Sunday Times Teaser 2769 – Coming Home

by Danny Roth

Published: 18 October 2015 (link)

George, Martha and their daughter all drive at their own steady speeds (whole numbers of mph), the daughter’s speed being 10mph more than Martha’s. One day George left home to drive to his daughter’s house at the same time as she left her house to visit her parents: they passed each other at the Crossed Keys pub. Another time Martha left her daughter’s to return home at the same time as her daughter started the reverse journey: they too passed at the Crossed Keys. The distance from George’s to the pub is a two-figure number of miles, and the two digits in reverse order give the distance of the pub from their daughter’s.

How far is it from George’s home to the Crossed Keys?

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Brian Gladman permalink


# Let x and y be the distances from the crossing point 
# to the parent's and daughter's homes respectively

# Let m, f and d be the speeds of the mother, father
# and daughter (with d = m + 10)

# we hence have:
#
#   x / f = y / d
#   x / d = y / m  = y / (d - 10)
#
# So:
#
#   d = 10 * x / (x - y)
#   f = d * x / y

fs = ('{} from {} miles in total, speeds ' +
      '{}, {} and {} mph (mother, daughter, father)')

for x in range(10, 100):
  y = 10 * (x % 10) + x // 10
  if x > y:
    d, r = divmod(10 * x, x - y)
    if not r:
      f, r = divmod(d * x, y)
      if not r:
        print(fs.format(x, x + y, d - 10, d, f))

# Let x and y be the distances from the crossing point

# to the parent's and daughter's homes respectively

# Let m, f and d be the speeds of the mother, father

# and daughter (with d = m + 10)

# we hence have:

# x / f = y / d

# x / d = y / m = y / (d - 10)

# So:

# d = 10 * x / (x - y)

# f = d * x / y

fs = ('{} from {} miles in total, speeds ' +

'{}, {} and {} mph (mother, daughter, father)')

for x in range(10, 100):

y = 10 * (x % 10) + x // 10

if x > y:

d, r = divmod(10 * x, x - y)

if not r:

f, r = divmod(d * x, y)

if not r:

print(fs.format(x, x + y, d - 10, d, f))

For a manual solution it is easy to show that the distance is a multiple of nine, which means that only a few values need to be checked.

Sunday Times Teaser 2769 – Coming Home

by Danny Roth

Published: 18 October 2015 (link)

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