Plot summary

Nathan Ellis (Asa Butterfield) is quiet and mostly non communicative except when dealing with numbers. His pediatrician diagnoses him with autism. Nathan finds peace of mind in patterns. The nine-year-old math prodigy has just lost his father Michael Ellis (Martin McCann) in a car accident. His father was the only one who was able to connect normally with him. Although Nathan values his mother, Julie (Sally Hawkins), he shuns any physical contact with her, even at his father's funeral. Nathan treats his mother Julie as more of a caretaker than a parent. Wanting to make sure Nathan is not distracted from his studies, Julie enrolls him in advanced classes at a new school. There, he comes under the tutelage of teacher Martin Humphreys (Rafe Spall), also a math genius, who suffers from multiple sclerosis. Martin has just confiscated a joint from some boys and smoked it himself prior to meeting Nathan. Martin's initial instruction is a tree diagram of a coin toss that Nathan identifies perfectly. Nathan takes a great interest in the International Mathematical Olympiad which China has dominated. Martin sees himself in Nathan, once a promising young mind in the field of mathematics, who gave it all up once he was diagnosed with his illness. Seven years later, Martin is preparing Nathan to compete for a place representing the United Kingdom in the International Mathematical Olympiad, a prestigious high school competition consisting of the world's best young mathematicians. Nathan is a bit nervous when he is about to begin the exam to qualify for the British team. Martin reassures him he is brilliant. After some initial interruptions while Martin settles himself, Nathan focuses on the test. When he speaks, Nathan is very blunt in his commentary. He asks Martin why he never accomplished anything after his failure at the IMO. When his mother orders a takeout meal, Nathan insists it must be compromised of a prime number of prawn balls. He criticizes his mother's failure to comply perfectly. Nathan recalls a restaurant meal with his reassuring father who made Nathan feel comfortable and with whom he was able to laugh. One evening, Martin delivers the results of the qualifying test. Nathan fears he is not good enough to qualify but ends up doing well enough to accompany fifteen other British teenagers to Taiwan where Nathan will train with the United Kingdom team potentials. Traveling to the two-week camp in Taiwan will be Nathan's first trip on an airplane. Nathan's response to the news is to go and study. Julie and Martin enjoy a glass of wine in a sun room. Julie asks Martin to join them when they take Nathan to the airport. Martin admits to Julie that he was let go by the school, but would still like to tutor Nathan. Nathan spies on them with his telescope until it falls through the glass ceiling. A lonely Julie, sits in the damaged car where her husband died. She recalls when she found her husband Michael and Nathan were playing with sparklers in the shed. She instructed Nathan to change the gold fish water. While sharing a romantic moment with her husband, Nathan let the water overflow the sink and lets the fish die. Before he departs, Martin presents Nathan with a sets of compasses and math tools. At the station where the team gathers, Martin greets a former teammate Richard (Eddie Marsan) from the Olympiad who is now the team's coach. Martin presents Nathan as his own. After an awkward goodbye with his mother, Nathan is off. Prior to their flight, Nathan meets other boys on the team. Isaac Cooper (Alex Lawther) is welcoming. Pav Kamdar's (Suraj Rattu) origin is from India. Luke Shelton (Jake Davies) seems brilliant but obsessed about splitting infinitive. They all board a bus. Richard describes that the camp will reduce the team from the current sixteen students to six who will actually compete in the Olympiad. Richard's main goal is to defeat the Chinese. This year, the actual competition is to be held at Cambridge. After a two-week math camp in Taiwan, the students will test to make their national teams. Rebecca Dunn (Alexa Davies) sits behind Nathan on the plane tells him he is neither weird or the best mathematician in the peer group but painstakingly average. The team arrives on Taiwan. Richard greets Chinese coach Deng Laoshi (Orion Lee) as a friendly rival. The UK team is each paired with a Chinese counterpart. Nathan is matched with Zhang Mei (Jo Yang), a Chinese girl and the niece of the Chinese coach. Nathan is quite shy and reserved. Zhang Mei shows Nathan how to use chopsticks at dinner. As they are walking around, Nathan speaks in Chinese which impresses Zhang Mei. She says Nathan does not speak much in English at all. Nathan is assigned Luke as his roommate. Luke is not appreciative of their forced proximity. Nathan ignores a phone call from his mother. He does call Martin who tells him first nights are always shit and refocuses him on numbers. Nathan falls asleep thinking of the nursery rhyme his father spoke to him. The next day during a brisk walk, Richard chides his counterpart about having his niece on the team. Luke defends his theory that counting should be based on a base twelve rather than the primitive base ten system. When Nathan has no opinion on the matter, Luke says Nathan would be better off not being there. Nathan is hesitant to participate in class. His work shows that he overcomplicates the problems when a simpler answer will do. Richard tells Nathan that Martin's failure in the Olympiad was not due to his disease but his lack of character. Back in England, Martin sleeps and pops pills much of the time. Julie is lonely and looks up Martin's phone number. Just then, Martin is with his doctor