The Approximation of the Distance of Two (ふたりの距離の概算, Futari no Kyori no Gaisan) is the fifth installment of the Hyouka novel series. It was released on June 25, 2010, with a novel version and, after the release of the anime series, an anime version with cover illustrations based on the character designs used on the anime series.
The subtitle for the volume is It walks by past, a reference to the 1930 detective novel It Walks By Night by John Dickson Carr.
Welcoming the arrival of spring, Houtarou and the others welcome the first year student Tomoko Oohinata as a provisional member of the Classics Club. But just before she became an official part of the club, she quickly announced her withdrawal. On the deadline for joining clubs, which is also the day of a marathon event, Houtarou, while running the long way, tries to find out the reason for the newcomer's change of heart!
Welcoming the arrival of spring, Houtarou and the others welcome the first year student Tomoko Oohinata as a provisional member of the Classics Club. Chitanda and the others immediately got along well with Oohinata, but one day, leaving behind enigmatic words, she informed them that she cannot join the club anymore. It seems that a conversation with Chitanda in the clubroom is the cause, but Houtarou doesn't agree. She is not the kind of person who would hurt others like that. Houtarou, as he runs on the marathon event that lies on the same day as the deadline for joining clubs, he tries finds out the reason for the newcomer's change of heart!
- Prologue - Too Long When Simply Running
- Prologue-1 - Present: 0km
- Prologue-2 - Past: 1 Day Ago
- Prologue-3 - Present: 1.2km
- 1 - The Reception Table is Right Over Here
- 1-1 - Present: 1.4km; 18.6km Remaining
- 1-2 - Past: 42 Days Ago
- 1-3 - Present: 4.1km; 15.9km Remaining
- 2 - Friends Need to be Celebrated
- 2-1 - Present: 5.2km; 14.8km Remaining
- 2-2 - Past: 27 Days Ago
- 2-3 - Present: 6.9km; 13.1km Remaining
- 3 - A Very Wonderful Shop
- 3-1 - Present: 8.0km; 12.0km Remaining
- 3-2 - Past: 13 Days Ago
- 3-3 - Present: 11.5km; 8.5km Remaining
- 4 - Easier to Just Let Go
- 4-1 - Present: 14.3km; 5.7km Remaining
- 4-2 - Past: Approximately 19 Hours and 30 Minutes Ago
- 4-3 - Present: 14.5km; 5.5km Remaining
- 4-4 - Present: 14.6km; 5.4km Remaining
- 4-5 - Present: 14.6km; 5.4km Remaining
- 5 - Approximating the Distance between Two People
- 5-1 - Present: 17.0km; 3.0km Remaining
- 5-2 - Present: 18.6km; 1.4km Remaining
- 5-3 - Present: 18.9km; 1.1km Remaining
- Epilogue - You Should Be Able to Extend Your Reach Anywhere
- Epilogue-1 - Present: 19.1km; 0.9km Remaining
Despite wishing for rain on that day, Houtarou Oreki, now a second year student, found that the year's marathon event in Kamiyama High School, also known as the Hoshigaya Cup, was going according to plan. As he grudgingly started the 20-kilometer course across town, a conversation with his fellow Classics Club member Satoshi (who is part of the General Committee for the event) made him recall the club's ongoing problem involving a provisional member named Tomoko Oohinata, who immediately withdrew her membership to the club, causing the club's morale to turn sour. Satoshi suspected that something made Oohinata sad or angry the day before, and although he considered Chitanda as the culprit, he thought that it would be contrary to what he heard from Mayaka where Oohinata described Eru as "like Buddha" (meaning that she is kind and understanding). Regardless, he decided that respecting Tomoko's decision is a better action, although he also knew that Houtarou is the kind of person who will try to resolve Tomoko's problem. True enough, Houtarou decided to recall the events that happened during and before Tomoko's withdrawal of her club membership.
By using his lead over the second-years Mayaka and Eru and the first-year Tomoko, who left the start line in respective order behind him, he was planning to talk to the three before the end of the day. However, as Satoshi ended the conversation with him, Houtarou realized that catching up to the girls would not be an easy task as their pace would constrain him to a few questions. For that reason, Houtarou started recalling his last conversation with Mayaka, when she commented on how distant he was with other people, all while gauging his own distance with her, who might be 6 or 7 minutes away from him, as well as remembering the conversation he and Eru had with Tomoko about the "unusual decorations" the Confectionary Society used during the club recruitment period during April. That day was also the day where Tomoko filled up her provisional club membership form.
