Problem 061 Project Euler

project_euler/problem_061/sol1.py

@@ -0,0 +1,388 @@
+"""
+Project Euler Problem 061: https://projecteuler.net/problem=61
+
+Problem statement -
+Triangle, square, pentagonal, hexagonal, heptagonal, and octagonal numbers are all
+figurate (polygonal) numbers and are generated by the following formulae:
+
+Triangle       P(3,n) = n(n+1)/2    1,3,6,10,15,...
+Square         P(4,n) = n^2         1,4,9,16,25,...
+Pentagonal     P(5,n) = n(3n-1)/2   1,5,12,22,35,...
+Hexagonal      P(6,n) = n(2n-1)     1,6,15,28,45,...
+Heptagonal     P(7,n) = n(5n-3)/2   1,7,18,34,55,...
+Octagonal      P(8,n) = n(3n-2)     1,8,21,40,65,...
+
+The ordered set of three 4-digit numbers: 8128, 2882, 8281,
+has three interesting properties.
+
+1. The set is cyclic, in that the last two digits of each number is the first two
+digits of the next number (including the last number with the first).
+2. Each polygonal type: triangle(P(3,127)), square(P(4,91)), and pentagonal(P(5,44)),
+is represented by a different number in the set.
+3. This is the only set of 4-digit numbers with this property.
+
+Find the sum of the only ordered set of six cyclic -digit numbers for which each
+polygonal type: triangle, square, pentagonal, hexagonal, heptagonal, and octagonal,
+is represented by a different number in the set.
+
+
+Solution explanation-
+The solution is actually pretty simple using a brute force approach.
+We first do some precomputations to get all the relevant sets of polygon numbers.
+Individual helper functions have been made for the same.
+We also have one more helper function to check if two given numbers are cyclic or not.
+
+Now the approach is to pick one number from a set then try to find a number in another
+set that is cyclic to the previous. Now consider this new number as previous and find
+a new number in another set(set that has not been used yet).Continue doing this until
+you either reach your last set, in which case you need only check for cyclic against
+the first number, if it matches you have an answer. If you cannot find a cyclic number
+for previous in your current set, you backtrack to the prev set and chose the next
+number from it.
+
+So code wise, we generate all possiblee permutations of the sets using itertools built
+in permutations method. Then its justa bunch of for loops trying to find the answer.
+Once we have a valid ordered set, we return its sum.
+
+"""
+
+import itertools
+
+
+def fill_triangle(int_range: tuple[int, int] = (1000, 9999)) -> list[int]:
+    """
+    Populates the triangle array with relevant integers i.e greater than
+    range[0] and less than range[1]+1.
+    The range provided should be inclusive.
+
+    >>> fill_triangle((1,20))
+    [1, 3, 6, 10, 15]
+
+    """
+
+    def triangle(num: int) -> int:
+        """
+        Gives the Nth triangle number.
+        >>> triangle(2)
+        3
+        >>> triangle(5)
+        15
+        """
+        return num * (num + 1) // 2
+
+    st, end = int_range
+    arr = []
+    i = 1
+    while True:
+        k = triangle(i)
+        if k < st:
+            i += 1
+            continue
+        if k > end:
+            break
+        arr.append(k)
+        i += 1
+    return arr
+
+
+def fill_square(int_range: tuple[int, int] = (1000, 9999)) -> list[int]:
+    """
+    Populates the square array with relevant integers i.e greater than
+    range[0] and less than range[1]+1.
+    The range provided should be inclusive.
+
+    >>> fill_square((1,30))
+    [1, 4, 9, 16, 25]
+
+    """
+
+    def square(num: int) -> int:
+        """
+        Gives the Nth square number.
+        >>> square(2)
+        4
+        >>> square(5)
+        25
+        """
+        return num**2
+
+    st, end = int_range
+    arr = []
+    i = 1
+    while True:
+        k = square(i)
+        if k < st:
+            i += 1
+            continue
+        if k > end:
+            break
+        arr.append(k)
+        i += 1
+    return arr
+
+
+def fill_pentagonal(int_range: tuple[int, int] = (1000, 9999)) -> list[int]:
+    """
+    Populates the pentagonal array with relevant integers i.e greater than
+    range[0] and less than range[1]+1.
+    The range provided should be inclusive.
+
+    >>> fill_pentagonal((1,40))
+    [1, 5, 12, 22, 35]
+
+    """
+
+    def pentagon(num: int) -> int:
+        """
+        Gives the Nth pentagon number.
+        >>> pentagon(2)
+        5
+        >>> pentagon(5)
+        35
+        """
+        return num * (3 * num - 1) // 2
+
+    st, end = int_range
+    arr = []
+    i = 1
+    while True:
+        k = pentagon(i)
+        if k < st:
+            i += 1
+            continue
+        if k > end:
+            break
+        arr.append(k)
+        i += 1
+    return arr
+
+
+def fill_hexagonal(int_range: tuple[int, int] = (1000, 9999)) -> list[int]:
+    """
+    Populates the hexagonal array with relevant integers i.e greater than
+    range[0] and less than range[1]+1.
+    The range provided should be inclusive.
+
+    >>> fill_hexagonal((1,50))
+    [1, 6, 15, 28, 45]
+
+    """
+
+    def hexagon(num: int) -> int:
+        """
+        Gives the Nth hexagon number.
