The day of Sagittarius is the second short story from the 5th book in the series, rampage, the third of which was not adapted and someday in the rain is an anime original finally as they did not go on to fully adapt the light novel.
The day of Sagittarius has the president of the computer club, a character introduced in melancholy and important in mysterique sign, challenging the brigade to a team-based game coded by his club titled the day of Sagittarius.
The stakes are that if the club lose they give them 4 laptops, but if the brigade lose they give back the PC they took in melancholy as well as a member. They give a week to practice but on the day Nagato sees the club are cheating.
Kyon had previously asked Nagato to cheat but after seeing the club are cheating he lets her block their maps putting the brigade at an advantage and allowing them to win. Kyon seeing throughout the game the usually emotionless Nagato was having fun, gets Haruhis permission to sometimes let Nagato go to the computer club.
Someday in the rain is the final episode of the anime as Kyoto animation, the studio behind the shows, decided to stop adapting the light novels at that point.
The episode is anime only and has no such counterpart from the books. It mostly acts as a farewell episode having Kyon run into multiple characters from the series.
The plot has Haruhi distracting Kyon by having him leave school to pick up a heating unit for the clubroom so that she can take pictures of asahina with Koizumis help while Nagato continues reading in the clubroom.
When the others have left Kyon and Haruhi walk home together, ending the episode and closing the series.
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