical_fig_nums.py b/061_cyclical_fig_nums.py
new file mode 100644
index 0000000..fdf7a52
--- /dev/null
+++ b/061_cyclical_fig_nums.py
@@ -0,0 +1,53 @@
+#!/usr/bin/env python3
+
+from math import sqrt
+
+def cynical():
+    #main
+    i=1+int((2+sqrt(4+12*1009))/6)
+    o=3*i**2-2*i
+    while o<10000:
+        base=(o-(o//100)*100)*100
+        if base>=1000:
+            a=curse(base,o//100,[])
+            if a!=False:
+                return a+[o]
+        i+=1
+        o=3*i**2-2*i
+    return False
+
+def curse(base,last,sit):
+    #recursive
+    if len(sit)==4:
+        if fig(base+last)!=False and not(fig(base*100+last) in sit):
+            return [base+last]
+    else:
+        i=10
+        while i<100:
+            if fig(base+i)!=False and not(fig(base+i) in sit):
+                a=curse(i*100,last,sit+[fig(base+i)])
+                if a!=False:
+                    return a+[base+i]
+            i+=1
+    return False
+
+def fig(x):
+    #pocekira ce je x katera izmed figurative stevil
+    if int(sqrt(x))==sqrt(x):
+        return 2
+    k=(1+sqrt(1+24*x))/3
+    if int(k)==k:
+        return 3
+    k=sqrt(1+8*x)
+    if int(k)==k:
+        k=(1+sqrt(1+8*x))/4
+        if int(k)==k:
+            return 4
+        return 1
+    k=(3+sqrt(9+40*x))/5
+    if int(k)==k:
+        return 5
+    return False
+
+if __name__ == '__main__':
+    print(sum(cynical()))
diff --git a/062_cubic_permutations.py b/062_cubic_permutations.py
new file mode 100644
index 0000000..81e14e0
--- /dev/null
+++ b/062_cubic_permutations.py
@@ -0,0 +1,31 @@
+#!/usr/bin/env python3
+
+def cube(x):
+    i=3
+    lim=100
+    while True:
+        cubes=[]
+        while i**3<lim:
+            cubes.append(i**3)
+            i+=1
+        lim*=10
+        perm={}
+        ans=[]
+        ans_route={}
+        for j in cubes:
+            a="".join(sorted(str(j)))
+            try:
+                perm[a][0]+=1
+                if perm[a][0]==x:
+                    ans.append(perm[a][1])
+                    ans_route[perm[a][1]]=a
+            except:
+                perm[a]=[1,j]
+
+        if len(ans)>0:
+            for i in sorted(ans):
+                if perm[ans_route[i]][0]==x:
+                    return i
+
+if __name__ == '__main__':
+    print(cube(5))
diff --git a/064_odd_period_square_roots.c b/064_odd_period_square_roots.c
new file mode 100644
index 0000000..25dcd07
--- /dev/null
+++ b/064_odd_period_square_roots.c
@@ -0,0 +1,39 @@
+#include <stdio.h>
+#include <math.h>
+
+int sq_frac(int x);
+
+int N = 10000;
+
+int main(int argc, char *argv[])
+{
+    int i, c;
+    
+    c = 0;
+    for(i = 1; i <= N; ++i){
+        if(sq_frac(i) % 2 == 0){
+            ++c;
+        }
+    }
+    printf("%d\n", c);
+    return 0;
+}
+
+int sq_frac(int x)
+{
+    int m, n, d, a, c;
+    
+    if((m = sqrt(x)) * m == x){
+        return 1;
+    }
+    d = c = 1;
+    n = m;
+    do{
+        d = (x - n * n) / d;
+        a = (n + m) / d;
+        n = a * d - n;
+        ++c;
+    }while(d != 1);
+    return c;
+}
+
diff --git a/065_convergents_of_e.py b/065_convergents_of_e.py
new file mode 100644
index 0000000..3912451
--- /dev/null
+++ b/065_convergents_of_e.py
