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While many top class American players prefer the international rules of Association Croquet, the official USCA game remains dominant in North America. Under Association rules, a ball may hit each of the other balls just once per turn, unless a hoop is scored. Under American rules, that quota isn't reset from turn to turn. Once a player has used up all three hits, his ball becomes "dead" on the other three balls, and will find it hard to make further progress.
Many outsiders find this version very defensive and exasperating. Nevertheless, it has had great popularity, particularly in Hollywood in the 1940s and 1950s. Leading movie figures David O Selznick, Samuel Goldwyn and George Cukor held regular croquet parties. Among the top players of the time was Harpo Marx.
Gateball was invented in Japan in 1946, and is a form of five-a-side speed croquet. It's played on a hard surface with just three hoops. The key feature is the 'spark' shot - a croquet stroke in which a foot is placed on the striker's ball, often used to send the opponent's ball out of bounds. Gateball has become very popular throughout the Far East, with around two million devotees in Japan.
Originally, Gateball was conceived as a game for children, but has become popular with retired people in Japan. In rural China, it seems to have developed a status similar to that of village cricket in England - league matches take place between neighbouring villages, with fierce rivalry.
Roque emerged early in the 20th Century, and was played at the 1904 Olympics in St Louis, Missouri. The USA provided the only competitors, and it has remained an exclusively American game since. It's played on an octagonal hard court, bordered by a low concrete wall. The balls are rubber, and may be bounced off the sides. The game faded from popularity in the 1960s, and is now thought to be extinct.
Roque features heavily in Stephen King's novel The Shining. Alas, the roque mallet failed to feature in Stanley Kubrick's film version, famously replaced with an axe.
Extreme Croquet has grown since the 1970s, and is played by American college students. Rules are variable and, in many ways, irrelevant. The crucial aspect of the game is the placement of hoops, which may be placed a great distance apart, and on severe slopes, under bushes, or in the middle of streams.
Care should be taken not to make the placement too extreme. There's a danger of making the game impossible (and boring) or downright dangerous, as well as presenting a nuisance to wildlife / bystanders / law enforcement officers.
At any rate, players should expect much of the equipment to become lost or broken during a session. For this reason, playing on a regular basis is likely to become prohibitively expensive very quickly.
Many countries have their own version of the recognised rules. A game called "Ancient Croquet" has been popular in Russia, where cosmonauts were encouraged to play the game as rehabilitation after long periods of weightlessness. Leo Tolstoy was a keen player, and described the game in his novel Anna Karenina.
Denmark has, per capita, by far the largest community of competitive croquet players, with around 5,000 registered participants. They play one-handed, with short mallets held golf-style. Apparently, they regard players from overseas with some disdain for the need to use both hands to play.
This seems to be a throwback to the early days of Victorian croquet. The use of two hands was seen as unsporting, as it disadvantaged ladies, who had to play whilst holding a parasol in the other hand.
The Danish game has nine hoops and two pegs. The croquet stroke, as in Gateball, is played with a foot on the striker's ball.