Teaser 2678: Map snap
From The Sunday Times, 19th January 2014 [link] [link]
I have two rectangular maps depicting the same area, the larger map being one metre from west to east and 75cm from north to south. I’ve turned the smaller map face down, turned it 90 degrees and placed it in the bottom corner of the larger map with the north-east corner of the smaller map touching the south-west corner of the larger map. I have placed a pin through both maps, a whole number of centimetres from the western edge of the larger map. This pin goes through the same geographical point on both maps. On the larger map 1cm represents 1km. On the smaller map 1cm represents a certain whole number of kilometres …
… how many?
[teaser2678]
S2T2 
Jim Randell 9:27 am on 6 August 2020 Permalink |
See also: Enigma 1177.
If we suppose the smaller map has a scale of k kilometres per centimetre, then the dimensions of the smaller map are (100 / k) (west to east) and (75 / k) (south to north).
If the point we are interested on the map has an easting of x and a northing of y (measured from the SW corner of the maps), then we have the following equations:
If we consider possible integer values for k, we can determine values for x and y.
So we can look for values of k that give an integer value for x.
This Python program runs in 49ms.
Run: [ @replit ]
from enigma import (irange, div, fdiv, printf) for k in irange(2, 75): x = div(25 * (3 * k - 4), k * k - 1) if x is None: continue y = fdiv(100 - x, k) printf("k={k} x={x} y={y:g}")Solution: The scale of the smaller map is 1cm = 6km.
So the smaller map measures 16.67 cm by 12.5 cm.
The marked point is 10 km from the western edge of the maps, and 15 km from the southern edge of the maps.
On the larger map the distances are (10cm, 15cm) on the smaller map the distances are (1.67cm, 2.5cm).
LikeLike