Sunday Times Teaser 2992 – Crystal Cleavage Calamity!

by Stephen Hogg

Published January 26 2020 (link)

My “mystic” crystal stood on its triangular end face, with every side of the triangle being less than 10cm. It had three vertical sides and an identical triangular top end face, with the height being under 50cm. Unfortunately, during dusting, it fell and broke in two. The break was a perfect cleavage along an oblique plane that didn’t encroach on the end faces, so I then had two, pointed, pentahedral crystals.

Later, I found that the nine edge lengths, in cm, of one of these were all different whole numbers — the majority prime — and their total was also a prime number. Curiously, the total of the nine, whole-number, edge lengths, in cm, of the other piece was also a prime number.

What was the total for this latter piece?

2 Comments Leave one →

Brian Gladman permalink


from itertools import permutations, product
from collections import defaultdict
from number_theory import Primes, pythag

# primes less than the maximum sum of nine edge lengths
prms = Primes().set(304)

# The prism's vertical sides are rectangles with pythagorean 
# triangles above them - index possible triangles with sides
# less than 51 on their non-hypotenuse sides (less than 10)
b_to_vt = defaultdict(list)
for a, b, c in pythag(51):
  if a < 10:  
    b_to_vt[a].append((b, c))
  if b < 10:
    b_to_vt[b].append((a, c))

# consider possible combinations of edges for the base triangle 
for b3 in permutations(b_to_vt.keys(), 3):
  
  # avoid duplicate solutions and ensure a valid base triangle
  if b3[0] < b3[1] and b3[1] - b3[0] < b3[2] < b3[1] + b3[0]:
    
    # consider possible pythagorean triangles for the prism sides
    for p3 in product(*(b_to_vt[x] for x in b3)):
  
      # the vertical triangle heights and top edge lengths  
      h3, t3 = zip(*p3)
  
      # the triangles must form the prism
      if h3[0] + h3[1] == h3[2]:
        
        # now consider the prism height
        for h in range(h3[2] + 1, 50):
          # form the three vertical edges
          v3 = h - h3[2], h - h3[0], h
          edges = b3 + t3 + v3
          
          # the edge lengths are all different, their sum is a 
          # prime and the majority of them are prime
          if (len(set(edges)) == 9 and sum(edges) in prms and 
              sum(x in prms for x in edges) > 4):
  
            # consider the unbroken prism height
            for ht in range(h + 1, 50):
              # calculate the (prime) edge sum of the other broken piece
              sms = sum(b3) + sum(t3) + 3 * ht - sum(v3)
              if sms in prms:
                print(f"sum = {sms}, height = {ht}, edges: base = {b3}, "
                      f"top = {t3}, vertical = {v3}")

from itertools import permutations, product

from collections import defaultdict

from number_theory import Primes, pythag

# primes less than the maximum sum of nine edge lengths

prms = Primes().set(304)

# The prism's vertical sides are rectangles with pythagorean

# triangles above them - index possible triangles with sides

# less than 51 on their non-hypotenuse sides (less than 10)

b_to_vt = defaultdict(list)

for a, b, c in pythag(51):

if a < 10:

b_to_vt[a].append((b, c))

if b < 10:

b_to_vt[b].append((a, c))

# consider possible combinations of edges for the base triangle

for b3 in permutations(b_to_vt.keys(), 3):

# avoid duplicate solutions and ensure a valid base triangle

if b3[0] < b3[1] and b3[1] - b3[0] < b3[2] < b3[1] + b3[0]:

# consider possible pythagorean triangles for the prism sides

for p3 in product(*(b_to_vt[x] for x in b3)):

# the vertical triangle heights and top edge lengths

h3, t3 = zip(*p3)

# the triangles must form the prism

if h3[0] + h3[1] == h3[2]:

# now consider the prism height

for h in range(h3[2] + 1, 50):

# form the three vertical edges

v3 = h - h3[2], h - h3[0], h

edges = b3 + t3 + v3

# the edge lengths are all different, their sum is a

# prime and the majority of them are prime

if (len(set(edges)) == 9 and sum(edges) in prms and

sum(x in prms for x in edges) > 4):

