Sunday Times Teaser 3227 – Chess Club Cupboard

by BRG on July 28, 2024

by Phil Jackson

Published Friday July 19 2024 (link)

When installing the Chess Club cupboard in its new room, the members carelessly jammed it on the low ceiling and back wall, as shown from the side (not to scale).

Measurements were duly taken and the following were noted. The floor and ceiling were level and the back wall was vertical. The cupboard was a rectangular cuboid. The ceiling height was just 2cm greater than the cupboard height. Also, the depth of the cupboard from front to back was 148cm less than the cupboard height. Finally the point on the ceiling where the cupboard jammed was a distance from the back wall that was 125cm less than the cupboard height.

What is the cupboard height?

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BRG permalink


# <cd> is the integer cupboard depth
cd = 0
while True:
  cd += 1
  # <ch> is the integer cupboard height
  ch = cd + 148
  
  # <hc2w> is the integer horizontal distance from the top
  # of the cupboard to the wall
  hc2w = ch - 125
  
  # <vc2c> is the (possibly non-integer) vertical distance
  # from the point where the cupboard touches the back wall
  # to the ceiling
  vc2c = (ch ** 2 - hc2w ** 2) ** (1 / 2)
  
  # <vc2f> is the (possibly non-integer) vertical distance
  # from the point where the cupboard touches the back wall
  # to the floor
  vc2f = cd * hc2w / ch
  
  # the floor to ceiling distance is 2cm more than the cupboard height
  if vc2c + vc2f == ch + 2:
    print(f"Cupboard (d x h) {cd}cm x {ch}cm, room height {ch + 2}cm "
          f"({vc2c = }cm, {vc2f = }cm).")
    break

# <cd> is the integer cupboard depth

cd = 0

while True:

cd += 1

# <ch> is the integer cupboard height

ch = cd + 148

# <hc2w> is the integer horizontal distance from the top

# of the cupboard to the wall

hc2w = ch - 125

# <vc2c> is the (possibly non-integer) vertical distance

# from the point where the cupboard touches the back wall

# to the ceiling

vc2c = (ch ** 2 - hc2w ** 2) ** (1 / 2)

# <vc2f> is the (possibly non-integer) vertical distance

# from the point where the cupboard touches the back wall

# to the floor

vc2f = cd * hc2w / ch

# the floor to ceiling distance is 2cm more than the cupboard height

if vc2c + vc2f == ch + 2:

print(f"Cupboard (d x h) {cd}cm x {ch}cm, room height {ch + 2}cm "

f"({vc2c = }cm, {vc2f = }cm).")

break

For those who prefer manual solutions, I have posted mine here.

Frits permalink


# return divisors of <n> between <mn> and <mx> (except 1 and n)
divs = lambda n, mn, mx: [d for d in range(mn, n // 2 + 1) 
                          if n % d == 0 and d <= mx]
  
# - only the integer solution is reported as the non-integer solution
#   is invalid (less than 148 cm)
# - difference between ceiling and floor horizontal jamming point/wall 
#   distances is equal to sqrt((h - 150)^2 - 600) --> h >= 175 
# - a low ceiling height, so less than the normal 8 feet (243,8 cm) ?
# - formula after analysis: (55 - (10 * h - 625)**.5) * h = 3700

for h in divs(3700, 175, 244): 
  if 10 * h - 625 == (3700 // h - 55)**2: 
    print(f"answer: the cupboard height is {h} cm")

# return divisors of <n> between <mn> and <mx> (except 1 and n)

divs = lambda n, mn, mx: [d for d in range(mn, n // 2 + 1)

if n % d == 0 and d <= mx]

# - only the integer solution is reported as the non-integer solution

# is invalid (less than 148 cm)

# - difference between ceiling and floor horizontal jamming point/wall

# distances is equal to sqrt((h - 150)^2 - 600) --> h >= 175

# - a low ceiling height, so less than the normal 8 feet (243,8 cm) ?

# - formula after analysis: (55 - (10 * h - 625)**.5) * h = 3700

for h in divs(3700, 175, 244):

if 10 * h - 625 == (3700 // h - 55)**2:

print(f"answer: the cupboard height is {h} cm")

Sunday Times Teaser 3227 – Chess Club Cupboard

by Phil Jackson

Published Friday July 19 2024 (link)

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