Merge remote-tracking branch 'origin/master'

day13.py

@@ -20,17 +20,27 @@ def part1(test_mode=False):
 def part2(test_mode=False):
     my_input = aoclib.getInputAsArraySplit(day=DAY, split_char=",", test=test_mode)
 
-    bus_ids = {-i: int(v) for i, v in enumerate(my_input[1]) if v != 'x'}
-    lowest_common_multiple = math.lcm(*bus_ids.values())
-    base_multipliers = [lowest_common_multiple // interval for interval in bus_ids.values()]
-    modular_multiplicative_inverses = [
-        pow(base_multiplier, -1, interval)
-        for base_multiplier, interval
-        in zip(base_multipliers, bus_ids.values())
-    ]
+    bus_ids = {i: int(v) for i, v in enumerate(my_input[1]) if v != 'x'}
+    # utilizing the Chinese Remainder Theorem we are searching for x = i (mod bus_ids[i]) for all i
+    # watch https://www.youtube.com/watch?v=ru7mWZJlRQg for an easy explanation
+
+    # finding the "left" part of each (mod x) part
+    base_multiplier = {i: math.prod(bus_ids.values()) // v for i, v in bus_ids.items()}
+
+    # finding the "right" part of each (mod x) part utilizing the Extended Euclidean Algorithm
+    # s. Python documentation on pow(x, -1, y)
+    ext_multiplier = {i: pow(base_multiplier[i], -1, bus_ids[i]) for i in bus_ids}
+
+    # sum all multiplications together and add our offset
+    # EEA gives us base_multiplier[i] * x == 1(one!) (mod bus_ids[i])
+    # but we need base_multiplier[i] * x == i (mod bus_ids[i])
+    answer = sum(i * base_multiplier[i] * ext_multiplier[i] for i in bus_ids)
+
+    # and shrink it down
+    # for the -answer see pythons behaviour when calculating the mod of negative numbers with positive divisor
+    # also: http://python-history.blogspot.com/2010/08/why-pythons-integer-division-floors.html
+    lowest_common_multiple = math.lcm(*bus_ids.values())
+    answer = -answer % lowest_common_multiple
+
+    return answer
 
-    return sum(
-        offset * multiplier * mmi
-        for offset, multiplier, mmi
-        in zip(bus_ids.keys(), base_multipliers, modular_multiplicative_inverses)
-    ) % lowest_common_multiple