Here are some of my favorite puzzles that happen to have Japanese names:
Sudoku. Sudoku is so popular that my readers have almost certainly heard of it. Here are some unconventional versions: Math Games: Sudoku Variations.
Tentaizu. A good example and explanation is found at this blog: y of x (the puzzle originally comes from Southwest’s in-flight magazine).
Kenken. The Wikipedia article gives the history, rules and some sample puzzles: Kenken on Wikipedia.
Kakuro. I’ll refer you to Wikipedia for this one also: Kakuro on Wikipedia.
Takegaki. Those geniuses who write the puzzles for Southwest Airlines have done it again: Southwest Magazine Takegaki.
Thanks go to Evelyn for pointing me to the last one, and for reminding me of this post that had been sitting in my draft-box for months.