# A network to catch a fish in

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- A network to catch a fish in

## A network to catch a fish in

‘See what you can do with this’ – these words, or certainly these sentiments, have preceded many significant mathematical breakthroughs. What if you were given the ‘problem’ of (re) designing a mathematics curriculum and nothing but that phrase? What might you do?

William Thomas Tutte (1917 – 2002; generally pronounced ‘tut’) spent most of his life as a research mathematician working in the field of graph theory. Born in Newmarket, Suffolk, he completed his PhD thesis *An Algebraic Theory of Graphs* at Cambridge University in 1948. He worked at the University of Toronto and then the newly founded University of Waterloo, Ontario, in 1962, becoming one of the first members of its Department of Combinatorics and Optimization in 1967, where ‘he advanced graph theory from a subject with one text (D. König’s) toward its present extremely active state, and he developed Whitney’s definitions of a matroid into a substantial theory’ (Hobbs & Oxley, 2004).

Tutte’s ground-breaking early work in graph theory resonates with those of us working on the Cambridge Maths project as we are using a graph database for the development of our Framework. This is enabling us not only to capture the content of Mathematics education but, more importantly, to identify and show the connections and progression within that content as well as generating subsets of the content depending on many different criteria. Our use of nodes, connected by edges, which may show direction, means our Framework is in fact a large, complex graph; Tutte’s work in this area helped to make it a more mainstream part of mathematics which underpins much of modern research in all fields.

However, there was another part of Tutte’s life for which he should be far better known. ‘As a young mathematician codebreaker, he deciphered a series of German military encryption codes known as Fish’* (Order of Canada citation, quoted in Younger, 2002). What the citation neglects to mention, however, is that this accomplishment was ‘probably the single biggest intellectual achievement at Bletchley during World War Two’ (Murrell, 2014).

Most of us have heard of Alan Turing and the remarkable work he and his team at Bletchley Park carried out in breaking the German codes created by the Enigma machine. What is far less well known, however, is that there was a second machine – Lorenz – which was used by the German Army High Command for its most top-secret communications. Where Enigma used three and eventually four rotors for encryption, Lorenz used 12 rotors (two sets of five coding wheels and two motor wheels) with 1.6 million billion possible combinations. Throughout WWII – and for decades after the war – the codes it created were believed by its users to have been unbreakable.

A Lorenz machine with its covers removed to show the rotors. Source: Matt Crypto [Public domain], from Wikimedia Commons

When the Lorenz-coded messages were first intercepted at Bletchley they certainly seemed to be so. Although they had been able to break one message which had carelessly been sent twice using the same settings, Bletchley’s top cryptanalysts were unable to work out anything about the machine which had produced them. Unlike with Enigma, they had no captured machine or manual to tell them its design or system – just this one broken message of 4,000 characters. They had ‘no clue as to how the Lorenz cypher worked, other than it produced a stream of key letters that were added to the message letters … the problem of how those key letters were generated from the info entered by the operator remained’ (Farr, 2017). After several months of trying, they passed the job on to the quiet, unassuming 24-year-old Bill Tutte with exactly the words we considered previously: ‘See what you can do with this’ (Tutte, 1998).

Captain Jerry Roberts worked in the same office as Tutte and remembered that he ‘saw him staring into the middle distance for extended periods, twiddling his pencil and making endless counts on reams of paper for nearly three months, and I used to wonder whether he was getting anything done.’ (2017, p. 73). Suffice to say that he was. Using paper, pencils and his mathematical intellect alone, Bill Tutte found the patterns in the coded sequence which enabled him to establish not only that Lorenz had twelve rotors but also their functions and how many teeth each had. The most challenging part of cracking Lorenz was broken, but as Tutte himself later remarked somewhat wryly, his feat was hardly recognised at the time: ‘I suppose I would have been said to have broken the key by pure analytic reasoning. As it was I was thought to have a stroke of undeserved good luck. There must be a moral in this.’ (1998).

Nor has his feat been sufficiently recognised to date. One reason for this is that the breaking of Lorenz remained classified until the 1990s: when Tutte visited Germany after the war and was shown a Lorenz machine he had to grit his teeth and pretend to agree with the German intelligence officer showing it to him that it was unbreakable (Roberts, 2017, p. 74). Even now information about it is still only slowly coming to light. What is known is that the broken Lorenz messages allowed the Allies to ‘read Hitler’s intentions and gave insight into his whole military planning and decision making’ (ibid, p. 135). Roberts asserts that intercepted and decoded Lorenz messages helped to decide the successful Allied strategies for the Battle of Kursk and for the D-Day landings, among other campaigns, and that General Dwight D. Eisenhower acknowledged they helped to shorten WWII by two years (ibid, p. 220).

It will doubtless never be known exactly how great the contribution the breaking of Lorenz was to the outcome of WWII. The proud tradition of mathematicians who rise tremendously to the occasion of ‘see what you can do with this’ remains, however, and the joy of tackling a problem with only one’s wits, paper and a pencil is an important part of mathematical creativity unlikely to change despite the developments in technology (which Tutte’s efforts actually helped to bring about) that eventually revolutionised the codebreaking process.

*The workers at Bletchley Park did not know that the machine which produced these codes was called Lorenz. The official name given to it at Bletchley was “Tunny”, and the codebreakers called the codes it produced “Fish”.

**References:**

Roberts, J (2017) *Lorenz: Breaking Hitler’s Top Secret Code at Bletchley Park*. The History Press, Stroud, Gloucestershire

Younger, D (2002) ‘Biography of Professor Tutte’ in *Combinatorics and Optimization*, University of Waterloo

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