Sunday Times Teaser 3103 – Empowered

by BRG on March 11, 2022

by Howard Williams

Published Sunday March 13 2022 (link)

I have a rectangular garden whose sides are whole numbers of feet. Away from the edge, an exposed strip of ground, again a whole number of feet in width, runs straight across (not diagonally) from a shorter side of the garden to the other shorter side. I need to run an electrical cable along the ground, between two opposite corners of the garden. Where the cable crosses the exposed area, it has to be threaded through expensive linear ducting to avoid damage. Because of costs, whilst seeking to minimise the length of cable, my overriding concern is to minimise the length of ducting used.

The straight-line distance between the two corners to be connected is 123 feet, but in minimising costs, the length of cable needed is a whole number of feet longer than this.

What is the length of cable needed?

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Brian Gladman permalink


from itertools import count
from math import isqrt

# garden width w, length l, diagonal D (123),
# cable length c, duct length d such that: 
#
#   l^2 +    w^2     =  D^2
#   l^2 + (w - d)^2  =  (c - d)^2
#
# hence d = (1/2).(c^2 - D^2) / (c - w)

# the garden diagonal
D = 123

# consider the width of the garden
for w in range(1, 87):
  # find the length of the garden
  l2 = D ** 2 - w ** 2
  l = isqrt(l2)
  # which must be an integer
  if l ** 2 == l2:
    # consider the cable length
    for c in count(D + 1):
      # find the duct length
      d, r = divmod(c ** 2 - D ** 2, 2 * (c - w))
      # which is an integer less than the garden width
      if d >= w:
        break
      if not r:
        print(f"Cable = {c}', Duct = {d}', Garden {w}' x {l}'.")

from itertools import count

from math import isqrt

# garden width w, length l, diagonal D (123),

# cable length c, duct length d such that:

# l^2 + w^2 = D^2

# l^2 + (w - d)^2 = (c - d)^2

# hence d = (1/2).(c^2 - D^2) / (c - w)

# the garden diagonal

D = 123

# consider the width of the garden

for w in range(1, 87):

# find the length of the garden

l2 = D ** 2 - w ** 2

l = isqrt(l2)

# which must be an integer

if l ** 2 == l2:

# consider the cable length

for c in count(D + 1):

# find the duct length

d, r = divmod(c ** 2 - D ** 2, 2 * (c - w))

# which is an integer less than the garden width

if d >= w:

break

if not r:

print(f"Cable = {c}', Duct = {d}', Garden {w}' x {l}'.")

John Z permalink


# Find rectangle sides
for i in range(1, 123):
  for j in range(i, 123):
    if i * i + j * j == 123 * 123:

      break # out of both loops (Break the European Central Bank)
  else: # European 
    continue # Central
  break # Bank
# sides are i and j

# ducting traverses perpendicular to strip  
# Limit case is when strip approaches long edge
# consider strip width (also duct length)
for s in range(1, i): # 'i' is short side
  cable = s + ((i - s) ** 2 + j * j) ** 0.5 # strip width + diagonal
  if cable == int(cable):
    print(f'cable length: {int(cable)} sides: {i}, {j}')

# Find rectangle sides

for i in range(1, 123):

for j in range(i, 123):

if i * i + j * j == 123 * 123:

break # out of both loops (Break the European Central Bank)

else: # European

continue # Central

break # Bank

# sides are i and j

# ducting traverses perpendicular to strip

# Limit case is when strip approaches long edge

# consider strip width (also duct length)

for s in range(1, i): # 'i' is short side

cable = s + ((i - s) ** 2 + j * j) ** 0.5 # strip width + diagonal

if cable == int(cable):

print(f'cable length: {int(cable)} sides: {i}, {j}')

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Sunday Times Teaser 3103 – Empowered

by Howard Williams

Published Sunday March 13 2022 (link)

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