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49 lines
1.2 KiB
Python
49 lines
1.2 KiB
Python
"""
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Project Euler Problem 100: https://projecteuler.net/problem=100
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If a box contains twenty-one coloured discs, composed of fifteen blue discs and
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six red discs, and two discs were taken at random, it can be seen that
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the probability of taking two blue discs, P(BB) = (15/21) x (14/20) = 1/2.
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The next such arrangement, for which there is exactly 50% chance of taking two blue
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discs at random, is a box containing eighty-five blue discs and thirty-five red discs.
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By finding the first arrangement to contain over 10^12 = 1,000,000,000,000 discs
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in total, determine the number of blue discs that the box would contain.
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"""
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def solution(min_total: int = 10**12) -> int:
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"""
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Returns the number of blue discs for the first arrangement to contain
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over min_total discs in total
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>>> solution(2)
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3
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>>> solution(4)
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15
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>>> solution(21)
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85
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"""
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prev_numerator = 1
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prev_denominator = 0
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numerator = 1
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denominator = 1
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while numerator <= 2 * min_total - 1:
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prev_numerator += 2 * numerator
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numerator += 2 * prev_numerator
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prev_denominator += 2 * denominator
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denominator += 2 * prev_denominator
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return (denominator + 1) // 2
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if __name__ == "__main__":
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print(f"{solution() = }")
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