@@ -0,0 +1,16 @@
+#!/usr/bin/env python3
+
+def nth_con(x):
+    # 212 114 116 118
+    if not x % 3:
+        n = 1
+        d = 0
+    else:
+        n = x % 3
+        d = 1
+    for m in range(2 * (x // 3), 2, -2):
+        n, d = 2*m*n + 2*d + n, m*n + n + d
+    return 8*n + 3*d, 2*n + d + n
+
+if __name__ == '__main__':
+    print(sum(map(int, list(str(nth_con(100)[0])))))
diff --git a/066_diophantine_equation.py b/066_diophantine_equation.py
new file mode 100644
index 0000000..152abe4
--- /dev/null
+++ b/066_diophantine_equation.py
@@ -0,0 +1,58 @@
+#!/usr/bin/env python3
+
+from math import sqrt
+
+def main(d):
+    mx = 0
+    my = 0
+    md = 0
+    for cd in range(2, d + 1):
+        x, y = diop(cd)
+        #print("{} -> {}, {}".format(cd, x, y))
+        if x > mx:
+            mx = x
+            my = y
+            md = cd
+    return md, mx, my
+
+def diop(d):
+    sol = []
+    a, f = sq_frac(d)
+    f.reverse()
+    if len(f) == 0:
+        return 0, 0
+    for i in f:
+        for j in range(len(sol)):
+            sol[j][0], sol[j][1] = sol[j][0] * i + sol[j][1], sol[j][0]
+        sol.append([i, 1])
+    r = fits(a, sol, d)
+    while r == 0:
+        for i in f:
+            for j in range(len(sol)):
+                sol[j][0], sol[j][1] = sol[j][0] * i + sol[j][1], sol[j][0]
+        r = fits(a, sol, d)
+    return r
+
+def fits(a, pairs, d):
+    for x, y in pairs[::-1]:
+        x, y = a * x + y, x
+        if x**2 - d * y**2 == 1:
+            return x, y
+    return 0
+
+def sq_frac(x):
+    f = []
+    m = int(sqrt(x))
+    if m**2 == x:
+        return m, f
+    d = x - m**2
+    f.append((2 * m) // d)
+    n = f[-1] * d - m
+    while d != 1:
+        d = (x - n**2) // d
+        f.append((n + m) // d)
+        n = f[-1] * d - n
+    return m, f
+
+if __name__ == '__main__':
+    print(main(1000)[0])
diff --git a/068_magic_5gon_ring.py b/068_magic_5gon_ring.py
new file mode 100644
index 0000000..1634214
--- /dev/null
+++ b/068_magic_5gon_ring.py
@@ -0,0 +1,40 @@
+#!/usr/bin/env python3
+
+def m5gon(x, i, mx):
+    r = []
+    m = [[] for _ in range(5)]
+    for a, b in enumerate(range(i-x-1, (i-x)//2, -1), 1):
+        if a > mx:
+            break
+        elif b > mx or a == x or b == x:
+            continue
+        m[0] = [x, a, b]
+        for ci, c in enumerate(range(i-b-1, 0, -1), 1):
+            if ci > mx:
+                break
+            elif c > mx or len(set([a, b, x, ci, c])) < 5:
+                continue
+            m[1] = [ci, b, c]
+            for di, d in enumerate(range(i-c-1, 0, -1), 1):
+                if di > mx:
+                    break
+                elif d > mx or len(set([a, b, x, c, ci, d, di])) < 7:
+                    continue
+                m[2] = [di, c, d]
+                for ei, e in enumerate(range(i-d-1, 0, -1), 1):
+                    if ei > mx:
+                        break
+                    elif e > mx or len(set([a, b, x, c, ci, d, di, e, ei, i-e-a])) < 10 or i-e-a > mx:
+                        continue
+                    m[3] = [ei, d, e]
+                    m[4] = [i-e-a, e, a]
+                    st = m.index(sorted(m, key = lambda y: y[0])[0])
