Sunday Times Teaser 3264 – Saving Rum

by BRG on April 13, 2025

by Mark Valentine

Published Sunday April 13 2025 (link)

Captain Noah was tired of spilling his rum at sea. His glass (twice the density of the rum) is a hollow cylinder with a solid flat base. The outer diameter is 72 mm. The glass thickness and base height are both 4 mm. After testing the stability of the upright glass, he instructed the steward to keep his glass topped up to one-third full, where it has the lowest centre of mass.

What is the overall height of the glass (including the base)?

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BRG permalink


# secant root finder (not general purpose)
def find_root(f, a, b):
  while True:
    m_n = a - f(a) * (b - a) / (f(b) - f(a))
    f_m_n = f(m_n)
    if abs(f_m_n) < 1e-13:
      return t if abs((t := round(m_n)) - m_n) < 1e-12 else m_n
    elif f(a) * f_m_n < 0:
      b = m_n
    elif f(b) * f_m_n < 0:
      a = m_n

# calculate the centre of mass for the values: internal height
# of glass <h>, height of rum <g>, external glass radius (r), 
# glass thickness (b) and glass density/rum density (d)
r, b, d = 36, 4, 2
def com(h, g):
  mb = d * b * r ** 2
  mc = d * h * (r ** 2 - (r - b) ** 2)
  mr = g * (r - b) ** 2
  return (mb * (b / 2) + mc * (b + h / 2) + mr * (b + g / 2)) / (mc + mb + mr)

# When the rum level is below the centre of mass (CoM),  the CoM lowers
# as the rum rises; when the rum is above the CoM, the CoM rises as the
# rum rises; hence the minimum level of the CoM occurs when the CoM and
# the rum are at the same level; we require this for a 1/3 filled glass
g = find_root(lambda g: com(3 * g, g) - (g + 4), 10, 50)
# where the CoM and the rum are at the same level
print(f"The overall glass height is {3 * g + 4}mm [CoM {g + 4}mm, rum {g}mm].")

# secant root finder (not general purpose)

def find_root(f, a, b):

while True:

m_n = a - f(a) * (b - a) / (f(b) - f(a))

f_m_n = f(m_n)

if abs(f_m_n) < 1e-13:

return t if abs((t := round(m_n)) - m_n) < 1e-12 else m_n

elif f(a) * f_m_n < 0:

b = m_n

elif f(b) * f_m_n < 0:

a = m_n

# calculate the centre of mass for the values: internal height

# of glass <h>, height of rum <g>, external glass radius (r),

# glass thickness (b) and glass density/rum density (d)

r, b, d = 36, 4, 2

def com(h, g):

mb = d * b * r ** 2

mc = d * h * (r ** 2 - (r - b) ** 2)

mr = g * (r - b) ** 2

return (mb * (b / 2) + mc * (b + h / 2) + mr * (b + g / 2)) / (mc + mb + mr)

# When the rum level is below the centre of mass (CoM), the CoM lowers

# as the rum rises; when the rum is above the CoM, the CoM rises as the

# rum rises; hence the minimum level of the CoM occurs when the CoM and

# the rum are at the same level; we require this for a 1/3 filled glass

g = find_root(lambda g: com(3 * g, g) - (g + 4), 10, 50)

# where the CoM and the rum are at the same level

print(f"The overall glass height is {3 * g + 4}mm [CoM {g + 4}mm, rum {g}mm].")

F rits permalink


# golden section function minimiser (return x where f(x) is minimal)
def gs_search(f, a, b, tol=1.0e-13):
  R = 0.5 * (5.0 ** 0.5 - 1.0)
  C = 1.0 - R
  x1, x2 = R * a + C * b, C * a + R * b
  f1, f2 = f(x1), f(x2)
  while b - a > tol:
    if f1 > f2:
      a, x1, x2 = x1, x2, C * x1 + R * b
      f1, f2 = f2, f(x2)
    else:
      b, x1, x2 = x2, R * a + C * x2, x1
      f1, f2 = f(x1), f1
  return x1 if f1 < f2 else x2


d = 72 # diameter
w = 4  # thickness glass
p = 2  # density glass

# find height <H> where the minimum center of mass occurs at rum height <h>
def find_height(H):
 
  # find center of mass for h = rum height
  def find_com(h):
    # calculate mass and relative vertical positions (bottom = 0)
    # surface area is pi * r^2 but we can ignore pi (in both parts of fraction)
    s = ((d - 2 * w) / 2)**2
    m1 = w * s * p              # the base (without part of the wall)
    #y1 = w / 2
 
    s = (d / 2)**2 - ((d - 2 * w) / 2)**2
    m2 = H * s * p              # outside wall (including part of base)
    #y2 = H / 2
 
    s = ((d - 2 * w) / 2)**2
    m3 = h * s                  # rum (with unit density)
    #y3 = w + h / 2
 
