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ISSN en línea: 2789-3855, marzo, 2025, Volumen VI, Número 1 p 875

DOI: https://doi.org/10.56712/latam.v4i6.1468

Is P equal to NP? Answer and Method to solve puzzles with
God's algorithm

¿Es P igual a NP? Respuesta y Método para resolver
rompecabezas con el algoritmo de Dios

Nilson Rafael Bolívar Barrios

nilsonbolivarbarrios@gmail.com
https://orcid.org/0000-0002-3933-0969

Independent Researcher
Barranquilla – Colombia

Artículo recibido: 24 de noviembre de 2023. Aceptado para publicación: 11 de diciembre de 2023

Conflictos de Interés: Ninguno que declarar.

Abstract
The complexity of any type of problem can be from the easiest to the infinitely complex, there are
problems where there are shortcuts with God's algorithms and they can be solved in polynomial time,
and there are problems where with God's algorithms there are no shortcuts and no are solved in
polynomial time, if a non-deterministic Turing machine finds solutions to problems in polynomial time
this means that a deterministic Turing machine with God's algorithm can also find solutions to the
same problems in polynomial time, when we have a problem we will solve it with God's algorithm, we
can have 1, 2 or more God's algorithms that solve the same problem with the same number of steps.
As a result, we use God's algorithm, and we observe how many steps this problem is solved in. It can
be many steps or few steps, and we make the decision if it is worth spending time to solve that
problem. The author presented the results that it is impossible to solve the problem in fewer steps
than the number of steps of God's algorithm.

Keywords: God’s algorithm, complexity, solution, P, NP

Resumen
La complejidad de cualquier tipo de problema puede ser desde el más fácil hasta el infinitamente
complejo, hay problemas donde hay atajos con los algoritmos de Dios y se pueden resolver en tiempo
polinómico, y hay problemas donde con los algoritmos de Dios no hay atajos y no se resuelven en
tiempo polinómico, si una máquina de Turing no determinista encuentra soluciones a problemas en
tiempo polinómico esto significa que una máquina de Turing determinista con el algoritmo de Dios
también puede encontrar soluciones a los mismos problemas en tiempo polinómico, cuando
tengamos un problema lo resolveremos con el algoritmo de Dios, podemos tener 1, 2 o más algoritmos
de Dios que resuelven el mismo problema con la misma cantidad de pasos. Como resultado,
utilizamos el algoritmo de Dios y observamos en cuántos pasos se resuelve este problema, pueden
ser muchos pasos o pocos pasos, y tomamos la decisión si vale la pena dedicarle tiempo a resolver
ese problema. El autor presento los resultados de que es imposible resolver el problema en menos
pasos que la cantidad de pasos del algoritmo de Dios.

Palabras clave: algoritmo de Dios, complejidad, solución, P, NP

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ISSN en línea: 2789-3855, marzo, 2025, Volumen VI, Número 1 p 876

Todo el contenido de LATAM Revista Latinoamericana de Ciencias Sociales y Humanidades,
publicados en este sitio está disponibles bajo Licencia Creative Commons .

Como citar: Bolívar Barrios, N. R. (2023). Is P equal to NP? Answer and Method to solve puzzles with
God’s algorithm. LATAM Revista Latinoamericana de Ciencias Sociales y Humanidades 4(5), 875 –
897. https://doi.org/10.56712/latam.v4i6.1468

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ISSN en línea: 2789-3855, marzo, 2025, Volumen VI, Número 1 p 877

INTRODUCTION

In this research we talk about God's algorithm, which is the optimal solution to puzzles. As an example,
a puzzle with a cylinder divided into 4 parts is given as an example. Problems of type P and NP are also
discussed, and the answer to the millennium question is given, which is: Is P equal to NP? The purpose
of this research is to answer one of the seven millennium questions for the advancement of science,
especially in the advancement of mathematics and computer systems.

