Arithmetic Chain (数学的連鎖, read アリスマティック・チェイン) is the seventh episode of the spin-off anime series Mahouka Koukou no Yuutousei. This spin-off follows the events of the main series from the perspective of the girls, especially that of Shiba Miyuki.
The Nine Schools Competition begins, and Shizuku goes head-to-head with Shiori in the Speed Shooting tournament. Shiori utilizes the Arithmetic Chain magic sequence, which allows her to perform mathematical calculations faster than a supercomputer. Shiori left her own broken family to join Airi’s team, and in her determination to win attempts to drive Shizuku into a corner. Honoka and her friends nervously watch the match, but Miyuki is confident smile due to who Shizuku's engineer is, and just as she has predicted, Tatsuya has a secret tactic up his sleeve to bring about victory.
Honoka then has her first match in Battle Board, and bothe she and Touko become aware of one another's skill and prepare to face off against in the final.
It’s August sixth, and the Speed Shooting Qualifiers are getting underway. Shizuku is up first and the audience is awed by her performance. Watching on, Airi and Shiori discuss the match and Shiori is confident she can counter the spell Shizuku is using. The match ends with Shizuku getting a perfect score.
After the match, Tatsuya and Shizuku bump into Airi and Shizuku. Shiori and Shizuku both resolve to face-off in the semi-finals.
Shiori takes to the field, with Airi watching on, and is determined not to let her down.
The match gets underway and Shiori’s performance awes the audience with her supercomputer level calculations allowing her to hit one target and use the shrapnel to hit others.
Watching on, Airi thinks back to how she met Shiori in a fencing match and recognising her talent, recruited her. Shiori’s ability is to turn visual information into mathematical formula and utilise it accordingly.
The match ends with Shiori scoring a perfect score.
Before their match, Shiori meets with Shinkurou to discuss a counter for Shizuku. They discuss her specialised CAD and are confident Shiori can account for the nine different activation sequences it holds.
Shiori and Shizuku’s match begins and a fierce high-level contest begins, with Shiori slightly ahead. As the battle continues, Shiori reflects on how she and Shizuku have polar opposite backgrounds, with Shizuku being raised in a wealthy and renowned, loving family, whilst Shiori was raised in an abusive family of Extras until she was saved that day by Airi.
Shiori is feeling more and more fatigued and cannot understand why she is getting so taxed, until she realises something and seeing Shizuku’s smile confirms it.
Watching on, Shinkurou and Masaki have realised it as well, Shizuku’s CAD isn’t a specialised type, it is a general type with an auxiliary sight. Realising she didn’t use it in the previous matches as part of a cunning plan to trick them, they realise Shiori has been getting completely overworked without even realising it.
In the match, Shiori is missing more and more targets as the fatigue hits and Shizuku takes the lead. The fear of having to return to her home sinks into her mind and Shiori misses the last targets, allowing Shizuku to win the match at 96 points to 92.
Speed Shooting ends with Shizuku taking first place, Eimi in second, Kazumi in third and Shiori ending in fourth place, having lost to Kazumi.
After the match, Third High’s team members review the result. Airi is shocked Shiori lost to Kazumi when normally she would have crushed her. Masaki gives the others the bad news, that First High have an engineer who is a beast who can raise a CAD’s level by two or three generations.
In First High’s tent, Mayumi is elated by the result and congratulates Tatsuya, though he simply says it was the competitor’s efforts. Active Air Mine is brought up, as is the fact it is to be added to the Magic Index. Tatsuya however, asks not to be named as the inventor, much to the shock of the others.
Meanwhile, Airi and Touko visit Shiori who is wallowing in despair at her terrible loss, crushed by having let Airi down. Airi tries to talk Shiori out of her room, but her words cannot reach her. Airi is concerned they may have to replace Shiori in her remining matches, but Touko is confident she will snap out of it.
Battle Board is getting underway. Honoka’s first match is approaching, and the nerves are getting to her. Touko gives Honoka a shock and reassures Honoka there won’t be any accidents so she should stop worrying.
Tatsuya arrives as Touko is leaving and Tatsuya tells Honoka that Touko is the one expected to win. The pair watch her match and Honoka is bewildered by Touko’s magic, which Tatsuya explains is Shinto-Type Ancient Magic. Honoka is intimidated but Tatsuya reassures Honoka, telling her she can compete after all the training she did. This makes Honoka think back to the training they did together and an embarring moment, as well as a conversation where she confessed her lack of confidence and Tatsuya offered to make a spell for her.
Honoka’s match begins, and she uses the spell to blind her opponents at the starting line, allowing her to take a huge lead and win. Watching on, Touko looks forwards to facing Touko in the final.
Shiori is due to compete the next day and Airi once again tried to speak to Shiori, telling her she’ll have to be replaced if she doesn’t get herself together, but her words fail to elicit a positive response. Airi tells Shiori she has until 6am to come to the tent or she’ll be replaced.
Mizuo-senpai visits Shiori, using a key card to enter the room. Mizuo then tells Shiori that Airi got as much out of their relationship as Shiori did, explaining that Airi was isolated due to her bloodline and talent, making Shiori the first person she could have a friendly rivalry with. Mizuo goes on to state giving up after one loss will stop her advancing as a magician.
The pep talk from Mizuo-senpai having worked, Shiori comes to the tent on the morning of the seventh, much to Airi’s relief.