Add Python solutions for problems 88, 90, 91

Python/p088.py

@@ -0,0 +1,74 @@
+#!/usr/bin/env python3
+
+# A natural number, N, hat can be written as the sum and product of a given set of at least two natural numbers, {a1,a2,...,ak} is called a product sum number: N = a1 + a2 + ... + ak = a1 x a2 x ... ak.
+#
+# For example, 6 = 1 + 2 + 3 = 1 x 2 x 3
+#
+# For a given set of size, k, we shall call the smallest N with this property a minimal product-sum number. The minimal product-sum numbers for sets of size, k = 2,3,4,5, and 6 are as follows.
+#
+# k = 2: 4 = 2 x 2 = 2 + 2
+# k = 3: 6 = 1 x 2 x 3 = 1 + 2 + 3
+# k = 4: 8 = 1 x 1 x 2 x 4 = 1 + 1 + 2 + 4
+# k = 5: 8 = 1 x 1 x 2 x 2 x 2 = 1 + 1 + 2 + 2 + 2
+# k = 6: 12 = 1 x 1 x 1 x 1 x 2 x 6 = 1 + 1 + 1 + 1 + 2 + 6
+#
+# Hence,for 2<=k<=6, the sum for all the minimal product-sum numbers is 4 + 6 + 8 + 12 = 30; note that 8 is only counted once in the sum.
+#
+# In fact, as the complete set of minimal product-sum numbers for 2<=k<=12 is {4,6,8,12,15,16}, the sum is 61.
+#
+# What is the sum of all the minimal product-sum numbers for 2<=k<=12000?
+
+from math import floor, sqrt, prod
+
+from projecteuler import sieve, timing
+
+
+N = 12000
+factorizations = {}
+product_sums = {}
+
+
+def factorize(n, factors, start, max_):
+    if n == 1:
+        factorizations[start].append(sorted(factors))
+
+        return
+
+    for i in range(max_, 1, -1):
+        if n % i == 0:
+            factors.append(i)
+            factorize(n // i, factors, start, i)
+            factors.pop()
+
+
+@timing
+def p088():
+    primes = sieve(2*N + 1)
+
+    for i in range(2, 2*N + 1):
+        if not primes[i]:
+            factorizations[i] = []
+            factorize(i, [], i, i)
+
+    for _, f in factorizations.items():
+        for i in f:
+            l = len(i) + (prod(i) - sum(i))
+
+            if l > N:
+                continue
+
+            if l not in product_sums:
+                product_sums[l] = prod(i)
+            elif prod(i) < product_sums[l]:
+                product_sums[l] = prod(i)
+
+    min_prod_sums = list(set(list(product_sums.values())))
+
+    res = sum(min_prod_sums)
+
+    print('Project Euler, Problem 88')
+    print(f'Answer: {res}')
+
+
+if __name__ == '__main__':
+    p088()

