Sunday Times Teaser 3262 – Log Cabins

by BRG on 30 March 2025

by John Owen

Published Sunday March 30 2025 (link)

Ann, Bert, Chloe and Dave own log cabins on the shore of a circular lake. Ann often rows across the lake (in a straight line) to visit Bert, and vice-versa. Chloe and Dave also visit each other in the same way. The two routes cross in the lake at point X. Interestingly, the four distances from the log cabins to X are all whole numbers of metres (greater than 20m), but the list of distances contains no digit more than once. Chloe’s distance from her log cabin to X is twice Ann’s, while Bert’s distance is three times Ann’s.

In ascending order, what are the four distances from the log cabins to X?

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BRG permalink


# See: https://en.wikipedia.org/wiki/Cyclic_quadrilateral
# In a cyclic quadrilateral, the two products of the lengths
# of opposite semi-diagonals are equal. 

# Lengths (n = AX/2 > 10): AX = 2.n, BX = 6.n, CX = 4.n, DX = 3.n 
n = 10
while True:
  n += 1
  # consider the four lengths ...
  ss = [str(i * n) for i in (2, 3, 4, 6)]
  # ... while they use at most ten digits
  if (ln := len(t := ''.join(ss))) <= 10:
    # if all digits are different
    if len(set(t)) == ln:
      print(f"Solution: [{', '.join(ss)}]")
  else:
    break

# See: https://en.wikipedia.org/wiki/Cyclic_quadrilateral

# In a cyclic quadrilateral, the two products of the lengths

# of opposite semi-diagonals are equal.

# Lengths (n = AX/2 > 10): AX = 2.n, BX = 6.n, CX = 4.n, DX = 3.n

n = 10

while True:

n += 1

# consider the four lengths ...

ss = [str(i * n) for i in (2, 3, 4, 6)]

# ... while they use at most ten digits

if (ln := len(t := ''.join(ss))) <= 10:

# if all digits are different

if len(set(t)) == ln:

print(f"Solution: [{', '.join(ss)}]")

else:

break

Frits permalink

Not assuming there is a single solution.


# assume AX = a, BX = b, CX = c, DX = d
# a.b = c.d as triangles AXC and BXD are similar (same angles)

# consider distance a = 2x  (a must be even as d must be eqyal to 3.a/2)
# 4x < 100 otherwise both b and c are of the form 1xx if d remains below 100
# (b, c, and d can't be all 3-digit numbers)
for x in range(12, 25):  
  # not duplicate digits for <a> or multiple numbers ending on 0
  if x % 11 == 0 or x % 5 == 0: continue
  (a, b, c, d) = (2 * x, 6 * x, 4 * x, 3 * x)
  ss = ''.join((str(x) for x in (a, b, c, d)))
  # list of distances contains no digit more than once
  if len(ss) == len(set(ss)):
    print("answer:", sorted((a, b, c, d)))

# assume AX = a, BX = b, CX = c, DX = d

# a.b = c.d as triangles AXC and BXD are similar (same angles)

# consider distance a = 2x (a must be even as d must be eqyal to 3.a/2)

# 4x < 100 otherwise both b and c are of the form 1xx if d remains below 100

# (b, c, and d can't be all 3-digit numbers)

for x in range(12, 25):

# not duplicate digits for <a> or multiple numbers ending on 0

if x % 11 == 0 or x % 5 == 0: continue

(a, b, c, d) = (2 * x, 6 * x, 4 * x, 3 * x)

ss = ''.join((str(x) for x in (a, b, c, d)))

# list of distances contains no digit more than once

if len(ss) == len(set(ss)):

print("answer:", sorted((a, b, c, d)))

Sunday Times Teaser 3262 – Log Cabins

by John Owen

Published Sunday March 30 2025 (link)

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