What motivated me to carry out this research is that I like the extreme and the complexities reach
infinity, and in this research there is a connection from the least complex cylinder puzzle to the infinitely
more complex one and using the same method these puzzles can be solved with God's algorithm.

Use God's algorithm to solve the problems. Remember that the complexity of problems can reach
infinity. Below we will use as an example a puzzle of a cylinder divided into 4 parts that will verify if it is
well assembled or not in approximately one minute, the cylinder will have a spiral that will help us know
if the cylinder is well assembled or not, a table will be made to assemble the cylinder with God's
algorithm, and the question will be answered: Is P equal to NP?

OBJECTIVES

Knowing that if a non-deterministic Turing machine finds solutions to problems in polynomial time, that
implies that it is also possible for a deterministic Turing machine with God's algorithm to find solutions
to those same problems in polynomial time. Also know that if it is possible quickly verify the solutions
to the problems, in some cases that implies that it is also possible to find the solution with the same
speed and in other cases, if it is possible, quickly verify the solutions to the problems, that does not
imply that it is also possible to find the solution just as quickly. Use God's algorithm to solve the
problems. Remember that the complexity of problems can reach infinity.

DEVELOPMENT

According to Nilson Bolivar (2020), it is explained how to assemble a puzzle called Pyraminx Duo with
God's algorithm, also the method to assemble the Rubik's cube or other puzzles in 3 dimensions with
God's algorithm is given.

According to Elvira Mayordomo (2012) P is usually identified with the class of feasible problems, or
problems that can be solved in practice. Clearly, a polynomial time limit can be huge due to both the
multiplicative constant and the degree, but there are two practical reasons why we think of P as
efficiently solvable problems.

According to Fortnow, Lance (2009) What is the P versus NP problem? Suppose we have a large group
of students who need to pair up to work on projects. We know which students are compatible with each
other and we want to put them into compatible groups of two. We could search as many pairings as
possible, but even for 40 students we would have over three hundred billion trillion possible pairings.

Continuing with the article, if we have a puzzle of a cylinder that is divided in half into equal parts with
the following rules: 1) that half of the number of divisions of the cylinder is an even number, 2) that each
movement of the cylinder is a 180 degree turn, in other words, half turn 3) that one of the divisions of
the cylinder never moves, which will be differentiated because it will be blue at the top and green at the
bottom, it will have the number 1, and black lines. This part of the cylinder will never move because it is
the same to move half of the cylinder with the numbers 1 and 2 than to move half of the cylinder with
the numbers 3 and 4, so it is decided that the half of the cylinder with the blue color with the number 1
and the black lines will never move, no matter how many divisions the cylinder has, the blue part with
the number 1 and the black lines will never move.

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4) The fourth rule is that for example if the cylinder has 4 divisions the cylinder will be assembled if
clockwise after the division of the cylinder that has the blue color, the number 1 and the black lines,
follows the division of the cylinder with the number 2, then the division of the cylinder with the number
3, and finally the division of the cylinder with the number 4, as shown in the following image.

Figure 1

Cylinder with 4 divisions

Source: self made.

Each division of the cylinder will have a number, for example, 1, 2, 3, 4, etc. The cylinder will be
assembled when, after the part of the cylinder that has the blue color, the number 1, and the black lines,
the numbers of each division of the cylinder will increase by 1 in 1 clockwise, as seen in the previous
image.

The total number of combinations for this cylinder in the previous image is 3 factorial = 3x2x1 = 6
combinations. Recall that half of its number of divisions is an even number; the cylinder has 4 divisions,
and half of 4 is 2, which is an even number.

Then, the cylinder with half of its even number of divisions divided into 8 parts, its number of
combinations is 7 factorial = 5,040.

The cylinder with half its even number of divisions divided into 16, its number of combinations is 15
factorial, and so on to infinity.

