Sunday Times Teaser 2760 – Prime Location

by Andrew Skidmore

Published: 16 August 2015 (link)

I have a rectangular garden of area just over one hectare. It is divided exactly into three parts — a lawn, a flower bed, and a vegetable patch. Each of these three areas is a right-angled triangle with sides a whole numbers of metres in length. A fence runs along two adjacent sides of the rectangular garden. The length of the fence is a prime number of metres.

What are the dimensions of the rectangular garden?

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from number_theory import divisor_pairs, is_prime

fs = 'The garden is {} by {} metres ({} square metres).'

def is_square(x):
  t = int(x ** 0.5 + 0.5)
  return t if x == t ** 2 else None

# Let the length of the sides of the garden be a and b;
# let a point on b split it into two parts of length c
# and d;  join this point to the opposite corners with
# lines of length e and f.  It is easy to show that:
#
#    c / a = a / d = e / f  ->  a^2 = c.d
#
#    a^2 + c^2 = e^2   a^2 + d^2 = f^2

for a in range(100):
  # c and d are the divisor pairs of a^2
  for c, d in divisor_pairs(a * a):
    b = c + d
    # check that the two diagonals are integer, that
    # the area is a bit higher than 10000 and that
    # half the perimeter is a prime
    if all(is_square(a * a + x * x) for x in (c, d)):
      area = a * b
      if 10000 < area < 11000 and is_prime(a + b):
        print(fs.format(a, b, area))

# An alternative layout is to join diagonal corners
# with a line, split it at a point into two parts
# (c and d) and join the point to one of the other
# corners with a line e. Here we have:
#
#    (c + d)^2 = a^2 + b^2 
#
#    c.(c + d) = b^2   d.(c + d) = a^2

for a in range(100):
  for d, cpd in divisor_pairs(a * a):
    b = is_square((cpd - d) * cpd)
    if b and a < b and is_prime(a + b):
      area = a * b
      if 10000 < area < 11000:
        print(fs.format(a, b, area))

from number_theory import divisor_pairs, is_prime

fs = 'The garden is {} by {} metres ({} square metres).'

def is_square(x):

t = int(x ** 0.5 + 0.5)

return t if x == t ** 2 else None

# Let the length of the sides of the garden be a and b;

# let a point on b split it into two parts of length c

# and d; join this point to the opposite corners with

# lines of length e and f. It is easy to show that:

# c / a = a / d = e / f -> a^2 = c.d

# a^2 + c^2 = e^2 a^2 + d^2 = f^2

for a in range(100):

# c and d are the divisor pairs of a^2

for c, d in divisor_pairs(a * a):

b = c + d

# check that the two diagonals are integer, that

# the area is a bit higher than 10000 and that

# half the perimeter is a prime

if all(is_square(a * a + x * x) for x in (c, d)):

area = a * b

if 10000 < area < 11000 and is_prime(a + b):

print(fs.format(a, b, area))

# An alternative layout is to join diagonal corners

# with a line, split it at a point into two parts

# (c and d) and join the point to one of the other

# corners with a line e. Here we have:

# (c + d)^2 = a^2 + b^2

# c.(c + d) = b^2 d.(c + d) = a^2

for a in range(100):

for d, cpd in divisor_pairs(a * a):

b = is_square((cpd - d) * cpd)

if b and a < b and is_prime(a + b):

area = a * b

if 10000 < area < 11000:

print(fs.format(a, b, area))

Sunday Times Teaser 2760 – Prime Location

by Andrew Skidmore

Published: 16 August 2015 (link)

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