Python/p090.py

@@ -0,0 +1,82 @@
+#!/usr/bin/env python3
+
+# Each of the six faces on a cube has a different digit (0 to 9) written on it; the same is done to a second cube. By placing the two cubes side-by-side in different positions we can form a variety of 2-digit numbers.
+#
+# For example, the square number 64 could be formed:
+#                       |6||4|
+#
+# In fact, by carefully choosing the digits on both cubes it is possible to display all of the square numbers below one-hundred: 01, 04, 09, 16, 25, 36, 49, 64, and 81.
+#
+# For example, one way this can be achieved is by placing {0,5,6,7,8,9} on one cube and {1,2,3,4,8,9} on the other cube.
+#
+# However, for this problem we shall allow the 6 or 9 to be turned upside-down so that an arrangement like  {0,5,6,7,8,9} and  {1,2,3,4,6,7} allows for all nine square numbers to be displayed; otherwise it would be impossible to obtain 09.
+#
+# In determining a distinct arrangement we are interested in the digits on each cube, not the order.
+#
+#   {1,2,3,4,5,6} is equivalent to {3,6,4,1,2,5}
+#   {1,2,3,4,5,6} is distinct from {1,2,3,4,5,9}
+#
+# But because we are allowing 6 and 9 to be reversed, the two distinct sets in the last example both represent the extended set {1,2,3,4,5,6,9} for the purpose of forming 2-digit numbers.
+#
+# How many distinct arrangements of the two cubes allow for all of the square numbers to be displayed?
+
+from copy import copy
+from itertools import product
+
+from projecteuler import timing
+
+
+N = 10
+K = 6
+COMBINATIONS = []
+
+
+def combination(i, k, comb):
+    if len(comb) == K:
+        COMBINATIONS.append(copy(comb))
+
+        return
+
+    for j in range(i, N - k + 1):
+        comb.append(j)
+        combination(j + 1, k - 1, comb)
+        comb.pop()
+
+
+@timing
+def p090():
+    squares = [1, 4, 9, 16, 25, 36, 49, 64, 81]
+    combination(0, K, [])
+
+    for c in COMBINATIONS:
+        if 6 in c and 9 not in c:
+            c.append(9)
+        if 9 in c and 6 not in c:
+            c.append(6)
+
+    count = 0
+
+    for i, c in enumerate(COMBINATIONS):
+        for j in range(i + 1, len(COMBINATIONS)):
+            cur_squares = copy(squares)
+
+            prods = list(product(c, COMBINATIONS[j]))
+            prods.extend(list(product(COMBINATIONS[j], c)))
+
+            for elem in prods:
+                val = int(str(elem[0]) + str(elem[1]))
+
+                if val in cur_squares:
+                    cur_squares.remove(val)
+
+                    if len(cur_squares) == 0:
+                        count += 1
+
+                        break
+
+    print('Project Euler, Problem 90')
+    print(f'Answer: {count}')
+
+
+if __name__ == '__main__':
+    p090()

Python/p091.py

@@ -0,0 +1,51 @@
+#!/usr/bin/env python3
+
+# The points P(x1,y1) and Q(x2,y2) are plotted at integer co-ordinates and are joined to the origin, O(0,0), to form ΔOPQ.
+#
+# There are exactly fourteen triangles containing a right angle that can be formed when each co-ordinate lies between 0 and 2 inclusive; that is, 0<=x1,y1,x2,y2<=2.
+#
+# Given that 0<=x1,y1,x2,y2<=50, how many right triangles can be formed?
+
+from itertools import product
+
+from projecteuler import timing
+
+
+N = 50
+
+
+def check_triangle(p1, p2, p3):
+    a = int((p2[0] - p1[0]) ** 2 + (p2[1] - p1[1]) ** 2)
+    b = int((p3[0] - p2[0]) ** 2 + (p3[1] - p2[1]) ** 2)
+    c = int((p3[0] - p1[0]) ** 2 + (p3[1] - p1[1]) ** 2)
+
+    if ((a > 0 and b > 0 and c > 0) and
+        (a == (b + c) or b == (a + c) or
+         c == (a + b))):
+        return True
+
+    return False
+
+
+@timing
+def p091():
+    combinations = []
+
+    for i in range(N + 1):
+        for j in range(N + 1):
+            combinations.append((i, j))
+
+    prods = product(combinations, repeat=2)
+
+    count = 0
+
+    for p in prods:
+        if check_triangle((0, 0), p[0], p[1]):
+            count += 1
+
+    print('Project Euler, Problem 91')
+    print(f'Answer: {int(count / 2)}')
+
+
+if __name__ == '__main__':
+    p091()

README.md

@@ -4,3 +4,4 @@ These are my solutions in C and Python, not necessarily the best solutions. I've
 # Notes
 - Solutions for problems 82 and 145 in Python are really slow.
 - Solutions for problems 84, 85, 86, 87, 89, 92, 95, 96, 97. 99, 102, 112, 124 and 357 have been implemented in C but not in Python.
+- Solutions for problems 88, 90 and 91 have been implemented in Python but not in C.