+                    m = m[st:] + m[:st]
+                    if m[0][1] < m[0][2]:
+                        m = list(map(lambda y: (y[0], y[2], y[1]), [m[0]] + m[::-1][:-1]))
+                    r.append(''.join([''.join(map(str, y)) for y in m]))
+    return r
+
+if __name__ == '__main__':
+    print(max(m5gon(10, i, 10) for i in range(10, 25))[0])
+
diff --git a/074_digit_factorial_chains.c b/074_digit_factorial_chains.c
new file mode 100644
index 0000000..b949d66
--- /dev/null
+++ b/074_digit_factorial_chains.c
@@ -0,0 +1,84 @@
+#include <stdio.h>
+#include <math.h>
+
+#define LIM 3 * 1000000
+#define MAX_CHAIN 100
+
+void initf(int f[]);
+void init_sieve(char s[], int len);
+long next_term(long x, int f[]);
+
+int main()
+{
+    int x = pow(10, 6);
+    int y = 60;
+    int c, len, count;
+    long i, n;
+    long chain[MAX_CHAIN];
+    int f[10];
+    char sieve[LIM];
+    
+    initf(f);
+    init_sieve(sieve, LIM);
+    count = 0;
+    
+    for(i = 3; i < x; ++i){
+        c = len = 0;
+        n = i;
+        while(n >= LIM || !sieve[n]){
+            if(c < MAX_CHAIN){
+                chain[c] = n;
+                ++len;
+            }
+            n = next_term(n, f);
+            ++c;
+        }
+        c += sieve[n];
+        if(c == y){
+            ++count;
+        }
+        for(n = 0; n < len; ++n){
+            if(chain[n] < LIM){
+                sieve[chain[n]] = c;
+            }
+            --c;
+        }
+        
+    }
+    printf("%d\n", count);
+    return 0;
+}
+
+long next_term(long x, int f[])
+{
+    long r;
+    int i, d;
+    r = 0;
+    for(i = pow(10, (int) log10(x)); i >= 1; i /= 10){
+        d = x / i;
+        r += f[d];
+        x -= d * i;
+    }
+    return r;
+}
+
+void initf(int f[])
+{
+    int i;
+    f[0] = f[1] = 1;
+    for(i = 2; i < 10; ++i){
+        f[i] = i * f[i - 1];
+    }
+}
+
+void init_sieve(char s[], int len)
+{
+    int i;
+    
+    for(i = 0; i < len; ++i){
+        s[i] = 0;
+    }
+    s[1] = s[2] = s[145] = s[40585] = 1;
+    s[871] = s[45361] = s[872] = s[45362] = 2;
+    s[169] = s[363601] = s[1454] = 3;
+}
diff --git a/075_singular_integer_right_triangles.c b/075_singular_integer_right_triangles.c
new file mode 100644
index 0000000..affbd01
--- /dev/null
+++ b/075_singular_integer_right_triangles.c
@@ -0,0 +1,42 @@
+#include <stdio.h>
+#include <math.h>
+
+int gcd(int a, int b);
+
+int main(int argc, char *argv[])
+{
+    int lim = 1500000;
+    int i, n, m, p, c, odd;
+    char sieve[lim + 1];
+    
+    for(i = 0; i <= lim; ++i){
+        sieve[i] = 0;
+    }
+    odd = c = 0;
+    for(n = 2; n <= (sqrt(2*lim + 1) - 1)/ 2; ++n){
+        for(m = odd + 1; m < n && m <= (lim / 2*n) - n; m += 2){
+            if(gcd(n, m) == 1){
+                p = 2*n * (n + m);
+                for(i = p; i <= lim; i += p){
+                    sieve[i] += 1;
+                }
+            }
+        }
+        odd = !odd;
+    }
+    for(i = 0; i <= lim; i += 2){
+        if(sieve[i] == 1){
+            ++c;
+        }
+    }
+    printf("%d\n", c);
+}
+
+int gcd(int a, int b)
+{
+  int c;
+  while (a != 0){
+     c = a; a = b%a;  b = c;
+  }
+  return b;
+}
diff --git a/076_counting_summations.c b/076_counting_summations.c
