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108 lines
4.2 KiB
Python
108 lines
4.2 KiB
Python
"""
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Project Euler Problem 85: https://projecteuler.net/problem=85
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By counting carefully it can be seen that a rectangular grid measuring 3 by 2
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contains eighteen rectangles.
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Although there exists no rectangular grid that contains exactly two million
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rectangles, find the area of the grid with the nearest solution.
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Solution:
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For a grid with side-lengths a and b, the number of rectangles contained in the grid
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is [a*(a+1)/2] * [b*(b+1)/2)], which happens to be the product of the a-th and b-th
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triangle numbers. So to find the solution grid (a,b), we need to find the two
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triangle numbers whose product is closest to two million.
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Denote these two triangle numbers Ta and Tb. We want their product Ta*Tb to be
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as close as possible to 2m. Assuming that the best solution is fairly close to 2m,
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We can assume that both Ta and Tb are roughly bounded by 2m. Since Ta = a(a+1)/2,
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we can assume that a (and similarly b) are roughly bounded by sqrt(2 * 2m) = 2000.
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Since this is a rough bound, to be on the safe side we add 10%. Therefore we start
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by generating all the triangle numbers Ta for 1 <= a <= 2200. This can be done
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iteratively since the ith triangle number is the sum of 1,2, ... ,i, and so
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T(i) = T(i-1) + i.
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We then search this list of triangle numbers for the two that give a product
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closest to our target of two million. Rather than testing every combination of 2
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elements of the list, which would find the result in quadratic time, we can find
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the best pair in linear time.
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We iterate through the list of triangle numbers using enumerate() so we have a
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and Ta. Since we want Ta * Tb to be as close as possible to 2m, we know that Tb
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needs to be roughly 2m / Ta. Using the formula Tb = b*(b+1)/2 as well as the
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quadratic formula, we can solve for b:
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b is roughly (-1 + sqrt(1 + 8 * 2m / Ta)) / 2.
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Since the closest integers to this estimate will give product closest to 2m,
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we only need to consider the integers above and below. It's then a simple matter
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to get the triangle numbers corresponding to those integers, calculate the product
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Ta * Tb, compare that product to our target 2m, and keep track of the (a,b) pair
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that comes the closest.
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Reference: https://en.wikipedia.org/wiki/Triangular_number
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https://en.wikipedia.org/wiki/Quadratic_formula
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"""
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from __future__ import annotations
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from math import ceil, floor, sqrt
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def solution(target: int = 2000000) -> int:
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"""
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Find the area of the grid which contains as close to two million rectangles
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as possible.
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>>> solution(20)
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6
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>>> solution(2000)
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72
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>>> solution(2000000000)
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86595
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"""
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triangle_numbers: list[int] = [0]
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idx: int
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for idx in range(1, ceil(sqrt(target * 2) * 1.1)):
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triangle_numbers.append(triangle_numbers[-1] + idx)
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# we want this to be as close as possible to target
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best_product: int = 0
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# the area corresponding to the grid that gives the product closest to target
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area: int = 0
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# an estimate of b, using the quadratic formula
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b_estimate: float
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# the largest integer less than b_estimate
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b_floor: int
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# the largest integer less than b_estimate
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b_ceil: int
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# the triangle number corresponding to b_floor
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triangle_b_first_guess: int
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# the triangle number corresponding to b_ceil
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triangle_b_second_guess: int
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for idx_a, triangle_a in enumerate(triangle_numbers[1:], 1):
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b_estimate = (-1 + sqrt(1 + 8 * target / triangle_a)) / 2
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b_floor = floor(b_estimate)
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b_ceil = ceil(b_estimate)
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triangle_b_first_guess = triangle_numbers[b_floor]
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triangle_b_second_guess = triangle_numbers[b_ceil]
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if abs(target - triangle_b_first_guess * triangle_a) < abs(
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target - best_product
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):
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best_product = triangle_b_first_guess * triangle_a
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area = idx_a * b_floor
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if abs(target - triangle_b_second_guess * triangle_a) < abs(
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target - best_product
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):
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best_product = triangle_b_second_guess * triangle_a
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area = idx_a * b_ceil
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return area
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if __name__ == "__main__":
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print(f"{solution() = }")
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