Sunday Times Teaser 3176 – Equal Shares

by BRG on August 6, 2023

by Howard Williams

Published Sunday August 06 2023 (link)

When I married, my wife insisted that everything be equally shared between us, which I readily agreed to, provided it included the gardening.

The garden is rectangular, with a smaller rectangular vegetable plot, of different relative dimensions, in one corner, the area of which is less than 7 per cent of that of the whole garden. The rest of the garden is lawned.

To satisfy my wife, I constructed the shortest straight length of partition that would split the garden in half, so that half the vegetable plot and half the lawn were each side of the partition. The length of the partition was 30 metres, which exactly equalled the perimeter of the vegetable plot. Both before and after partitioning, all side lengths were an exact number of metres.

What is the area of the lawn in each half?

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BRG permalink


from fractions import Fraction as RF

# To divide both the lawn and the vegetable plot in half, the
# partition has to include the centre points of the garden and 
# the vegertable plot; when the line between these two points is 
# scaled up to be 30cm in length, the scaled vertical distance
# between them is the garden height which must be an integer

# the partition and vegetable plot perimeter 
d = 30

# the possible dimensions of the vegetable plot (x, y)
for x in range(1, 15):
  if (y := 15 - x) > 0:
    # the garden width
    for w in range(x + 1, 40):
      # the garden height
      for h in range(y + 1, w + 1):
        # the ratio between the garden height and the vertical distance 
        # between the garden and vegatable plot centres   
        sf = RF(h, h - y)
        # scale the partition accordingly and check its length
        if ((h - y) ** 2 + (w - x) ** 2) * sf ** 2 == d ** 2:
          # check that possible non-integer side lengths are integer
          if not ((x * h - y * w) % (2 * (h - y)) or (d * y) % h):
            # check the area constraint
            if 100 * x * y < 7 * w * h:
              t = (w * h - x * y) / 2
              print(f"{t:.0f}m^2 [{w = }, {h = }, {x = :02}, {y = }].")

from fractions import Fraction as RF

# To divide both the lawn and the vegetable plot in half, the

# partition has to include the centre points of the garden and

# the vegertable plot; when the line between these two points is

# scaled up to be 30cm in length, the scaled vertical distance

# between them is the garden height which must be an integer

# the partition and vegetable plot perimeter

d = 30

# the possible dimensions of the vegetable plot (x, y)

for x in range(1, 15):

if (y := 15 - x) > 0:

# the garden width

for w in range(x + 1, 40):

# the garden height

for h in range(y + 1, w + 1):

# the ratio between the garden height and the vertical distance

# between the garden and vegatable plot centres

sf = RF(h, h - y)

# scale the partition accordingly and check its length

if ((h - y) ** 2 + (w - x) ** 2) * sf ** 2 == d ** 2:

# check that possible non-integer side lengths are integer

if not ((x * h - y * w) % (2 * (h - y)) or (d * y) % h):

# check the area constraint

if 100 * x * y < 7 * w * h:

t = (w * h - x * y) / 2

print(f"{t:.0f}m^2 [{w = }, {h = }, {x = :02}, {y = }].")

Frits permalink

A variation of Brian’s program (checking all the requirements).


is_square = lambda n: int(rt := n**.5) == rt
 
# To divide both the lawn and the vegetable plot in half, the
# partition has to include the centre points of the garden and 
# the vegetable plot; when the line between these two points is 
# scaled up to be 30m in length, the scaled vertical distance
# between them is a garden side which must be an integer
 
# the partition and vegetable plot perimeter 
L = 30
 
# possible dimensions of the vegetable plot with x > y
# the garden has dimension g1 by g2
for x in range(8, 15):
  y = 15 - x
  # g1 and g2 < 42, (g1 - 2.a)^2 + g2^2 == 900 and integer a <= 6
  for g1 in range(y + 1, 42):
    # the ratio between the sides of the big and small triangle 
    # sf = g1 / (g1 - y)
    
    # scale the partition accordingly and check its length
    # we now have z2 * sf^2 == L^2 with z2 = (g1 - y)^2 + (g2 - x)^2 
     
    (z2, r) = divmod(L**2 * (g1 - y)**2, g1**2)
    if r or not is_square(z2): continue 
    g2minx2 = z2 - (g1 - y)**2
    if g2minx2 <= 0 or not is_square(g2minx2): continue 
    
    # the other garden dimension
    g2 = int(g2minx2**.5) + x   
    
    # different relative dimensions
    if g1 * y == g2 * x or g1 * x == g2 * y: continue
    
    # the distance <a> from the corner to the point where the partition hits 
    # the border is an integer,  (g1 - y) / (g2 - x) = (y - 2 * a) / x
    a, r = divmod(x - y * (g2 - x) / (g1 - y), 2)
    if r: continue
    
    # plot area is less than 7 per cent of that of the whole garden
    if 100 * x * y < 7 * g1 * g2:
      t = (g1 * g2 - x * y) / 2
      print(f"{t:g}m^2 [{g2 = }, {g1 = }, {x = :02}, {y = }].")

is_square = lambda n: int(rt := n**.5) == rt

# To divide both the lawn and the vegetable plot in half, the

# partition has to include the centre points of the garden and

# the vegetable plot; when the line between these two points is

# scaled up to be 30m in length, the scaled vertical distance

# between them is a garden side which must be an integer

# the partition and vegetable plot perimeter

L = 30

# possible dimensions of the vegetable plot with x > y

# the garden has dimension g1 by g2

for x in range(8, 15):

y = 15 - x

# g1 and g2 < 42, (g1 - 2.a)^2 + g2^2 == 900 and integer a <= 6

for g1 in range(y + 1, 42):

# the ratio between the sides of the big and small triangle

# sf = g1 / (g1 - y)

# scale the partition accordingly and check its length

# we now have z2 * sf^2 == L^2 with z2 = (g1 - y)^2 + (g2 - x)^2

(z2, r) = divmod(L**2 * (g1 - y)**2, g1**2)

if r or not is_square(z2): continue

g2minx2 = z2 - (g1 - y)**2

if g2minx2 <= 0 or not is_square(g2minx2): continue

# the other garden dimension

g2 = int(g2minx2**.5) + x

# different relative dimensions

if g1 * y == g2 * x or g1 * x == g2 * y: continue

# the distance <a> from the corner to the point where the partition hits

# the border is an integer, (g1 - y) / (g2 - x) = (y - 2 * a) / x

a, r = divmod(x - y * (g2 - x) / (g1 - y), 2)

if r: continue

# plot area is less than 7 per cent of that of the whole garden

if 100 * x * y < 7 * g1 * g2:

t = (g1 * g2 - x * y) / 2

print(f"{t:g}m^2 [{g2 = }, {g1 = }, {x = :02}, {y = }].")

Sunday Times Teaser 3176 – Equal Shares

by Howard Williams

Published Sunday August 06 2023 (link)

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