3ring.py

from itertools import permutations

def main():
    A = [24, 3, 'A', 6, 27, 0, 9, 'AB', 17, 'ABC', 14, 'AC']
    B = [20, 4, 23, 7, 'B', 1, 26, 'BC', 16, 'ABC', 17, 'AB']
    C = [11, 5, 22, 8, 'C', 2, 25, 'AC', 14, 'ABC', 16, 'BC']

    # Figure out what numbers are missing.
    N = list(range(28))
    n = set(i for i in A + B + C if type(i) == int)
    missing = sorted(set(N) - n)
    print(missing)

    # we have 7 blanks:  'A', 'AB', 'ABC', 'AC', 'B', 'BC', 'C'
    # we have 7 missing numbers.
    # brute force means trying all the mappings of 7 to 7, 7! = 5040, so that's doable.
    partial_sum_a = sum([a for a in A if type(a) == int])
    partial_sum_b = sum([b for b in B if type(b) == int])
    partial_sum_c = sum([c for c in C if type(c) == int])
    print('A: %d' % (partial_sum_a))
    print('B: %d' % (partial_sum_b))
    print('C: %d' % (partial_sum_c))

    for (a, ab, abc, ac, b, bc, c) in permutations(missing):
        asum = partial_sum_a + a + ab + abc + ac
        bsum = partial_sum_b + b + bc + abc + ab
        csum = partial_sum_c + c + ac + abc + bc
        if asum == bsum == csum:
            print("3 ring circuit total: %d" % asum)
            print(f"'A': {a}, 'B': {b}, 'C': {c}, 'AB': {ab}, 'AC': {ac}, 'BC': {bc}, 'ABC': {abc}")
            print(f'ring A: 24 + 3 + {a} + 6 + 27 + 0 + 9 + {ab} + 17 + {abc} + 14 + {ac} = {asum}')
            print(f'ring B: 20 + 4 + 23 + 7 + {b} + 1 + 26 + {bc} + 16 + {abc} + 17 + {ab} = {bsum}')
            print(f'ring C: 11 + 5 + 22 + 8 + {c} + 2 + 25 + {ac} + 15 + {abc} + 16 + {bc} = {csum}')


if __name__ == '__main__':
    main()

description.md

3 Ring Circuit

Approach:

1. Label the blank spots in the diagram (see above).
2. Problem now boils down to:

Ring A Sum = 24 + 3 + A + 6 + 27 + 0 + 9 + AB + 17 + ABC + 14 + AC
Ring B Sum = 20 + 4 + 23 + 7 + B + 1 + 26 + BC + 16 + ABC + 17 + AB
Ring C Sum = 11 + 5 + 22 + 8 + C + 2 + 25 + AC + 14 + ABC + 16 + BC

Find A, AB, B, BC, C, AC, ABC such that Ring A Sum = Ring B Sum = Ring C Sum.

4. Find the numbers that are missing from the range that is used and are candidates for the solution:

Missing numbers: [10, 12, 13, 15, 18, 19, 21]

5. This is a mapping of 7 numbers onto 7 blank spots (A, AB, B, BC, C, AC, ABC); there are 7! = 5040 ways to do this.
6. Brute force is your friend, see code above :)

Solutions

• 3 ring circuit total: 164

  • A = 19, B = 13, C = 21, AB = 15, AC = 18, BC = 10, ABC = 12
  • Ring A: 24 + 3 + 19 + 6 + 27 + 0 + 9 + 15 + 17 + 12 + 14 + 18 = 164
  • Ring B: 20 + 4 + 23 + 7 + 13 + 1 + 26 + 10 + 16 + 12 + 17 + 15 = 164
  • Ring C: 11 + 5 + 22 + 8 + 21 + 2 + 25 + 18 + 15 + 12 + 16 + 10 = 164

• 3 ring circuit total: 167

  • A = 21, B = 10, C = 19, AB = 13, AC = 15, BC = 12, ABC = 18
  • Ring A: 24 + 3 + 21 + 6 + 27 + 0 + 9 + 13 + 17 + 18 + 14 + 15 = 167
  • Ring B: 20 + 4 + 23 + 7 + 10 + 1 + 26 + 12 + 16 + 18 + 17 + 13 = 167
  • Ring C: 11 + 5 + 22 + 8 + 19 + 2 + 25 + 15 + 15 + 18 + 16 + 12 = 167

• 3 ring circuit total: 168

  • A = 21, B = 12, C = 15, AB = 10, AC = 18, BC = 13, ABC = 19
  • Ring A: 24 + 3 + 21 + 6 + 27 + 0 + 9 + 10 + 17 + 19 + 14 + 18 = 168
  • Ring B: 20 + 4 + 23 + 7 + 12 + 1 + 26 + 13 + 16 + 19 + 17 + 10 = 168
  • Ring C: 11 + 5 + 22 + 8 + 15 + 2 + 25 + 18 + 15 + 19 + 16 + 13 = 168

• 3 ring circuit total: 170

  • A = 19, B = 10, C = 15, AB = 12, AC = 18, BC = 13, ABC = 21
  • Ring A: 24 + 3 + 19 + 6 + 27 + 0 + 9 + 12 + 17 + 21 + 14 + 18 = 170
  • Ring B: 20 + 4 + 23 + 7 + 10 + 1 + 26 + 13 + 16 + 21 + 17 + 12 = 170
  • Ring C: 11 + 5 + 22 + 8 + 15 + 2 + 25 + 18 + 15 + 21 + 16 + 13 = 170