+        >>> hexagon(2)
+        6
+        >>> hexagon(5)
+        45
+        """
+        return num * (2 * num - 1)
+
+    st, end = int_range
+    arr = []
+    i = 1
+    while True:
+        k = hexagon(i)
+        if k < st:
+            i += 1
+            continue
+        if k > end:
+            break
+        arr.append(k)
+        i += 1
+    return arr
+
+
+def fill_heptagonal(int_range: tuple[int, int] = (1000, 9999)) -> list[int]:
+    """
+    Populates the heptagonal array with relevant integers i.e greater than
+    range[0] and less than range[1]+1.
+    The range provided should be inclusive.
+
+    >>> fill_heptagonal((1,60))
+    [1, 7, 18, 34, 55]
+
+    """
+
+    def heptagon(num: int) -> int:
+        """
+        Gives the Nth heptagon number.
+        >>> heptagon(2)
+        7
+        >>> heptagon(5)
+        55
+        """
+        return num * (5 * num - 3) // 2
+
+    st, end = int_range
+    arr = []
+    i = 1
+    while True:
+        k = heptagon(i)
+        if k < st:
+            i += 1
+            continue
+        if k > end:
+            break
+        arr.append(k)
+        i += 1
+    return arr
+
+
+def fill_octagonal(int_range: tuple[int, int] = (1000, 9999)) -> list[int]:
+    """
+    Populates the octagonal array with relevant integers i.e greater than
+    range[0] and less than range[1]+1.
+    The range provided should be inclusive.
+
+    >>> fill_octagonal((1,70))
+    [1, 8, 21, 40, 65]
+
+
+    """
+
+    def octagon(num: int) -> int:
+        """
+        Gives the Nth octagon number.
+        >>> octagon(2)
+        8
+        >>> octagon(5)
+        65
+        """
+        return num * (3 * num - 2)
+
+    st, end = int_range
+    arr = []
+    i = 1
+    while True:
+        k = octagon(i)
+        if k < st:
+            i += 1
+            continue
+        if k > end:
+            break
+        arr.append(k)
+        i += 1
+    return arr
+
+
+def check_cyclic(num1: int, num2: int) -> bool:
+    """
+    This function checks if two 4 digit numbers are cyclic.
+    For this problem we are only concerned with 4 digit numbers.
+
+    Raises:
+        ValueError
+
+
+    >>> check_cyclic(8228, 2810)
+    True
+    >>> check_cyclic(8228, 2410)
+    False
+
+    """
+
+    if num1 < 1000 or num2 < 1000:
+        raise ValueError("Both integers must be greater than 999")
+    if num1 > 9999 or num2 > 9999:
+        raise ValueError("Both integers must be less than 10000")
+
+    return (num1 % 100) == (num2 // 100)
+
+
+def solution() -> int:
+    """
+    The function gives a solution to problem 061 of project Euler.
+    The function does some precomputations first to get all the
+    relevant sets of polygon numbers then applies a for loop on
+    all possible permutations of the sets and check for a valid
+    answer.
+
+    >>> solution()
+    28684
+
+    """
+
+    # Make initial arrays to store all types of numbers in individual arrays.
+    # This will make accessing them easier in the future.
+    # We will be fill each of these arrays only considering numbers greater
+    # than 999 and less than 10000.
+    # Since we will add them in a systematic manner, these will be present
+    # in a sorted order(increasing order).
+    triangle = fill_triangle()
+    square = fill_square()
+    pentagonal = fill_pentagonal()
+    hexagonal = fill_hexagonal()
+    heptagonal = fill_heptagonal()
+    octagonal = fill_octagonal()
+
+    # create a dictionary of all polygons for access later
+    polygon_dict = {
+        0: triangle,
+        1: square,
+        2: pentagonal,
+        3: hexagonal,
+        4: heptagonal,
+        5: octagonal,
+    }
+
+    answer = []
+
+    # The elements can be in any order from any of the sets,
+    # so we need to consider all possible permutations of the sets.
+    permutations = list(itertools.permutations(range(6)))
+
+    # Now we simply apply a for loop to consider all permutations and
+    # check if that permutation leads to a valid answer.
+    for perm in permutations:
+        first_set = polygon_dict[perm[0]]
+        second_set = polygon_dict[perm[1]]
+        third_set = polygon_dict[perm[2]]
+        fourth_set = polygon_dict[perm[3]]
+        fifth_set = polygon_dict[perm[4]]
+        sixth_set = polygon_dict[perm[5]]
+        for first in first_set:
+            for second in second_set:
+                prev = first
+                if not (check_cyclic(prev, second)):
+                    continue
+                for third in third_set:
+                    prev = second
+                    if not (check_cyclic(prev, third)):
+                        continue
+                    for fourth in fourth_set:
+                        prev = third
+                        if not (check_cyclic(prev, fourth)):
+                            continue
+                        for fifth in fifth_set:
+                            prev = fourth
+                            if not (check_cyclic(prev, fifth)):
+                                continue
+                            for sixth in sixth_set:
+                                prev = fifth
+                                if not (check_cyclic(prev, sixth)):
+                                    continue
+                                if check_cyclic(sixth, first):
+                                    answer = [
+                                        first,
+                                        second,
+                                        third,
+                                        fourth,
+                                        fifth,
+                                        sixth,
+                                    ]
+    # print(answer)
+    return sum(answer)
+
+
+if __name__ == "__main__":
+    # import time
+
+    # start_time = time.perf_counter()
+
+    print(f"{solution() = }")
+
+    # end_time = time.perf_counter()
+
+    # execution_time = end_time - start_time
+    # print(f"Execution time: {execution_time:.6f} seconds")