Mathematical Principles in Cube

Document directory

• [Conversion] mathematical principles in Cube

[] Mathematical principle Kim In Cube was published:

Mathematical Principles in Cube

I. Cube shape and number of rooms

Cube is composed of a huge Cube and a layer of shell outside the Cube. There is space between the two. The Cube also contains many small Cube rooms, similar to Cube. Cube has only one exit. Only the room that has reached the connected housing and the internal Cube can exit the Cube. This room is called a bridge in the film ". Each room has a length of 14 feet (slightly longer than 4 meters ). Each side of the cube has a length of 26 rooms, so the total length is 26*26*26 = the size of the 17576 rooms. (But in fact there are not so many rooms, because the room has to be moved)

II. How to identify traps in a room

· Identify room security

Each room in the Cube is marked with three-digit numbers. Because the numbers in each room are different, Holloway initially thought this was the number of the room (which she thought was hundreds of millions of rooms, but she was wrong ). Leaven then thought they could use these three numbers to identify whether there were traps in the room. Leaven had a good memory. She noted down the numbers in every room they passed, after summing up, she concluded that there was a trap in a room with a prime number in three numbers (this theory was useful at the beginning, but there was also a trap in a room without a prime number, so far, this theory has been overturned ). At the end of the film, the truth is finally mined: Identifying traps is not the prime number, but the multiplication of the prime number. Leaven asked Kazan to report the prime number factor of each number.

· Multiplication of prime numbers

Each natural number (1, 2, 3, 4...) can be multiplied by a prime number if it is not itself, for example, 120 = 2*2*2*3*5. This representation is unique, regardless of the order before or after the prime number. 2*2*2 is expressed as 2 ^ 3 here, So 120 = (2 ^ 3) * 3*5. If a number contains only one prime number factor, it is the multiplication of prime numbers. Obviously, each prime number is also the multiplication of prime numbers (This also explains why Leaven's theory has not started to go wrong ). However, the multiplication of a prime number is not necessarily a prime number. For example, 27 = 3*3*3 = 3 ^ 3, but 27 is not a prime number because it can be expressed as 3 multiplied by 9, in this case, Leaven's theory becomes invalid.

III. Location and movement of room space

No matter whether there is a trap in the room, the three-digit number does not represent itself. After the introduction below, you will find that they represent the space position and moving track of the room.

· Room coordinates

The number of each room is actually a Cartesian coordinate. It represents the location of the room in space, but it is different from the Cartesian coordinate. The two coordinates can be converted to each other. For example, if the Cartesian coordinates of a room are 493,454,967, the X axis coordinates of the room are 4 + 9 + 3 = 16, and the Y axis coordinates are 4 + 5 + 4 = 13, the Z axis coordinates are 9 + 6 + 7 = 22, so the Cartesian coordinates of the room are (16, 13, 22). The coordinate unit is a room, so in the Z axis direction, this room is four rooms away from the housing. The coordinate value cannot be a negative number (because the three natural numbers cannot be a negative number), and the coordinate value of XYZ in each direction is not greater than 26 (except for the bridge "). Leaven they once reached a room with the Y axis of 27, which is actually a "bridge" to the outside of the Cube ". But they didn't find this secret at the time, because the room was still surrounded by other rooms, until the room where Rennes died before he was thrown by Quentin and there was no external channel, then he realized that the room would move. "We are not moving, but the room," he said ....... This explains why we have always felt the shock and we have been moving our room ." Cube is like a huge Cube that keeps rotating. Every room is moving from time to time, and every coordinate only indicates the position at the beginning of the room.

· Room Movement Mode

The moving trajectory of each room is also hidden in Cartesian coordinates. For example, the Cartesian coordinates of a room with the coordinates of 477,804,539 are (18, 12, 17 ). If you want to know the moving track of the room, you can do the following for each three-digit number:
1. Hundreds of digits minus ten digits
2. Ten digits minus single digits
3. single digit minus hundred digits

Perform the preceding operations on all three numbers, that is:
1. 477: 4-7 =-3 | 7-7 = 0 | 7-4 = 3
2. 804: 8-0 = 8 | 0-4 =-4 | 4-8 =-4
3. 539: 5-3 = 2 | 3-9 =-6 | 9-5 = 4

In this way, three vectors (-3, 8, 2), (0,-4,-6), and (3,-4, 4) are obtained ). These three vectors represent the moving trajectory of the room. They will be converted to Cartesian coordinates to represent the coordinates of the initial location of the room (which can be viewed as vectors), and these three vectors will be added in sequence:
(18, 12, 17) + (-3, 8, 2) = (15, 20, 19)
(15, 20, 19) + (0,-4,-6) = (15, 16, 13)
(15, 16, 13) + (3,-4, 4) = (18, 12, 17)

We can see that after three changes, the original initial coordinates (18, 12, 17) are returned ). Each room is based on this rule (18, 12, 17) --> (15, 20, 19) --> (15, 16, 13) --> (18, 12, 17) -->... .

· Room location changes within a period of time

According to the coordinate changes, each room is actually moving on a regular track repeatedly. To know the location of the space, you must also have a reference object, that is, you must know at least the coordinates of a neighboring room. For example:
The coordinates of the room are 320,176,223 (recorded as room 1), and the Cartesian coordinates are (5, 14, 7), with (5, 14, 7) --> (6, 8, 7) --> (8, 9, 6) --> (5, 14, 7) -->... Trajectory Movement
The room on the right is 214,168,104 (as room 2), and the Cartesian coordinates are (7, 15, 5), with (7, 15, 5) --> (8, 10, 6) --> (5, 8, 2) --> (7, 15, 5) -->... Trajectory Movement
The room above it is 254,303,017 (recorded as Room 3), the Cartesian coordinates are (11, 6, 8), to (11, 6, 8) --> (8, 9, 7) --> (9, 6, 1) --> (11, 6, 8) -->... Trajectory Movement

From the three moves of the three rooms, we can see that they are not always adjacent. In other words, only when Room 1 arrives (8, 9, 6), room 2 arrives (8, 10, 6) the two are left and right adjacent, and only when Room 1 arrives (8, 9, 6), Room 3 arrives (8, 9, 7) the two are adjacent to each other, and the three rooms are separated from each other in other time periods. Not all rooms are moved together, but they are moved independently of each other. In this way, the Cube will have an initial state. At this time, all the rooms will stay on their initial coordinates, and the rooms will move and return to the initial state after several times, this cycle may take several days, depending entirely on the size of the Cube, which also affects the time required to reach the bridge.

· Bridge and exit

"Bridge" is actually a room, as we have already said above. At its initial position, it connects the shell and the big cube inside, and the exit is inside the "bridge. The Y axis of the bridge is 27, and the Y axis of other rooms is not greater than 26. The "bridge" moves like other rooms on a fixed track, which means that it is the real "bridge" only when it reaches its initial position, and talents can walk out of the Cube through it, it is located elsewhere in the big cube in other time periods. Therefore, you must seize the opportunity and wait for another round after you miss the initial position. Leaven compares Cube to a safe deposit box lock. The lock can be opened only when all rooms reach their initial position. However, when the room moves, the lock is closed. Therefore, to find the exit, you must first find a room at the boundary of the cube (a coordinate is 26), then select the room along the boundary, and finally find the bridge ", wait until it returns to the initial position to exit the Cube.