Sunday Times Teaser 3304 – Roaming Bull

by BRG on January 18, 2026

by Mark Valentine

Published Sunday January 18 2026 (link)

Farmer Quentin keeps his prize bull in a circular field of diameter 50m. To improve the layout, he decides to switch to a square field with side length equal to the old diameter. He builds the new fencing around the old, with fence posts every metre starting at the corner. After completing the new fencing, he tethers the bull outside the fence to one of these fence posts with a rope of integer metre length, to allow the removal of the inner circular fence. As a result, the bull’s temporary roaming area is now four times what it was inside the circle.

What is the length of the rope?

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BRG permalink

It took me ages looking at diagrams and even longer sorting out units before I found an answer using Jim Randell’s definition of a ‘standard quadrant’.


# With <r> as the rope length, work with a unit area of pi.r^2 / 4, the
# quadrant of a circle with radius <r>; with <q> units making 4.pi.25^2:
# q.(pi.r^2 / 4) = 4.pi.25^2  ==> q.r^2 = 16 x 25^2 = 100^2
target_area = 100 ** 2

# solve the quadratic equation a.x^2 + b.x + c = target_area
# for a positive integer root
def quadratic_root(a, b, c):
  d2 = b ** 2 - 4 * a * (c - target_area)
  if (d := round(d2 ** 0.5)) ** 2 == d2:
    s, _ = divmod(d - b, 2 * a)
    if not _:
      return s

# Assume that the Bull is tethered on a post on the west N/S 
# fence <p> metres south of the NW corner with 0 <= p <= 25
for p in range(26):

  # with a rope of length <r> buid the coefficents of a quadratic in
  # <r> (a.r^2 + b.r + c) giving the area that the bull can reach

  # add two full quadrants (the bull's area west of west N/S fence)
  # when 0 <= r <= p [add two quadrants of area r^2]
  a = 2; b = 0; c = 0

  # add one smaller quadrant (the bull's area north of the north W/E fence) 
  # when p <= r <= p - 50 [add a quadrant of area (r - p)^2]
  a += 1; b += -2 * p; c += p ** 2

  # add a second smaller quadrant (the bull's area south of the south W/E fence)
  # when p - 50 <= r <= p + 50 [add a quadrant of area (r - (50 - p))^2]
  a += 1; b -= 2 * (50 - p); c += (50 - p) ** 2

  # and a third smaller quadrant (the bull's area east of the east N/S fence)
  # when p + 50 <= r <= 100 - p [add a quadrant of area (r - (50 + p))^2]
  a += 1; b -= 2 * (50 + p); c += (50 + p) ** 2
  # note: the quadratic in r here is now (5.r^2 - 2.(p + 100).r  + 3.p^2 + 5000)
  
  if (r := quadratic_root(a, b, c)):
    print(f"A {r}m rope with Bull tethered {p}m south of NW corner.")

# With <r> as the rope length, work with a unit area of pi.r^2 / 4, the

# quadrant of a circle with radius <r>; with <q> units making 4.pi.25^2:

# q.(pi.r^2 / 4) = 4.pi.25^2 ==> q.r^2 = 16 x 25^2 = 100^2

target_area = 100 ** 2

# solve the quadratic equation a.x^2 + b.x + c = target_area

# for a positive integer root

def quadratic_root(a, b, c):

d2 = b ** 2 - 4 * a * (c - target_area)

if (d := round(d2 ** 0.5)) ** 2 == d2:

s, _ = divmod(d - b, 2 * a)

if not _:

return s

# Assume that the Bull is tethered on a post on the west N/S

# fence <p> metres south of the NW corner with 0 <= p <= 25

for p in range(26):

# with a rope of length <r> buid the coefficents of a quadratic in

# <r> (a.r^2 + b.r + c) giving the area that the bull can reach

# add two full quadrants (the bull's area west of west N/S fence)

# when 0 <= r <= p [add two quadrants of area r^2]

a = 2; b = 0; c = 0

# add one smaller quadrant (the bull's area north of the north W/E fence)

# when p <= r <= p - 50 [add a quadrant of area (r - p)^2]

a += 1; b += -2 * p; c += p ** 2

# add a second smaller quadrant (the bull's area south of the south W/E fence)

# when p - 50 <= r <= p + 50 [add a quadrant of area (r - (50 - p))^2]

a += 1; b -= 2 * (50 - p); c += (50 - p) ** 2

# and a third smaller quadrant (the bull's area east of the east N/S fence)

# when p + 50 <= r <= 100 - p [add a quadrant of area (r - (50 + p))^2]

a += 1; b -= 2 * (50 + p); c += (50 + p) ** 2

# note: the quadratic in r here is now (5.r^2 - 2.(p + 100).r + 3.p^2 + 5000)

if (r := quadratic_root(a, b, c)):

print(f"A {r}m rope with Bull tethered {p}m south of NW corner.")

Frits permalink

@Brian, I didn’t know you could read out the underscore variable. I prefer to only use underscore if I am not interested in the contents of the variable.

Reply
- BRG permalink
  
  I add to ‘not interested’ situations where I want a local variable that won’t clash
  with one I am using elsewhere.
  
  Reply

John Z. permalink


# original surface area: 25 ** 2 * pi
# as all surfaces are fractions of a circle and have pi as a common multiplier
# we leave it out of the calculations

# distance from corner
for d in range(50):
  # rope may wrap around 2 corners in addition to its nearest corner
  # but must not wrap around all 4 corners, otherwise surface overlap
  for r in range(d, 100):

    area = (
      # semicircular area centered on post with r = rope length
      r ** 2 / 2  +

      # quarter circle wrapping around nearest corner
      (r - d) ** 2 / 4 +

      # quarter circle wrapping around nearest corner and next corner

      (r - d - 50) ** 2 / 4 +

      # quarter circle wrapping around farther corner

      (r + d - 50) ** 2 / 4)

    if area == 4 * 25 ** 2:
      print('rope length:', r, 'distance from corner:', d)

# original surface area: 25 ** 2 * pi

# as all surfaces are fractions of a circle and have pi as a common multiplier

# we leave it out of the calculations

# distance from corner

for d in range(50):

# rope may wrap around 2 corners in addition to its nearest corner

# but must not wrap around all 4 corners, otherwise surface overlap

for r in range(d, 100):

area = (

# semicircular area centered on post with r = rope length

r ** 2 / 2 +

# quarter circle wrapping around nearest corner

(r - d) ** 2 / 4 +

# quarter circle wrapping around nearest corner and next corner

(r - d - 50) ** 2 / 4 +

# quarter circle wrapping around farther corner

(r + d - 50) ** 2 / 4)

if area == 4 * 25 ** 2:

print('rope length:', r, 'distance from corner:', d)

Sunday Times Teaser 3304 – Roaming Bull

by Mark Valentine

Published Sunday January 18 2026 (link)

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