new file mode 100644
index 0000000..393cca5
--- /dev/null
+++ b/076_counting_summations.c
@@ -0,0 +1,28 @@
+#include <stdio.h>
+
+int sums(int x);
+
+int main(int argc, char *argv[])
+{
+    printf("%d\n", sums(100));
+}
+
+int sums(int x)
+{
+    int i, n;
+    int s[x + 1][x];
+    
+    for(n = 0; n <= x; ++n){
+        s[n][0] = 1;
+        for(i = 2; i < n; ++i){
+            s[n][i - 1] = s[n][i - 2];
+            if(n - i <= i){
+                s[n][i - 1] += s[n - i][n - i - 1];
+            }else{
+                s[n][i - 1] += s[n - i][i - 1];
+            }
+        }
+        s[n][i - 1] = s[n][i - 2] + 1;
+    }
+    return s[x][x - 1] - 1;
+}
diff --git a/077_prime_summations.py b/077_prime_summations.py
new file mode 100644
index 0000000..d20085d
--- /dev/null
+++ b/077_prime_summations.py
@@ -0,0 +1,30 @@
+#!/usr/bin/env python3
+
+from math import sqrt
+
+def main(lim, x):
+    sieve = [False, True] * (lim//2 + 1)
+    primes = []
+    sub_lim = int(sqrt(lim) + 1)
+    count = [[1], [0], [1]]
+    even = False
+    for n in range(3, lim + 1):
+        if sieve[n]:
+            primes.append(n)
+            for j in range(n**2, lim + 1, 2*n):
+                sieve[j] = False
+        count.append([int(even)])
+        for i, p in enumerate(primes, 1):
+            if p > n - p:
+                count[n].append(count[n - p][-1] + count[n][-1])
+            else:
+                count[n].append(count[n - p][i] + count[n][-1])
+        if count[n][-1] - int(sieve[n]) > x:
+            return n, count[n][-1] - int(sieve[n])
+        even = not even
+    return False
+
+if __name__ == '__main__':
+    lim = 10**6
+    x = 5000
+    print(main(lim, x)[0])
diff --git a/078_coin_partitions.py b/078_coin_partitions.py
new file mode 100644
index 0000000..07a10b4
--- /dev/null
+++ b/078_coin_partitions.py
@@ -0,0 +1,29 @@
+#!/usr/bin/env python3
+
+def main(d):
+    tbl = [1]
+    n = 1
+    while p(n, tbl) % d: n+=1
+    return n, p(n,tbl)
+
+def p(n, tbl=[1]):
+    if n < 0:
+        return 0
+    if n >= len(tbl):
+        r = -sum((-1)**i *
+                (p(n-a, tbl) +
+                 p(n-b, tbl))
+                 for a,b,i in pentnum(n))
+        tbl.append(r)
+    return tbl[n]
+
+def pentnum(bound):
+    n = 1
+    a,b = 1,2
+    while a <= bound:
+        yield a,b,n
+        a += 3*n + 1
+        b += 3*n + 2
+        n += 1
+
+print(main(10**6)[0])
diff --git a/makefile b/makefile
index 6424554..be1cf88 100644
--- a/makefile
+++ b/makefile
@@ -1,6 +1,18 @@
+all : 064 074 075 076 085
 
-85 : 085_counting_rectangles.c
+064 : 064_odd_period_square_roots.c
+	gcc 064_odd_period_square_roots.c -lm -o 064_odd_period_square_roots.exe
+
+074 : 074_digit_factorial_chains.c
+	gcc 074_digit_factorial_chains.c -lm -o 074_digit_factorial_chains.exe
+
+075 : 075_singular_integer_right_triangles.c
+	gcc 075_singular_integer_right_triangles.c -lm -o 075_singular_integer_right_triangles.exe
+
+076 : 076_counting_summations.c
+	gcc 076_counting_summations.c -o 076_counting_summations.exe
+
+085 : 085_counting_rectangles.c
 	gcc 085_counting_rectangles.c -lm -o 085_counting_rectangles.exe
-
 clean : 
 	rm -f *.exe *.out