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In the shadow of Colossus

A crack at code breaking gives a fascinating mathematical dimension to history studies, says Jenny Houssart.

As a resident of Milton Keynes for some years I have taken a scholarly interest in the many new pubs built and the names chosen for them. Until recently my favourite was "The Eager Poet", for a pub bearing a picture of John Milton on its sign, intended as a pun on the name of the city. Recently another new pub with an original name has opened. "The Enigma Tavern" is not a pun but rather a reminder of a local claim to fame. For it was at nearby Bletchley Park that Axis codes created by the famous Enigma machine were broken. The team of code breakers included many gifted mathematicians. Yet the basic ideas behind their work are accessible in a simplified form to many children at key stage 2, and can provide a fascinating mathematical dimension to their study of history.

Part of the code breaking at any level is considering all the different possibilities and grasping just how many there are. One of the ways the German encoders tried to make the Enigma coding machine unbeatable was by daily variations on the machine's wheels. Each Enigma operator had five wheels available, with three to be used on any given day. The order of the wheels mattered, and this, too, was specified on a daily basis. The problem of working out how many possible permutations exist is similar to many investigations which pupils may already have carried out.

In reality, the problem was harder. Naval encoders, for example, had an extra three wheels in addition to the five available to the other operators. The Germans intended to create further difficulty by the introduction of the four-wheel Enigma in 1942. Operators were given a monthly order of settings, and no setting could be repeated in a month. Children could be asked to speculate whether this actually made the code breakers' job harder, or whether it in fact provided them with an extra clue.

Choosing the wheels and their order was only part of the Enigma encoding process. Another part was the pairing of letters known as "steckering" from the German word for the plugs used in the mechanical process. Children could consider a code based on this principle when letters are swapped with each other in pairs. Again the number of possibilities can be calculated and is far too large to be tackled by considering all options even if, as was usual, six letters were "self-steckered" and therefore not swapped at this stage of the process.

Linguists worked with mathematicians in the code breaking teams. Their expertise included a knowledge of such things as common letters, letter combination or word endings and likely word lengths. Children can attempt to obtain such information by tactics like letter counts using text in German or Italian. Japanese codes, of course, posed additional problems.

There are many children's books on codes currently available. They tend to be stamped "TOP SECRET" and to include pictures of men in raincoats and dark glasses. They also include some quite mathematical activities. For example, one idea suggested is to make a simple code based on arranging the letters of the alphabet in a five-by-five grid. (Letter frequency experts will be able to decide which letter to leave out or combine with another.) Letters can then be represented by co-ordinates. The German "double-playfair" codes were actually based on such a grid but with added complications such as key words and letter swaps. Apparently the Germans derived this idea from the British Playfair codes which they had successfully broken in the First World War.

Accounts of their work by the real code breakers make a fascinating read for adults. Mixed in with the high-level mathematics it is possible to find methods based on the same principles as those used in the children's books. For example much use was made of squared paper and paper with holes punched in it. Other techniques involved writing out the alphabet repeatedly and moving strips of letters across each other.

The latter technique is popular in children's books, as is the idea of coding wheels made from circles of card joined at the centre by split pins. In making these children will need to divide the circles equally as well as to decide how many divisions are needed. Again, 26 is not a convenient number and one solution is to include the digits 0 to 9 as well.

As children may discover, methods of code breaking based on trying out a range of possibilities can be very time consuming and mechanical aids can speed the process. At Bletchley Park, where codes had to be re-broken every day, there was a clear need for such assistance. First came small machines called "bombs" operated by Wrens in a claustrophobic building which became known as the "hell hole". But still more complex machinery was required, and this gave rise to the Colossus machine, now regarded by many as the first genuine computer. The significance of the Bletchley Park code breaking operation to the history of computing is now widely recognised.

Sadly, Colossus was dismantled, though pictures of it survive and are an important aid in understanding how computers have developed over the past 50 years. The name seems especially appropriate given its size compared with modern machines. It is also easy to see why the machine from which it was developed was known as "Heath Robinson".

Plans are now under way to preserve Bletchley Park and, importantly, to rebuild Collosus. A daunting task, but if completed it will provide a fitting tribute to the code breakers' ingenuity and determination. Perhaps it will also provide inspiration for a new generation of investigators and problem solvers.

o Code Breakers, The Inside Story of Bletchley Park. Edited by F H Hinsley and Alan Stripp. Oxford University Press. Codebreakers; Super Sleuths Code Cracker both in the Fun Fax series available from Henderson Publishing Limited.

o Bletchley Park is open to the public on alternate weekends: Bletchley Park Trust, The Bungalow, Bletchley Park, Bletchley, Milton Keynes MK3 6EF. Tel: 01908 640404 Jenny Houssart is a lecturer in mathematics at Nene College, Northampton

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