Sunday Times Teaser 3223 – Shaping Up

by BRG on June 28, 2024

by Peter Good

Published Friday June 28 2024 (link)

Clark wondered how many different shapes he could draw with identical squares joined edge to edge and each shape containing the same number of squares. He only drew five different shapes containing four squares, for example, because he ignored rotations and reflections (see diagram, above). He drew all of the different shapes containing 1, 2, 3, 4, 5 and 6 squares and wrote down the total number of different shapes in each case. He took the six totals in some order, without reordering the digits within any total, and placed them end to end to form a sequence of digits which could also be formed by placing six prime numbers end to end.

In ascending order, what were the six prime numbers?

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Using jigsaw.py.


from itertools import permutations
from jigsaw import generate_pieces, print_pieces

show_pieces = False

# the set of primes less than 1000
prms = {3, 5, 7, 11, 13, 17, 19, 23, 29, 31}
prms = {2} | prms | set(n for n in range(37, 1000, 2) if all(n % p for p in prms))

# generate sets of up to six fully connected unit squares 
# from within a six by six rectangle
d = generate_pieces(6, 6, use_all=False, two_sided=True, max_sq=6)
nbrs = [0] * 6
for v in d.values():
  nbrs[len(v[0][0]) - 1] += 1

if show_pieces:
  print_pieces(d)
  print()

# print the numbers of shapes made for 1 .. 6 squares
print(f"Numbers of shapes for 1..6 squares: {nbrs}\n")

# if possible break a digit sequence in string <ns>
# into a sequence of primes
def prime_seq(ns, pseq=[]):
  if not ns:
    if len(pseq) == 6:
      yield pseq
  else:
    for i in range(1, len(ns) + 1):
      if (pp := int(ns[:i])) in prms:
        yield from prime_seq(ns[i:], pseq + [pp])

sol = set()
# consider all orders of the numbers of shapes
for n6 in permutations(nbrs, 6):
  # look for sequences that are concatenations of six primes
  sn6 = ''.join(str(n) for n in n6)
  for pr in prime_seq(sn6):
      print(f"{sn6} -> {pr} -> {sorted(pr)}")
      sol.add(tuple(sorted(pr)))

print(f"\nAnswer: {sol.pop()}")

from itertools import permutations

from jigsaw import generate_pieces, print_pieces

show_pieces = False

# the set of primes less than 1000

prms = {3, 5, 7, 11, 13, 17, 19, 23, 29, 31}

prms = {2} | prms | set(n for n in range(37, 1000, 2) if all(n % p for p in prms))

# generate sets of up to six fully connected unit squares

# from within a six by six rectangle

d = generate_pieces(6, 6, use_all=False, two_sided=True, max_sq=6)

nbrs = [0] * 6

for v in d.values():

nbrs[len(v[0][0]) - 1] += 1

if show_pieces:

print_pieces(d)

print()

# print the numbers of shapes made for 1 .. 6 squares

print(f"Numbers of shapes for 1..6 squares: {nbrs}\n")

# if possible break a digit sequence in string <ns>

# into a sequence of primes

def prime_seq(ns, pseq=[]):

if not ns:

if len(pseq) == 6:

yield pseq

else:

for i in range(1, len(ns) + 1):

if (pp := int(ns[:i])) in prms:

yield from prime_seq(ns[i:], pseq + [pp])

sol = set()

# consider all orders of the numbers of shapes

for n6 in permutations(nbrs, 6):

# look for sequences that are concatenations of six primes

sn6 = ''.join(str(n) for n in n6)

for pr in prime_seq(sn6):

print(f"{sn6} -> {pr} -> {sorted(pr)}")

sol.add(tuple(sorted(pr)))

print(f"\nAnswer: {sol.pop()}")

This teaser is partly about counting the numbers of plane figures that can be formed using a number of edge connected unit squares. Such figures, which are known as Polyominoes, are well researched and are described in some detail at Wikipedia and Mathworld.

Sunday Times Teaser 3223 – Shaping Up

by Peter Good

Published Friday June 28 2024 (link)

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