Mr. Kaji says he is now negotiating with American publishers to introduce several puzzles to the United States, including hashiwokakero and slitherlink, a popular game in Japan where lines are connected in snakelike shapes around numbers.

He says he also wants to promote more heavily the half dozen or so of Nikoli’s puzzles that have already followed sudoku overseas. The best known is kakuro, a mathematical version of a crossword puzzle that uses sums instead of spellings.

Still, none of Nikoli’s other games has approached the popularity of sudoku, which has been carried, at one time or another, in more than 600 newspapers in 66 countries. It has also been the topic of more than 200 books, which have sold 20 million copies worldwide, according to puzzle authors and publishers.

While no one knows how much revenue is generated by the global sudoku business, most agree it has easily topped $250 million over the last two years from an estimated 80 million devotees. The New York Times syndicate provides a variety of logic puzzles, including sudoku, kakuro and others, for newspapers and Web sites around the world.

Nikoli received only a sliver of that money. Mr. Kaji says his private company, with just 20 employees, had annual sales of about $4 million.

Sudoku’s popularity in the United States caught Mr. Kaji by such surprise that he did not try to get the trademark there until it was too late. As a result, Nikoli receives no royalties from sudoku-related sales overseas by other publishers.

In hindsight, though, he now thinks that oversight was a brilliant mistake. The fact that no one controlled sudoku’s intellectual property rights let the game’s popularity grow unfettered, Mr. Kaji says. Nikoli does not plan to trademark other new games, either, in hopes this will also help them take off.