Houtarou ultimately failed to properly measure the gap between him and Mayaka, and he found himself with her classmates at 2-C. He eventually found Mayaka regardless, and he began asking her about Tomoko quitting the club and what was said regarding Eru. Despite acting aloof toward him at first, Mayaka decided to approach Houtarou and clarified that Tomoko did not use the word "Buddha" to describe her. Instead, Tomoko used the word "bodhisattva". (The word "bodhisattva" refers to a person who has achieved enlightenment through one's desire to help others.)
As he continued a steep descent further down the course, Houtarou began mulling over the word "bodhisattva", and soon remembered the time when Satoshi, Eru, Mayaka and Tomoko came over to an unexpected visit for him on his birthday and Houtarou exposed Eru's attempt to hide her knowledge of his home address. Later, he once again encountered Satoshi, who had recently responded to a race-related "accident", and Houtarou took the chance to ask about the concept of "being a bodhisattva on the outside". Satoshi responded by saying that someone who is a bodhisattva outside may be a yaksa on the inside, but immediately clarified that Eru isn't someone as cruel as one.
Satoshi once more separated from Houtarou, and the latter began speculating that Tomoko, based on what he knows about her, had tried to criticize Eru, but she used a roundabout way in her attempt to preserve the friendship with the Classics Club. However, without anything to prove that something had caused relations between Eru and Tomoko to break down, Houtarou looked back on one day where the Classics Club gathered while he was trying to deal with his lack of money and food. The short snack session eventually became a plan for the entire club to go out on the Saturday that followed to a newly-opened cafe owned by Tomoko's cousin near Kaburaya Middle School.
The club became the first customers of the cafe, and Eru barely managed to catch up to the event after attending a formal kiju celebration. While Houtarou and the others tried out the cafe's house blend coffee and scones, Eru settled for water due to her difficulty to sleep after being exposed to caffeine. Meanwhile, the topic about the name of the cafe is opened by Houtarou, and Tomoko left guessing the name to her fellow members, and eventually was the only one left in answering the cafe owner's challenge. Then, as Tomoko was still puzzled about the cafe's name, Houtarou prepared to head out home with the rest of the club, but not until solving the three kanji characters that make up the cafe's name: 歩恋兔 or Blend (Burendo: 歩 = Ayumi, the cafe owner's wife, 恋 = love, and 兔 = rabbits, a favorite of Ayumi). Although the riddle of the cafe name was solved, Houtarou was certain that something strange happened during the cafe visit, but he considered a better way to confirm it, and this involved asking Eru himself.
Houtarou soon found that his legs were starting to ache after trying to gauge his pace throughout the course. Hearing the sour remarks by a schoolmate named Nanigashi reminded him of the ongoing issue involving the Classics Club, and he decided to confront Eru so that she could stop taking the responsibility for the issue, all while trying to recall what actually happened the previous afternoon when Tomoko decided to quit the Classics Club.
The previous afternoon, Houtarou encountered Tomoko, who was in the classroom beside the Earth Sciences lecture room. They talked about Satoshi and his little sister who is also a student in Kamiyama High School, and while Tomoko was reluctant to open the topic, Houtarou spoke in her behalf about Satoshi's relationship with Mayaka, which Tomoko had heard from rumors among the first-year students. Afterwards, they join Eru, who was studying alone, and a conversation passed by Houtarou, who was too immersed in reading until he saw Eru's dark expression. Mayaka later arrived to tell the news that Tomoko had decided to drop her membership.
As Houtarou waited in a bus stop for Eru to pass by, he gathered his thoughts about Tomoko's issue, which he was sure involved Mayaka and Satoshi. As for Eru, he was thinking that Tomoko had been harboring repressed feelings from the past 40 days that eventually culminated with Eru making Tomoko angry. Houtarou soon realized that waiting for Eru to come would risk him not catching her and possibly leaving him waiting for no one, and he was already preparing to take a bus to school when Eru suddenly passed by the bus stop, prompting him to catch up in haste.
Houtarou struggled in catching up to Eru, but she eventually started talking to him. There, Eru declared that she was not going to give up her responsibility for Tomoko's issue, and even refused to speak her side in it. But, Houtarou's assuring remark about Tomoko's phone forced Eru to finally divulge what she knew about the problem involving Tomoko's withdrawal. In Mizunashi Shrine along the marathon course, Houtarou revealed that the "yes" he heard from Eru the day before was a response to a call, which Eru clarified as coming from Tomoko's phone. Eru then explained that Tomoko left shortly with her phone among the former's reference books, and Tomoko coldly snatched the phone from Eru's hand after she tried answering a call from Tomoko's phone.
