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83 lines
2.7 KiB
Python
83 lines
2.7 KiB
Python
4 years ago
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"""Solution to 2020/13 part2
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Declarations:
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t: timestamp
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bus_id, bus: used as slope (a) in the linear function
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observation: bus_ids are distinct prime numbers
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offset: postion of the bus in the list
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t + offset = arrival of bus
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normalized_offset: used so f(0) is a correct (non-negative) solution
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= -offset % bus_id
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used as intercept (b) in linear function
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bus_solution: solutions for y in the scope of one bus.
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solution: smallest y that is a solution to all linear functions
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Functions:
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- The solutions per bus. linear (y=a*x+b)
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- t = bus_id * x + normalized_offset
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- x: natural numbers. xth solution (time of arrival - index)
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- y: solutions for t in the scope of one bus. t + delta = arrival of bus
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Combining two functions:
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- Given y = x0 * a0 + b0
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- Given y = x1 * a1 + b1
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- x0 * a0 + b0 = x1 * a1 + b1
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- x1 = (x0 * a0 + b0 - b1) / a1
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- Remember: x0 and x1 have to be whole numbers
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- Find linear function x0 = n * a + b that describes the possible solutions
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- Let x0 be in linear function x0 = n * a + b
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- x1 = ((n * a + b) * a0 + b0 - b1) / a1
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- x1 = (n * a * a0 + a0 * b + b0 -b1) / a1
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- x1 = (n * a * a0) / a1 + (a0 * b + b0 - b1) / a1
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- For x1 to be a whole number:
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- (n * a * a0) needs to be divisible by a1
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- since a's prime factors are distinct from a1 set a = a1
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- a0 * b + b0 - b1 needs to be divisible by a1
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- find b by brute force (b = magic_number)
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- => x0 = n * a1 + magic_number
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- Apply to y = x0 * a0 + b0
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- y = (n * a1 + magic_number) * a0 + b0
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- y = n * (a0 * a1) + (a0 + b0 + magic_number)
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- new a = a0 * a1
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- new b = a0 + b0 + magic_number
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Apply this schema recursively (or iteratively) to all functions. Each interim
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result represents a formula for all possible solutions that includes the
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processed bus_ids. At last set n = 0 (in y = n * a + b) to receive the smallest
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solution (earliest timestamp).
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"""
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with open('input', 'r') as f:
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f.readline()
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buses = f.readline().strip().split(',')
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departures = []
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i = 0
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for offset, bus in enumerate(buses):
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try:
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bus = int(bus)
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normalized_offset = -offset % bus
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print(f'y = x{i} * {bus:>3} + {normalized_offset:>3} ({offset=:>2})')
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departures.append((bus, normalized_offset))
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i += 1
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except ValueError:
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pass
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print()
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a = 1
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b = 0
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for a_cur, b_cur in departures:
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magic_nr = None
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i = 0
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while magic_nr is None:
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if (a * i + b - b_cur) % a_cur == 0:
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magic_nr = i
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i += 1
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b = b + a * magic_nr
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a = a * a_cur
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print(f'y = n * {a} + {b} | {magic_nr=}')
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print()
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print('Solution:')
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print(b)
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