Sunday Times Teaser 2678 – Map Snap

by Ian Duff

I have a rectangular map with dimensions 1 metre east-west, 75 centimetres north-south and a scale of 1 centimetre to a kilometre.

I have a smaller map covering the same geographical area that I have turned face down, rotated 90 degrees and placed on the larger map with its north-east corner on the south-west corner of the latter.

A pin that is pushed through both maps a whole number of centimetres from the larger map’s western edge goes through the same geographical point on both maps.

On the smaller map each centimetre represents a whole number of kilometres. How many?

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brian gladman permalink


#  +-----------------------------------------+
#  |     75/k            100                 |
#  +---------------+                         |
#  |  x/k |        |                         |
#  |      +--------|                         |
#  | x       y/k   |                      75 |
#  |---+           | 100/k                   |
#  |   |           |                         |
#  |   | y         |                         |
#  |   |           |                         |
#  +---------------+-------------------------+
#
#  If the scale of the small map is k km/cm, we have
#
#    x + y / k =  75 / k  and  y + x / k = 100 / k
#
#  giving (with both x and k integer):
#
#    x = 25 * (3 * k - 4) / (k * k - 1)
#    y = 25 * (4 * k - 3) / (k * k - 1) = (100 - x) / k

for k in range(2, 10):
  t, b = 25 * (3 * k - 4), k * k - 1
  if not t % b:
    x, y = t // b, 25 * (4 * k - 3) / b
    print('{} km/cm (x = {}, y = {}).'.format(k, x, y))

# +-----------------------------------------+

# | 75/k 100 |

# +---------------+ |

# | x/k | | |

# | +--------| |

# | x y/k | 75 |

# |---+ | 100/k |

# | | | |

# | | y | |

# | | | |

# +---------------+-------------------------+

# If the scale of the small map is k km/cm, we have

# x + y / k = 75 / k and y + x / k = 100 / k

# giving (with both x and k integer):

# x = 25 * (3 * k - 4) / (k * k - 1)

# y = 25 * (4 * k - 3) / (k * k - 1) = (100 - x) / k

for k in range(2, 10):

t, b = 25 * (3 * k - 4), k * k - 1

if not t % b:

x, y = t // b, 25 * (4 * k - 3) / b

print('{} km/cm (x = {}, y = {}).'.format(k, x, y))

ahmet cetinbudaklar permalink

Taking into account of the following two equations

x+y/k=75/k
x/k+y=100/k

and , hence we can get:

x=(75k-100)/(k^2-1) as a result of which by trial and error we can manage to find the result.

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Sunday Times Teaser 2678 – Map Snap

by Ian Duff

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