# consider the unbroken prism height

for ht in range(h + 1, 50):

# calculate the (prime) edge sum of the other broken piece

sms = sum(b3) + sum(t3) + 3 * ht - sum(v3)

if sms in prms:

print(f"sum = {sms}, height = {ht}, edges: base = {b3}, "

f"top = {t3}, vertical = {v3}")

GeoffR permalink


from number_theory import is_prime

# return the square root of a perfect square otherwise 50
# (to signal failure)
def is_psqr(x):
  return r if (r := int(x ** 0.5 + 0.5)) ** 2 == x else 50

# choose base triangle numbers (a, b, c)
for a in range(1, 10):
  for b in range(a + 1, 10):
    # ensure a valid base triangle
    for c in range(1, a + b):
      
      # the height of the first vertical edge
      for x in range(1, 50):
        
        # the cleavage plane edges corresponding to (a, b, c)    
        # are (d, e, f)
        for d in range(a + 1, 50):
          # use the lengths of edges a and d to calculate the
          # height difference between x and y and hence y
          y = x + is_psqr(d * d - a * a)
          if y > 50:
            continue
          
          for e in range(max(d, b) + 1, 50):
            # use the lengths of edges b and e to calculate the
            # height difference between y and z and hence z
            z = y + is_psqr(e * e - b * b)
            if z > 50:
              continue
            
            # calculate the remaining cleavage edge from the base
            # edge c and the height difference between z and x
            f = is_psqr(c * c + (z - x) ** 2)
            if f >= 50:
              continue
          
            # the edge lengths are all different, their sum is prime
            # and the majority of them are prime
            edges = (a, b, c, d, e, f, x, y, z)
            if not ( len(set(edges)) == 9 and is_prime(sum(edges))
                    and sum(is_prime(x) for x in edges) > 4):
              continue
            
            # now consider the other prism 
            for ht in range(z + 1, 50):
              t = sum(edges[:6]) + 3 * ht - sum(edges[6:])
              if is_prime(t):
                print(f"Base triangle sides: {a},{b},{c}")
                print(f"Heights to fracture plane: {x},{y},{z}")
                print(f"Sides of fracture plane: {d},{e},{f}")
                print(f"Edge total length of 2nd piece: {t}")
                print()

from number_theory import is_prime

# return the square root of a perfect square otherwise 50

# (to signal failure)

def is_psqr(x):

return r if (r := int(x ** 0.5 + 0.5)) ** 2 == x else 50

# choose base triangle numbers (a, b, c)

for a in range(1, 10):

for b in range(a + 1, 10):

# ensure a valid base triangle

for c in range(1, a + b):

# the height of the first vertical edge

for x in range(1, 50):

# the cleavage plane edges corresponding to (a, b, c)

# are (d, e, f)

for d in range(a + 1, 50):

# use the lengths of edges a and d to calculate the

# height difference between x and y and hence y

y = x + is_psqr(d * d - a * a)

if y > 50:

continue

for e in range(max(d, b) + 1, 50):

# use the lengths of edges b and e to calculate the

# height difference between y and z and hence z

z = y + is_psqr(e * e - b * b)

if z > 50:

continue

# calculate the remaining cleavage edge from the base

# edge c and the height difference between z and x

f = is_psqr(c * c + (z - x) ** 2)

if f >= 50:

continue

# the edge lengths are all different, their sum is prime

# and the majority of them are prime

edges = (a, b, c, d, e, f, x, y, z)

if not ( len(set(edges)) == 9 and is_prime(sum(edges))

and sum(is_prime(x) for x in edges) > 4):

continue

# now consider the other prism

for ht in range(z + 1, 50):

t = sum(edges[:6]) + 3 * ht - sum(edges[6:])

if is_prime(t):

print(f"Base triangle sides: {a},{b},{c}")

print(f"Heights to fracture plane: {x},{y},{z}")

print(f"Sides of fracture plane: {d},{e},{f}")

print(f"Edge total length of 2nd piece: {t}")

print()

Sunday Times Teaser 2992 – Crystal Cleavage Calamity!

by Stephen Hogg

Published January 26 2020 (link)

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