who is encouraging him to join a support group and not give up. Julie invites Martin over to tutor her in math so she can better relate to her son. Martin and Julie seem to enjoy their first session. Nathan does not fit in well with the other boys. He wanders off and finds Rebecca who sat behind him on the plane playing piano. She teaches him that music is simply numeric ratios which Nathan easily replicates. He then gets a map and walks around Taipei. He finds fascination with patterns in the stop lights and architecture. Martin and Julie enjoy some wine and companionship. They do end up kissing but Martin reveals that he is physically unable to perform and does not want to disappoint Julie. One day Zhang Mei and Nathan are walking through the park. Nathan admits he is usually is not fond of change but appreciates the Chinese approach to math which make it an art form. In class, Richard requires to go to the board and prove that a sequence of turning playing cards must terminate. Zhang Mei encourages him. Nathan uses binary numbers which are decreasing in value as his proof which receives applause from the class. Martin is lonely and depressed, but finds empathy in an MS support group where he is able to talk about his attraction to Julie but is fearful that he may not even be able to hold hands or cuddle in a few years. In math camp, Nathan's roommate Luke is also autistic but has picked up many bad characteristics including being obsessive and social incompetence. Isaac and the rest of the team finds him constantly disagreeable. Zhang Mei and Nathan walk the outdoor market before the final test. Zhang Mei says Nathan is a bit like a turtle in his shell. Nathan learns that smart kids are not bullied, but revered in China. Zhang Mei wishes China was more like Taiwan where there is less pressure to succeed from the family. The Chinese and British teams take their exams on the same room. When Richard announces the scores of the exams to the team, Nathan learns that he beat out Luke for the last spot on the team by fractions of a point. Zhang Mei also barely made her team. On the last evening at camp, the group of students fish for their dinner of prawns. Luke reenacts the Monty Python Dead Parrot Sketch with his dinner. The team in not amused and has made Luke an outcast. That night Nathan finds Luke cutting himself in the bathroom. Luke tells Nathan that he has done it before but just got carried away. Luke admits to Nathan that he does not even enjoy mathematics. As they load the bus to leave, Richard pulls Nathan aside. He tells Nathan that there is great beauty in his math but he is unpredictable and inconsistent. He warms Nathan not to be distracted by his Chinese houseguest if he wants to achieve a medal in the competition. Nathan and Zhang Mei fly to England. Zhang Mei thinks England is beautiful. They see a rainbow together on the train. When Nathan and Zhang Mei return to Nathan's home, Martin and Julie have set up Nathan's trainset to carry a sign sending him off to Cambridge. Nathan is embarrassed. In England, Zhang Mei stays with Nathan and his mother. Julie is shocked to find that Nathan's behavior has transformed into something more normal. She becomes aware that Nathan may have feelings for Zhang, which she asks him about. Nathan offers his bed to Zhang Mei and sleeps on an air mattress below. Zhang Mei learns that Nathan's father is dead and feels sorry for him. Nathan recalls his father and himself setting up the train when he was younger. Not fully understanding the concept of love, Nathan is unsure how to express his feelings. He looks for the mathematical formula for love on his computer. Martin volunteers at Cambridge competition as an observer. Richard decides Panama hats would be appropriate attire for his team. Julie brings Zhang Mei and Nathan to Cambridge. Zhang Mei takes Nathan's hand as they climb the spiral staircase. Richard gives his team an inspirational boat tour of the university. The teams are introduced in a ceremony. That night Zhang Mei cannot sleep. She seeks out Nathan's room and climbs into bed with him. She tries to kiss Nathan but Nathan is awkward in response. She tries again and Nathan gets the idea. They hold hands and fall asleep. Zhang Mei's uncle catches her in Nathan's room in the morning. Zhang Mei and her uncle argue and Zhang Mei storms out of the room. Zhang Mei withdraws from the competition and leaves Cambridge believing others do not believe she has earned her way onto the Chinese team. She tells Nathan that even though she is leaving, it does not mean she does not find herself liking Nathan. On the way to the competition, Rebecca tells Nathan that it may have been her fault that Zhang Mei and Nathan were caught together. The room is set for the four hour and thirty-minute test. When Nathan sits down among hundreds of other students from around the world for the exam, the first question triggers memories of his dead father. Combined with his newly lost love, an emotional overload inside Nathan is created. Nathan bolts from the room. Martin and then Julie follows behind. Martin tells Julie that he could see in his eyes that Nathan wanted to go and he did not stop him. Julie kisses Martin. Julie finds Nathan in a Chinese cafe. Nathan speaks of his confusion about love and considers both the loss of his father and the girl who appreciated and loved him, Zhang Mei. Julie explains that love is like an added value when love is given and received. She says love is not without risk and is a loss of value to the self when love is lost or taken away. Mother and son embrace at last. Then Julie drives Nathan, now -willing to sitting the front seat, to the station to fetch back Zhang Mei.