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 [TM224 or search for “thatsmaths” at irishtimes.com].

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 10 km, the estimated length is accurate to within 10%. The method is statistical rather than deterministic and the estimate probable rather than precise.

Imagine a circle of diameter 1 km drawn on the map. It has circumference π and must cross the grid 4 times. More generally, a circle of circumference *L* will have 4*L*/π crossings. Put another way, the length equals the number of crossings multiplied by π/4. Since π 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 π/2 kilometres, we halve the number of crossings for the length in miles.

**George-Louis Leclerc, Comte de Buffon**

What has this to do with the Comte de Buffon? In 1733, he considered the question of dropping a stick onto 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 π or, knowing π as we do, to estimate the length of the stick. His discovery was one of the earliest results in geometric probability theory.

About thirty years later, another French mathematician, Joseph-Émile Barbier, showed that Buffon’s formula remains valid even when the needle is 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”

**Sources**

November issue of *The Summit.** *PDF here.

The original article, by Ben Craven, is here. Or Here.

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**That’s Maths II: A Ton of Wonders**

by Peter Lynch now available.

Full details and links to suppliers at

http://logicpress.ie/2020-3/

>> Review in *The Irish Times <<*

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