    # overall center of mass of multiple bodies
    # Z = (m1 * y1 + m2 * y2 + m3 * y3) / (m1 + m2 + m3)
    Z = (m1 * w + m2 * H + m3 * (2 * w + h)) / (2 * (m1 + m2 + m3))
    # return a value that should be minimised
    return Z
 
  h = gs_search(find_com, 0, H)
  # return a value that should be minimised
  return abs(H - w - 3 * h)

# find height <H> where the minimum center of mass occurs at rum height <h>
# and where rum is close to 100/3 percent
H = gs_search(find_height, 20, 200)

print("answer", round(H, 4))

# golden section function minimiser (return x where f(x) is minimal)

def gs_search(f, a, b, tol=1.0e-13):

R = 0.5 * (5.0 ** 0.5 - 1.0)

C = 1.0 - R

x1, x2 = R * a + C * b, C * a + R * b

f1, f2 = f(x1), f(x2)

while b - a > tol:

if f1 > f2:

a, x1, x2 = x1, x2, C * x1 + R * b

f1, f2 = f2, f(x2)

else:

b, x1, x2 = x2, R * a + C * x2, x1

f1, f2 = f(x1), f1

return x1 if f1 < f2 else x2

d = 72 # diameter

w = 4 # thickness glass

p = 2 # density glass

# find height <H> where the minimum center of mass occurs at rum height <h>

def find_height(H):

# find center of mass for h = rum height

def find_com(h):

# calculate mass and relative vertical positions (bottom = 0)

# surface area is pi * r^2 but we can ignore pi (in both parts of fraction)

s = ((d - 2 * w) / 2)**2

m1 = w * s * p # the base (without part of the wall)

#y1 = w / 2

s = (d / 2)**2 - ((d - 2 * w) / 2)**2

m2 = H * s * p # outside wall (including part of base)

#y2 = H / 2

s = ((d - 2 * w) / 2)**2

m3 = h * s # rum (with unit density)

#y3 = w + h / 2

# overall center of mass of multiple bodies

# Z = (m1 * y1 + m2 * y2 + m3 * y3) / (m1 + m2 + m3)

Z = (m1 * w + m2 * H + m3 * (2 * w + h)) / (2 * (m1 + m2 + m3))

# return a value that should be minimised

return Z

h = gs_search(find_com, 0, H)

# return a value that should be minimised

return abs(H - w - 3 * h)

# find height <H> where the minimum center of mass occurs at rum height <h>

# and where rum is close to 100/3 percent

H = gs_search(find_height, 20, 200)

print("answer", round(H, 4))

Frits permalink

Assuming that the overall height of the glass is a whole number.


d = 72 # diameter
w = 4  # thickness glass
p = 2  # density glass

# find center of mass for h = rum height and masses m1 and m2
def find_com(H, h, m1, m2):
  m3 = h * ((d - 2 * w) / 2)**2                    # rum (with unit density)

  # return overall center of mass of multiple bodies
  return (m1 * w + m2 * H + m3 * (2 * w + h)) / (2 * (m1 + m2 + m3))

for H in range(20, 201):
  # assume a glass 1/3 filled with rum has a minimal center of mass
  h = (H - w) / 3
  m1 = w * ((d - 2 * w) / 2)**2 * p                # the base (without part of the wall)
  m2 = H * ((d / 2)**2 - ((d - 2 * w) / 2)**2) * p # outside wall (including part of base)
  # find center of mass for h = rum height
  com = find_com(H, h, m1, m2)
  
  # check if center of mass is higher for values surrounding <h>
  m = 0.000001
  if all(find_com(H, x, m1, m2) > com for x in [h - m, h + m]):
    print("answer:", H)

d = 72 # diameter

w = 4 # thickness glass

p = 2 # density glass

# find center of mass for h = rum height and masses m1 and m2

def find_com(H, h, m1, m2):

m3 = h * ((d - 2 * w) / 2)**2 # rum (with unit density)

# return overall center of mass of multiple bodies

return (m1 * w + m2 * H + m3 * (2 * w + h)) / (2 * (m1 + m2 + m3))

for H in range(20, 201):

# assume a glass 1/3 filled with rum has a minimal center of mass

h = (H - w) / 3

m1 = w * ((d - 2 * w) / 2)**2 * p # the base (without part of the wall)

m2 = H * ((d / 2)**2 - ((d - 2 * w) / 2)**2) * p # outside wall (including part of base)

# find center of mass for h = rum height

com = find_com(H, h, m1, m2)

# check if center of mass is higher for values surrounding <h>

m = 0.000001

if all(find_com(H, x, m1, m2) > com for x in [h - m, h + m]):

print("answer:", H)

Sunday Times Teaser 3264 – Saving Rum

by Mark Valentine

Published Sunday April 13 2025 (link)

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