Sunday Times Teaser 2644 – Route Canal

by Peter G Chamberlain

A 70 mile canal trip is planned from the lock at Aytown to the lock at Beeswick with a stop at a pub next to a lock over half way along the route.

A list is created of the number of miles between Aytown and the pub, between the pub and Beeswick, and between adjacent locks.

A list is created of the number of miles between Aytown and the pub, between the pub and Beeswick, and between adjacent locks (where the longest is between the pub and the next lock towards Beeswick).

In the order visited, what are the numbers of miles between adjacent locks?

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# primes less than 70
primes = [2, 3, 5, 7]
primes += [x for x in range(11, 70, 2) if all(x % p for p in primes)]

# locks: list of distances between locks
# distances: distances from start to each lock
# left: unused (prime) lock distances
# pub_at: distance to the pub

def place(locks, distances, left, pub_at):
  if distances[-1] < 70:
    # place another lock
    for nxt in left:
      # its distance from the start
      l = distances[-1] + nxt
      # check that its distance from all other locks
      # is prime whenever it is odd
      for x in distances:
        if (l - x) % 2 and l - x not in primes:
          break
      else:
        # place more locks
        place(locks + [nxt], distances + [l], left - set((nxt,)), pub_at)

  # if we have reached the destination via the pub
  elif distances[-1] == 70 and pub_at in distances:
    # the maximum distance between locks must be
    # immediately after the pub
    if locks[distances.index(pub_at)] == max(locks):
      fs = 'lock gaps {:s}, at {:s}, pub {:d}'
      print(fs.format(tuple(locks), tuple(distances), pub_at))

for pub in primes:
  # the pub is over half way and its distances from the
  # start and the finish are both prime
  if pub > 35 and 70 - pub in primes:
    # primes for distances between adjacent locks - the maximum distance
    # between locks is less than half the total distance
    lock_primes = set(p for p in primes if p < 35) - set((pub, 70 - pub))
    # place the locks
    place([], [0], lock_primes, pub)

# primes less than 70

primes = [2, 3, 5, 7]

primes += [x for x in range(11, 70, 2) if all(x % p for p in primes)]

# locks: list of distances between locks

# distances: distances from start to each lock

# left: unused (prime) lock distances

# pub_at: distance to the pub

def place(locks, distances, left, pub_at):

if distances[-1] < 70:

# place another lock

for nxt in left:

# its distance from the start

l = distances[-1] + nxt

# check that its distance from all other locks

# is prime whenever it is odd

for x in distances:

if (l - x) % 2 and l - x not in primes:

break

else:

# place more locks

place(locks + [nxt], distances + [l], left - set((nxt,)), pub_at)

# if we have reached the destination via the pub

elif distances[-1] == 70 and pub_at in distances:

# the maximum distance between locks must be

# immediately after the pub

if locks[distances.index(pub_at)] == max(locks):

fs = 'lock gaps {:s}, at {:s}, pub {:d}'

print(fs.format(tuple(locks), tuple(distances), pub_at))

for pub in primes:

# the pub is over half way and its distances from the

# start and the finish are both prime

if pub > 35 and 70 - pub in primes:

# primes for distances between adjacent locks - the maximum distance

# between locks is less than half the total distance

lock_primes = set(p for p in primes if p < 35) - set((pub, 70 - pub))

# place the locks

place([], [0], lock_primes, pub)

Sunday Times Teaser 2644 – Route Canal

by Peter G Chamberlain

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