Add solution for Project Euler problem 93

project_euler/problem_093/sol1.py

@@ -0,0 +1,112 @@
+"""
+Project Euler Problem 93: https://projecteuler.net/problem=93
+
+By using each of the digits from the set {1, 2, 3, 4} exactly once, and making
+use of the four arithmetic operations (+, -, *, /) and brackets/parentheses, it
+is possible to form different positive integer targets.
+
+For example,
+
+    8 = (4 * (1 + 3)) / 2
+    14 = 4 * (3 + 1 / 2)
+    19 = 4 * (2 + 3) - 1
+    36 = 3 * 4 * (2 + 1)
+
+Note that concatenations of the digits, like 12 + 3 - 4, are not allowed.
+
+Using the set {1, 2, 3, 4}, it is possible to obtain thirty-one different
+target numbers, of which 28 is the highest, and the set of consecutive numbers
+1 to 28 can be obtained.
+
+Find the set of four distinct digits, a < b < c < d, for which the longest set
+of consecutive positive integers, 1 to n, can be obtained, giving your answer
+as a string: abcd.
+
+References:
+- https://en.wikipedia.org/wiki/Four_fours
+
+Solution
+--------
+Any expression built from four numbers with brackets can be seen as repeatedly
+merging two of the numbers into one with a single operation, three times in
+total.  Recursing over every unordered pair therefore covers every possible
+bracket structure without enumerating the brackets themselves, and because both
+orders of a pair are merged there is no need to permute the digits either.
+
+All arithmetic is done with ``fractions.Fraction`` so that divisions such as
+1 / 3 stay exact.  Floating point silently destroys the answer: results that
+should be whole numbers come out as 23.999999999999996 and get discarded.
+"""
+
+from collections.abc import Sequence
+from fractions import Fraction
+from itertools import combinations
+
+
+def _merge_all(numbers: Sequence[Fraction]) -> set[Fraction]:
+    """
+    Return every value obtainable by merging ``numbers`` with +, -, * and /.
+
+    >>> sorted(map(str, _merge_all([Fraction(1), Fraction(2)])))
+    ['-1', '1', '1/2', '2', '3']
+    """
+    if len(numbers) == 1:
+        return {numbers[0]}
+
+    results: set[Fraction] = set()
+    for i in range(len(numbers)):
+        for j in range(i + 1, len(numbers)):
+            left, right = numbers[i], numbers[j]
+            remaining = [
+                numbers[index] for index in range(len(numbers)) if index not in (i, j)
+            ]
+
+            candidates = [left + right, left - right, right - left, left * right]
+            if right:
+                candidates.append(left / right)
+            if left:
+                candidates.append(right / left)
+
+            for candidate in candidates:
+                results.update(_merge_all([*remaining, candidate]))
+    return results
+
+
+def _consecutive_length(values: set[Fraction]) -> int:
+    """
+    Return how far the run 1, 2, 3, ... reaches inside ``values``.
+
+    >>> _consecutive_length({Fraction(1), Fraction(2), Fraction(4)})
+    2
+    >>> _consecutive_length(set())
+    0
+    """
+    whole_numbers = {
+        int(value) for value in values if value.denominator == 1 and value > 0
+    }
+    length = 0
+    while length + 1 in whole_numbers:
+        length += 1
+    return length
+
+
+def solution(digits: str = "123456789") -> str:
+    """
+    Return the four digits that reach the longest run of consecutive integers.
+
+    >>> solution("1234")
+    '1234'
+    """
+    best_digits, best_length = "", 0
+    candidates = sorted({int(digit) for digit in digits})
+    for combination in combinations(candidates, 4):
+        numbers = [Fraction(digit) for digit in combination]
+        length = _consecutive_length(_merge_all(numbers))
+        if length > best_length:
+            best_digits = "".join(str(digit) for digit in combination)
+            best_length = length
+    return best_digits
+
+
+if __name__ == "__main__":
+    print(f"{solution() = }")