The Tower of Doors is both a location and game in the Kakegurui manga. It is a large structure owned by Kirari Momobami. The entire building is a "game" in itself filled with several tricks and mechanisms that allow a specific kind of gamble to take place. The game pits Yumeko Jabami against Sayaka Igarashi. Kirari herself serves as the judge.
The game is played from within the 5th floor. Players must descend to the 1st floor, take a lily as proof and return to the 5th floor. Whoever accomplishes this first is the winner.
Floors have several doors with panels on them, in the walls, the center of the tower and trap doors in the floor. To open them, players must respond correctly to whatever question is presented to them in each of them. The outside doors are basically windows, they lead nowhere and walking off them is basically suicide. The Central door is the only one that leads directly to the 1st floor and likewise from the 1st floor all the way up to the 5th. The Trap Doors can allow one to traverse 1 or more floors at once. Players are given turns of 5 minutes each, which they can use at their leisure, this includes the time they can use to travel between floors. Once the 5 minutes pass, all doors close.
Things are more complicated than this however. The questions and the answers provide hints as to the real workings of the Tower as well as how many Floors one can descend. It should be noted that a "correct answer" is not the same as a "correct solution". Even if one answers the question properly if said answer doesn't make sense for the aforementioned mechanics then it won't benefit the player. For example, if the "correct answer" to a question is 8, but the player is in the 5th floor, although a correct answer it doesn't make sense to "descend 8 floors".
Furthermore the tower in itself rotates 60° after every turn, but the numbers on the outside door do not. These numbers are a hint to the tower's true trick.
Both players bet "everything". For Yumeko this means being forbidden to gamble for the remainder of her life. For Sayaka, this means cutting all and any ties with Kirari and being considered strangers. However since she may still hold a one-sided admiration towards Kirari, this is still considered a "bond", therefore she is requested to jump from the tower to commit suicide, to become a stranger and a dead.
Before Yumeko's turn, the numbers on the wall doors, change. Kirari warns that "answering correctly" is the key to live. Yumeko has the first turn, she decides to open the center door. Yumeko causes a distraction for Sayaka and enters an answer on the panel. The Door opens but she encounters a wall. Kirari provides a hint, it is a "correct answer" but not a "correct solution".
Next is Sayaka's turn. She looks at the numbers on the walls and deduces they are hexadecimal numbers. She then translates these numbers into decimal notation, revealing they are angles. She further confirms this as the numbers changed but their order did not. 12c in the Center Door translates to 300°, however since Yumeko faced a wall, Sayaka decides instead to go for the Trap Doors as their number didn't change. Since 1 is the same in both formats, Sayaka answers with 1, leading to a correct answer.
Sayaka descends the stairs and reaches the 4th floor. Once there, she notices a stair that leads from the 5th floor to the 3rd. She then realizes that the number 1 and 2 were meant to represent the number of floors one can move. Sayaka begins to panic but regains her composure.
It is Yumeko's turn again. Yumeko voices the same conclusions reached by Sayaka and voices her praise for the Secretary for her logic. Kirari states that logic isn't the real way to describe it but that she will understand at the end. Yumeko then decides to open one of the wall doors which Kirari finds curious as it is an illogical choice. Yumeko voices her excitement at the tower Kirari prepared for them and her wishes to resolve the mysteries behind the gamble.
From this point the game changes to reflect Sayaka's perspective across several turns. She spots the numbers on the walls are in Radians now, but realizes they are meaningless now. She goes to the answer the door that allows her to descend 2 floors.
In the 2nd floor she encounters Mahjong related problems. One of the answers is 8, which she deems illogical as she cannot descend 8 floors from the 2nd floor. She answers 1 in the other Trap Door and is allowed to continue. She begins to feel anxious about Yumeko but regains her composure and descends.
She reaches the 1st floor and is the first one to reach there. Sayaka realizes what makes her uncomfortable; she has yet to realize the trick behind the center door. The numbers of the doors are in Abacus format. The center door has 120 so she answers accordingly, but faces a wall. She begins to wonder about the angles in the door walls and how they are not making sense.
It is Yumeko's turn and she catches up to Sayaka, Yumeko gets her flower. It is Sayaka's turn and she gets her flower and climbs up from the path opened by Yumeko. She faces a Chess problem at the Trap Door and answers with 1. The answer is correct and she advances. Just like before the game changes to Sayaka's perspective across multiple turns.
Sayaka faces more pressure time-wise as she won't be able to discern the question unless she climbs up. She is faces two problems with 23 and 2 as answers. Since the latter is illogical, Sayaka answers with 2 and moves forward with renewed confidence. She answers one more undisclosed problem with 2 and reaches the 5th floor. She believes she is the winner but her confidence is broken as she finds Yumeko to have reached there first. Sayaka is thus deemed the loser.
Yumeko then explains what happened. She reveals that the numbers on the wall doors were indeed angles and that they were rotating clockwise. Although seemingly useless in later floors they had a deeper meaning behind. She reasons that she wanted to see the Moon as they walked to the tower so she used a turn to open the door that after rotating, should have revealed the moon.
However she found nothing. Since the moon moving is illogical Yumeko reveals a shocking truth, it is the Tower, not the numbers that move every 5 minutes, the numbers remain static. Since the tower moves, this reveals something else, that the structure with the center door is a "core tower" that remains static. This is the reason for the "wall" they encountered before hand. The center door was always in the same place but since the tower moved the "opening" was concealed by the concrete of the "core tower". This leads to another point, that since the tower rotates 60° every turn, it takes a full 6 turns for the center door to be accessible again.
Therefore on the 6th turn, Yumeko uses the center door to climb all the way up to from the 1st floor to the 5th floor and win before Sayaka.
Sayaka realizes her loss and that her overly logical way of thinking averted her from the game that Kirari had set up. Seeing that Kirari and Yumeko seemingly understand each other much better, Sayaka is resigned to her fate.
She opens up one of the doors and jumps ready to commit suicide. Much to her surprise Kirari also jumps with her as they fall into a safety mat. Sayaka is confused but Kirari reveals what happened. Sayaka unconsciously opened up a door in braille code whose answer was 5. Since they were on the 5th floor, the door chosen by Sayaka had the "correct" answer. This coincides with Kirari's warning of "answer correctly and you live". Kirari then reveals that all other doors lead to an inevitable death and that she wouldn't jump with Sayaka, had she used another door.
As Sayaka remained logical to the very end, even in the face of sure death, Kirari finds newfound amusement for Sayaka and "hires" her as her secretary in their condition of "total strangers". Yumeko believes Sayaka lost the game but won the war.
After the game, Kirari announces to the student council that she will disband the Student Council as she will hold special elections, where every student is eligible to vote and to run for president, herself included as well. She also has the Momobami family participate as well.