Buffon’s Noodle and calculating the length of a hillwalk

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.

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George-Louis Leclerc

What has this to do with the Comte de Buffon? In 1733 he considered the question of dropping a stick on to a wooden floor with planks of fixed width. The stick might land on a single board, or it might cross between two or more boards. Buffon regarded this process as a game of chance, and he was interested in the odds of a given drop crossing a line.

The game can be played on a table-top with a needle and a ruled sheet of paper and is generally known as “Buffon’s Needle”. Buffon calculated the average number of crossings for a stick dropped many times. This number varies directly with the length of the stick and inversely with the width of the planks. His formula could be used as a (very inefficient) way to calculate pi or, knowing pi as we do, to estimate the length of the stick. His discovery was one of the earliest results in geometric probability theory.

About 30 years later, another French mathematician, Joseph-Émile Barbier, showed that Buffon’s formula remained valid even when the needle was bent. This surprising result is less perplexing when we realise that multiple crossings must be counted accordingly: for a needle bent double the chance of crossing a gap is halved, but each occurrence is a double cross, which must be counted twice. Since a straight line bent into an arbitrary curve resembles a noodle, this variation of the game is known as “Buffon’s Noodle”.

Peter Lynch is emeritus professor at UCD School of Mathematics and Statistics, University College Dublin. He blogs at thatsmaths.com