Everybody knows that Alekhine and Euwe played two matches for the world championship, but few realize they actually played three matches. In 1921, Alekhine left Soviet Russia and emigrated to France. In 1926, after a four-month stay in South America, he returned home in December and a week later was in The Netherlands to play an exhibition match against Euwe which had been arranged a year earlier.
For Alekhine, he was playing for the first time with the new time control of 40 moves in 2-1/2 hours which he felt was strange. Because of the new fast time limit, Alekhine committed a lot of mistakes. Even so, he took the lead by two points then Euwe equalized but lost the last game. Alekhine squeaked by with a score of +3 -2 =5.
Like Emanuel Lasker, who was World Champion for 27 years from 1894 to 1921, Alekhine has also been criticized for avoiding his most dangerous challengers and avoiding a rematch with Capablanca. However, Alekhine was only asking for the same conditions that Capablanca had wanted for their match: the challenger had to provide a stake of $10,000 (that's about $150,000 today). At the time though poor worldwide economic conditions made it impossible for Capa to raise the money.
During his Easter break from his teaching duties in 1928 and the Christmas break in 1928-29, Euwe lost two matches to Bogoljubow, both by a single point. Then in 1931, he lost a match to Capablanca (=0 -2 =8). According to Wiener Schachzeitung they were playing for the right to play a title match against Alekhine. The following year Euwe defeated Spielmann and drew Flohr in matches. Remember that Flohr's playing ability peaked in the mid-1930s and he became one of the world's strongest players and a leading contender for the World Championship.
Like Lasker, Alekhine has also been criticized for avoiding his most dangerous challengers and avoiding a rematch with Capablanca. With his old opponent, Bogoljubow, no longer a viable challenger, if he ever was one, Alekhine picked Euwe, possibly thinking he'd be an easy opponent because Euwe had lost two matches to Bogoljubow. The fact is, as Arnold Denker pointed out, as late as the 56th game between the two the score was even.
It was only when Alekhine won game 7 of their second match that Alekhine went ahead and their final record was 44-36 in Alekhine's favor. If Alekhine thought Euwe was going to be an easy mark, he was wrong because after 30 hard fought games, Euwe won the match. The match was close; after the 24th game the score was tied, but then Euwe won two games in row and Alekhine was only able to win one more. In the last game Euwe agreed to a draw in a winning position only because a draw was all he needed.
In 1937 Alekhine challenged Euwe to a rematch and always a gentleman and sportsman, Euwe agreed. The match was played in the Netherlands from October to December in 1937 and this time Alekhine did not underestimate his opponent and scored an overwhelming win, winning 10, losing only 4 with 11 draws.
Of Alekhine's play Euwe wrote that Alekhine exhibited perfect technique and combinative talent, his conduct of the endgame was shining and he admired most how he finished the adjourned games. Euwe wrote that he had great admiration for how Alekhine created ingenious ideas and finished them in unexpected ways.
What of Euwe's play? Alekhine himself wrote that the general chess playing public as well as the critics realize that Euwe almost never made an unsound combination even though he occasionally failed to take them into account for his opponent. Alekhine also wrote that when Euwe had the initiative in a tactical situation his calculation was impeccable. Reti probably put his finger on Euwe's weakness when he wrote that Euwe believed, perhaps a little too much, in immutable laws. Euwe may have had a dry and uninspired style, but he was a formidable opponent who was only a shade beneath Alekhine and Capablanca.
The following game is not widely known, but Fred Reinfeld called it a titanic struggle in which Euwe successfully withstood one of Alekhine's dangerous attacks. Alekhine lost because he tried to win the game at all costs, but it demonstrates Euwe's cool and resourceful defensive play. Also, in analyzing the game with Stockfish, it discovered some endgame errors by both sides that nobody seems to have mentioned before.