At Cambridge Alexander continued to divide his time between his chess and his study of mathematics. He was playing on the top board for Cambridge University with outstanding success by the time he was sitting the Mathematical Tripos in 1931. His performance in that examination was excellent but he missed out on the award of a fellowship which would have allowed him to continue to undertake mathematics research. Hardy knew that, but for the time he devoted to chess, he would have easily made his mark as a research mathematician. Hardy described Alexander as :-
... the only genuine mathematician he knew who did not become a professional mathematician.Alexander was appointed as a mathematics teacher at Winchester in 1932. He married Enid Constance Crichton Neate, the daughter of a sea captain, on 22 December 1934. They had two sons.
Being a mathematics teacher certainly allowed Alexander to play chess at the highest level. He was placed second in the British Chess Championship of 1932. In 1934 he took part in the Nottingham International Chess Tournament which was played at the University of Nottingham. The top tournament comprised of eleven leading international players, including four world champions, and four British players. Euwe and Lasker, two chess playing mathematicians with biographies in this archive, played. It was too strong a field for Alexander to excel at this point in his career but he did have victories over Flohr and Tartakover. Watts writes :-
Alexander was perhaps the surprise of the English contingent, and the good impression he made has since been more than confirmed in the Stockholm Team Tournament. He has shown himself able to devise attacking combinations and ideas that escape the most wary and the strongest of the masters, and a little more experience should put him in the front rank and once again place British Chess on the map.Indeed Alexander gained this experience and in 1938 he won the British Chess Championship and came equal second with Paul Keres in the Hastings Christmas International Congress. It was in 1938 that he left his job as a mathematics teacher to take up the position of head of research in the John Lewis Partnership, London. He was captain of the English team at the International Team Tournament in Benos Aires in 1939, and in fact the tournament was only half completed when news reached Buenos Aires that World War II had begun. Alexander did not remain to complete the tournament but immediately caught a boat back to England to offer to serve his country.
It was now that Alexander's mathematical and problem solving abilities were put to full use. In February 1940 he was sent to Bletchley Park where he joined Hut 6 at the Government Code and Cypher School. Here he worked on decoding messages sent through army and air force Enigma machines, then in March 1941 he moved to Hut 8 as second in command to Turing. Alexander and Turing worked on decoding naval Enigma messages, and by August they had solved the Kriegsmarine cipher (see ). However much time was still required to decode the messages and in October Turing and Alexander appealed directly to Winston Churchill for some junior assistants. Although Turing was the head of Hut 8, Alexander soon slotted into this role since he was an outstanding administrator while Turing was uninterested in organisational matters. When Turing went to the United States in November 1942, Alexander became the official head. His colleagues appreciated his leadership describing him as:-
... a quite splendid head of Hut 8and
... a model manager [who] treated us cryptographers as colleagues and was remarkably tolerant of our foibles.He made many outstanding contributions to the decoding work at Bletchley Park, devising new Bayesian scoring methods which became necessary since the Germans, realising that their code was being broken, continued to increase its complexity.
After the end of World War II, Alexander returned to his position as head of research at the John Lewis Partnership, but quickly realised that code breaking was a far more exciting occupation. In the summer of 1946 he joined GCHQ (Government Communications Headquarters), an organisation based on the teams at Bletchley Park. He became Head of Section H (the cryptanalysis section) in 1949 and remained with GCHQ as Head of Section H until he retired in 1971. In fact he was offered promotion at GCHQ but refused, and he was also offered a leading position by NSA, the equivalent organisation to GCHQ in the United States, and again refused but only after being seriously tempted to accept.
World War II put an end to any chances that Alexander might have had at becoming World Chess Champion. Mikhail Botvinnik, who was World Chess Champion during 1948-57, 1958-60 and 1961-63, wrote:-
... with his urge for overcoming and taming opposition, with his enthusiasm for uncompromising struggle, Alexander pioneered the way for British players to modern, complicated and daring chess; chess players will never forget him.In fact Alexander did play chess at the highest level again after the war, winning the Hastings International Tournament in 1946 and coming first equal with David Bronstein in the Hastings Tournament of 1953. He had victories over some of the top players in the world including Botvinnik, Euwe and Bronstein. He wrote chess columns for the Sunday Times, the Financial Times, the Evening News, and The Spectator. He was captain for the British Chess Federation and played in Olympiad competitions abroad, except when the competitions were in Eastern Europe. His job with GCHQ prevented him travelling to certain countries, for example he could not play in the 1956 Olympiad in Moscow.
It is interesting to note that, although he was born in Ireland, he played chess for England, something which was certainly noted when he played in the Dublin Zonal in 1957. He did, however, indicate clearly that despite living most of his life in England, he considered himself Irish. He wrote in 1957:-
I do think of myself as an Irishman, not an Englishman, in spite of my long time here.In  he is described as follows:-
Alexander was that rarest of men: a superbly skilled cryptanalyst who was also an excellent manager. He also combined a razor-keen intelligence with considerable energy and enthusiasm. His exceptional technical skills, and his gifts of leadership, man management, and administrative ability, made him an inspiring head of both Hut 8 at Bletchley and section H in GCHQ. He became an almost legendary figure to the intelligence communities of Great Britain and the USA. ... Hugh Alexander was a most vivid and attractive personality, who delighted his friends with his gaiety, humour, and warmth. A magnificent talker, he loved to argue but was ever ready to see his opponent's point of view.He had wide interests including bridge, croquet and philately.
For his outstanding contributions to his country Alexander was awarded an OBE in 1946, a CBE in 1955, and a CMG in 1970.
Article by: J J O'Connor and E F Robertson