Eru determined that Tomoko cherished her cellphone a lot, something that Eru does not understand because she did not own one, and decided to apologize to Tomoko for it, but Houtarou was thinking that it is not the only problem between the two girls. Nonetheless, Eru decided to tell everything to Houtarou, but not after she admitted that it involved what seemed like a normal conversation to her.
The day before, Eru engaged in conversation with Tomoko about several things, notably her weakness in English, the weather for the next day, and Mayaka quitting the Manga Research Society. Because of Mayaka's tendency to not talk about sensitive things connected to the Manga Research Society, Eru was forced to talk broadly about Mayaka's decision to leave the club, but this was met by disappointment from Tomoko, who called Mayaka's decision "abandoning the club".
After hearing Eru's side of the situation, Houtarou decided to look back to the person asked by Tomoko to Eru during the trip to the cafe of Tomoko's cousin, and Eru identified the person as Sachi Agawa, Class 1-A, and the first year representative, who Eru only knows for her extremely long hair. Houtarou then decided to return to the course, and Eru requested him to help out Tomoko in her situation.
Houtarou ran for a certain distance to the city's center, and proceeded to the course until he reached the foot of a bridge, where he remained to wait for the first year students to pass by. Tomoko soon caught him among the participants, and after dismissing her guesses for his presence there, Houtarou told Tomoko that Eru did not know anything about Agawa, and he soon moved their discussion to a nearby alleyway, away from the rest of the students. There, he led Tomoko into a shortcut leading to Arekusu Shrine, and the two began to talk about the 40 days that passed, from Tomoko's eavesdropping on the New Recruit Festival, to Houtarou's birthday at home, to the visit to her uncle's cafe and the conversations from the day before.
The recollection revealed a friend from middle school that Tomoko refused to talk about to the point of saying that she had not made any friends in high school. Houtarou, however, began to question why Tomoko did not call Satoshi's little sister, from whom she heard of rumors involving Satoshi and Mayaka, a friend despite their level of interaction. He also asked about Tomoko's conversation with Eru involving Mayaka's withdrawal from the Manga Research Club. Tomoko, however, denied that it was relevant to the issue. Instead, she returned to Sachi Agawa, who she revealed as a graduate from Kaburaya Middle School, causing Houtarou to remind Tomoko that Eru does not know everyone in town.
Houtarou was suspecting that Tomoko was somehow afraid of Eru for some time, but only because of a misunderstanding that caused her to see Eru as a "yaksa" that may know too much about her middle school friend and Mayaka as someone willing to "abandon" a group of friends on a whim. Houtarou then told Tomoko that Eru also misunderstood something after the issue involving Tomoko's cellphone and was about to apologize to her for it.
A more relaxed Tomoko decided on a detour to a dango shop near Arekusu Shrine. As she and Houtarou eat yomogi, Tomoko spoke about her "friend" from middle school and her trips across Japan, as well as the financial stress caused by their friendship. She also found out that her friend swindled money from her relative and Tomoko soon found herself being judged for "abandoning" their friendship as she became admitted to Kamiyama High School. Despite the separation, Tomoko promised to maintain her friendship. However, this made her cautious about people who might have enough connections to expose her friend's wrongdoings. Houtarou listened as Tomoko admitted her mistakes and her promise to make amends with Eru, but he stayed mum about her decision to not join the club until she had accomplished it. Tomoko, however, saw through Hotarou's lie.
Houtarou and Tomoko later returned to the course. Satoshi informed Houtarou that someone had seen students who went off-course, but he showed no plans in reprimanding him since the detour he took was Satoshi's plan himself. Houtarou then told Satoshi that Tomoko would not be joining the club, and left the story behind the issue unknown to Satoshi, something that he understood well since he had a hunch that the issue lied outside of Kamiyama High School.
Soon, Houtarou wondered about the incident that triggered Tomoko's distrust in Eru, and quickly recalled the conversation during his birthday about how Eru determined Houtarou's address, where the name "Souda" showed up. He also remembered hearing a name "Sonoko" during his talk with Tomoko and began wondering if a certain "Sonoko Souda" was Tomoko's friend from middle school.
Satoshi broke away from Houtarou once more as the home stretch leading to Kamiyama High School approached. As he gained speed, he tried to look for Tomoko in the crowd of exhausted first year students on the way to the finish line, but failed to do so.