Fermat's Room (Spanish: La habitación de Fermat) is a 2007 Spanish thriller film directed by Luis Piedrahita and Rodrigo Sopeña. Three mathematicians and one inventor are invited to a house under the premise of solving a great enigma, and told to use pseudonyms based on famous historical mathematicians. At the house, they are trapped in a room. They must solve puzzles given by the host, who calls himself "Fermat," in order to escape the slowly closing walls of the room.

The film opens with a small set of background pieces on the characters. One, a young mathematician, is shown discussing sums of prime numbers with some women, and it is revealed that he is scheduled to present a proof of Goldbach's conjecture. A friend approaches him letting him know that his office was broken into and his work destroyed, delaying any possibility of presenting it publicly. Another, an older mathematician, is shown playing chess with a friend who urges him to quit mathematics, citing many of the brilliant mathematicians eventually go insane. Lastly, a middle aged man is shown to receive a letter from an anonymous source using the name Fermat. The letter states that Fermat is organizing a meeting of brilliant intellectuals, and to attend, the recipient of the letter must solve a certain "enigma," which is to determine how the following numbers are ordered: 5, 4, 2, 9, 8, 6, 7, 3, 1. The man is shown to struggle until the deadline, when an offhand comment from a coworker leads him to realize that the sequence of numbers are in alphabetical order (in Spanish: cinco, cuatro, dos, nueve, ocho, seis, siete, tres, uno).

Upon completion of the puzzle, a second letter is received with instructions to show up at a given time and place, alone, without a cell phone, to work on the greatest enigma. All the recipients receive pseudonyms. The young mathematician is Galois, the older mathematician is Hilbert, and the middle aged man is Pascal. We are then introduced to a fourth character, known as Oliva. Hilbert's car breaks down on the way to the meeting, and Galois stops to give him a ride to the meeting when he realizes Hilbert is going to the same place he is. Hilbert admits that he is familiar with Galois since he saw him in a mathematics magazine as a brilliant mind who had proved Goldbach's conjecture. When they are all at the meeting place, at a river, a car on the other side flashes its lights, and they use a row boat named Pythagoras to reach it. Inside is a PDA with GPS directions that they follow to an abandoned warehouse in the middle of nowhere. Upon entering and searching, they find a room resembling what Hilbert describes as a room that intellectuals would meet in.

After a short waiting period, a man enters and introduces himself as Fermat. They begin to talk and discuss things, but eventually Fermat receives a phone call about his daughter, whom he claims is in a coma at a nearby hospital. Fermat leaves, but forgets his coat. Pascal chases after him, but Fermat has already driven off. Fermat's wallet drops out of a pocket in the confusion. Pascal picks up the open wallet, and notices a picture of a girl. He returns to the room.

Upon Pascal's return, the PDA emits a sound and displays the following enigma, along with a deadline of one minute for its solution: A candy merchant receives 3 opaque boxes. One box contains mint candies, another contains anise candies, and the last box contains a mixture of mint and anise. The boxes are labeled Mint, Anise, and Mixed. All of the boxes are labeled incorrectly. What is the minimum number of candies the merchant will have to sample to correctly label each box? Hilbert, Oliva, and Galois work on the problem, exceeding the one-minute deadline but eventually solving it. Meanwhile, Pascal notices that the room had begun to shrink when the deadline was reached. He notices an inventory sales order on a piano for 4 industrial strength presses. The group then realizes that the presses are set to compress the room when time on the enigmas is exceeded.

As the occupants of the room continue to work on more and more enigmas, details about their pasts begin to emerge. Pascal accidentally hit the girl in the photo he found with a car. Pascal believes that Fermat wants to kill him for this transgression, and that the others are collateral damage. Meanwhile, Fermat is shown at a gas station, and then at a hospital. He is shown upset when it turns out that the hospital didn't call him. The nurses suggest that he go home, but he realizes he left his keys in his jacket, and must return to the warehouse.

More details emerge about the characters' pasts. Oliva and Galois used to date, but broke up when Galois began to suspect she was having an affair. Oliva admits that she met a man online while playing chess, and she eventually met him and would go on boat trips out to international waters, to do things illegal in their country. She then reveals to the group that this man is Hilbert. The group then tries to use the furniture in the room to stop the moving walls, but the furniture shatters. At this point, Pascal makes the suggestion that the person who set all of this up could not have been Fermat, since he acted strangely, and they all assumed he was the host. They also realize that he could not have faked a phone call to himself. Pascal theorizes that since all this was a revenge plot, the person wanting revenge would want to see it happen. As there were no visible cameras or viewpoints, he suggests that the person trying to kill them is in the room. First, suspicion is thrown upon Galois for his anger at Oliva and Hilbert, but then Pascal realizes it couldn't have been him since Hilbert was the only one not with the group when Fermat received his phone call. Hilbert admits that he is the one responsible for their current predicament, and that he did it because he spent his entire career working on Goldbach's conjecture, and Galois solved it at such a young age. Galois admits that he faked the proof to impress Oliva, and that when the pressure and spotlight became too great, he sabotaged his own work. Hilbert states he'd been driven even harder by Galois, and that he'd actually solved the conjecture, and hands a folder with his work to Galois. Galois comments that the mathematics contained within is "brilliant". They try to call Fermat from Hilbert's cell phone, but Hilbert placed a chemical to kill Fermat, who dies en route back to gathering. Hilbert says they should continue working on his enigmas, but Galois becomes angry, and attacks him, knocking Hilbert unconscious. Pascal then realizes that all of the pseudonym characters died at the ages of the people they are named after, except for Hilbert. A furious search for Hilbert's escape route begins, and they find it behind the chalkboard. Pascal, Oliva, and Galois escape with the folder containing Hilbert's solution to Goldbach's conjecture, leaving behind an unconscious Hilbert. They make their way back to their cars. While crossing the river, Galois comments that he doesn't want to release Hilbert's solution under Hilbert's name, because then he would have won, but releasing it under his own name would be unethical but solve his problems. Pascal throws the solution into the river, remarking that the world is still the same without the proof.