# Buffon’s Noodle and calculating the length of a hillwalk

## That’s Maths: Times and distances are of vital importance in the mountains

In addition to some beautiful photos and maps and descriptions of upland challenges in Ireland and abroad, the November issue of the Summit, the Mountain Views Quarterly Newsletter for hikers and hillwalkers, describes a method to find the length of a walk based on ideas originating with the French naturalist and mathematician George-Louis Leclerc, Comte de Buffon.

Times and distances are of vital importance in the mountains, and this article, by hillwalker Ben Craven, gives a simple way to estimate walking distance. It uses the grid of kilometre squares printed on OSI maps: to estimate the length of the route, count the number of gridlines that it crosses, and divide by 2 to get the length in miles.

The longer a walk is, the more gridlines it crosses and the more accurate the estimate becomes. Moreover, the fewer convolutions and zig-zags, the better. The results are best for well-rounded routes and poor for walks that follow a fixed compass bearing. Generally, for routes of more than about 10km, the estimated length is accurate to within 10 per cent. The method is statistical rather than deterministic and the estimate probable rather than precise.

Imagine a circle of diameter 1km drawn on the map. It has circumference pi and must cross the grid four times. More generally, a circle of circumference L will have 4L/pi crossings. Put another way, the length equals the number of crossings multiplied by pi/4. Since pi is close to 3, we may just take three-quarters of the number of crossings to get the length in kilometres. The distance in miles is simpler still. Since 1 mile is close to pi/2km, we halve the number of crossings for the length in miles.