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The Intercalated Olympic Games were to be a series of International Olympic Games half-way between what we now call Games of the Olympiad. This proposed series of games, intercalated in the Olympic Games cycle, was to always be held in Athens, and were to have equal status with the international games. However, the only such games were held in 1906.
The first Intercalated Games had been scheduled by the IOC in 1901 as part of a new schedule, where every four years, in between the internationally organised games, there would be intermediate games held in Athens. This was apparently a bit of a compromise: After the successful games of Athens 1896 the Greeks suggested they could organise the games every four years. Since they had the accommodations, and had proven to be able to hold well-organised games, they received quite a bit of support. However, Pierre de Coubertin did not like this at all, he had intended the first games to be in Paris in 1900 and had no intention of losing not only the première for Paris but the games as well. Thus the second games became the Paris 1900 games.
When these games turned out less than perfect and were overshadowed by the Exposition Universelle, the IOC supported the Greek idea by granting them a second series of quadrennial games, in between the first series. All of the games would be International Olympic Games; the difference was that only half of them would follow De Coubertin's idea of organising them in different countries to make the Olympic Movement more international, while the other half would follow the Greeks' idea of a permanent home with the Greek NOC as experienced organisers. This was a departure of the ancient schedule, but it was expected that if the ancient Greek could keep a four year schedule, the modern Olympic Movement could keep a two year schedule. As 1902 was now too close, and Greece was experiencing internal difficulties, the 2nd Olympic Games in Athens were scheduled for 1906.
As the 1904 Summer Olympics in St. Louis, Missouri were overshadowed by the Louisiana Purchase Exposition, and as a result met with a fate similar to that of Paris 1900, the Olympic Movement was not in good shape. It desperately needed to recapture the spirit of Athens 1896. It also needed to do so quickly, as to all those who didn't participate in St. Louis, Rome 1908 meant an 8-year gap. By that time there would not be much goodwill left for the Games. And on top of that, Rome was planning an Exhibition. The Athens games being just around the corner must have seemed like a lifeline. De Coubertin still disliked the idea, and didn't do anything more than his function required him to. But the IOC as a whole gave the Greek NOC full support for the organisation.
First Intercalated Games 
The 1906 games were quite successful. Unlike the 1900, 1904 or 1908 games, they were neither stretched out over months nor overshadowed by an international exhibition. Their crisp format was most likely instrumental in the continued existence of the games.
These Games also were the first games to have all athlete registration go through the NOCs. They were the first to have the Opening of the Games as a separate event; an event at which for the first time the athletes marched into the stadium in national teams, each following its national flag. They introduced the closing ceremony, and the raising of national flags for the victors, and several more less visible changes now accepted as traditional.
The Greeks were unable to keep the schedule for 1910. On the one hand problems in the Balkans made things difficult, but on the other, the modern Greeks found out the ancient Greeks were right: A two year interval was too short. Where there had been a gap of six years before Athens 1906 (because of the almost all-American nature of the 1904 St Louis games), a gap of two years after London 1908 did not leave people enough time to prepare.
With Athens 1910 a failure, the faith in Athens diminished, and as a result Athens 1914 got even less support. And then World War I started, and any further Intercalated Games had to wait until after the war. But after the war was over it had been more than a decade since Athens 1906, and the idea of Intercalated Games was given up entirely.
Since the 2nd International Olympic Games in Athens now had become an exception, the personal views of various IOC chairmen caused the IOC to retroactively downgrade the 1906 games, and the explanation for the games became that they had been a 10th anniversary celebration. As more stress was placed on the continuing sequence of four-year Olympiads, the games of 1906 did not fit in. Hence, today the IOC does not recognize Athens 1906 as Olympic Games, and does not regard any events occurring there, such as the setting of new records or the winning of medals, as official.
The success of Athens 1906, however, may have been what kept the Olympics alive. And as the next games are always built on the successes of the last, the innovations of Athens were used again in London, and eventually became Olympic tradition. In fact, the influence of the First Intercalated Games pervades the Olympics, with the holding of the Games concentrated in a small time period, at a small area, and with good organization.
Winter sports 
Since there were no winter sports at the First Intercalated games, the idea has arisen that this was because the IOC had made such an explicit requirement. In reality, though the IOC had intended some winter sports to be Olympic, before London 1908 none of the games included any winter sports, and Athens 1906 was no exception.
It is worth noting, however, that one of the ideas of the Intercalated Games - namely that the games took place two years either side of Summer Olympics - has been revived for the Winter Olympic Games since 1994.