The more divisions the cylinder has, the more complex the problem will become, because the cylinder
will have a greater number of combinations, and because it will have more movements options. And
you can divide the cylinder in half, to infinity.

Now we have to make sure that this puzzle can be checked whether it is well assembled or not in a
short time, for example in about a minute. To do this, we place something like the hand of a clock that
will almost go around the spiral of the cylinder in approximately one minute. Obviously this needle is
going to be stuck in the center of the circle so that it can almost go around, starting clockwise on the
right side of the part of the cylinder that is blue and ending on the left side of the part of the cylinder
which has blue color, as shown in the following image.

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Figure 2

Cylinder with many divisions

Source: self made.

In the previous image, we can also see a spiral. In the following image, we see an example of the length
of what is similar to the needle of a watch, and we see that inside the needle, there is a sphere that has
a welded rod that enters the spiral. As the spiral is getting closer to the center, the sphere with the
welded rod is also getting closer to the center.

Figure 3

Similar to the hand of a clock

Source: self made.

In the following image, you can see that inside what is similar to the needle of the watch, there is a
sphere, and the welded rod in the sphere is smaller; for example, it measures 1 mm, and the spiral
measures 1.5 millimeters wide. We also observe that the welded rod will not rub against the floor of the
spiral because the floor of the spiral is deeper.

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Figure 4

interior view of what is similar to the hand of a clock

Source: self made.

In the following image we see, as we already know, that the welded rod is 1 millimeter in diameter, and
the spiral is 1.5 millimeters wide. If the cylinder is properly assembled, the spiral will approach, for
example, 1 millimeter to the center of the cylinder circle in each part into which the cylinder is divided.
For example, if the 8-division cylinder is well assembled clockwise after the blue part of the cylinder,
the spiral of the 7 parts of the cylinder approach the center of the circle by 1 millimeter in each division.
For example, if you next to the blue part of the cylinder clockwise, the spiral that is on the side of the
cylinder clockwise would be 7 millimeters from the center. The part of the cylinder that follows would
be 6 millimeters from the center of the circle, the next part would be 5 millimeters away, the next part
at 4, the next at 3, the next at 2, and the last part at 1 millimeter from the center of the circle. This
happens if the cylinder is well assembled, and the welded rod would reach the end of the spiral.
However, if the cylinder is badly assembled for example, if the part of the cylinder that is 3 millimeters
from the center follows the part that is 1 millimeter from the center, then the welded rod cannot pass
because it is 0.5 millimeters wide, as shown in the following image below.

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Figure 5

Poorly assembled cylinder

Source: self made.

In the previous image, we check in more or less 1 minute that the cylinder is badly assembled, even if
the cylinder has many divisions. We would check if it is well or badly assembled depending on whether
the needle reaches or does not reach the end of the spiral. Then, it would be an NP problem because it
can be easily checked in polynomial time.

The puzzles have a God algorithm, and the cylinders in this paper also have a God algorithm. Another
way to quickly check if the cylinder is well assembled or not is to place a button at the beginning of the
spiral so that the welded rod when pressed turns on a red light, and at the end of the spiral place another
button so that the rod welded when pressing a green light comes on.

METHOD TO ASSEMBLE THE CYLINDER WITH 4 DIVISIONS WITH THE GOD´S ALGORITHM

It seems impossible for a human being to assemble a puzzle with God's algorithm, but it is possible
using this method. The cylinder is divided into 4 parts, and to each part of the cylinder, we are going to
place a letter: A, B, C, D. Also, to each part of the cylinder, we are going to place a number, which are 1,
2, 3, 4, as seen in the following image below.

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ISSN en línea: 2789-3855, marzo, 2025, Volumen VI, Número 1 p 882

Figure 5

Front view of assembled cylinder

Source: self made.

The letters A, B, C, and D are like shadows, that is to say, they are not going to move. If the cylinder
moves, the letters are not going to move; only the numbers are going to move. The part of the cylinder
that has the number 2 also has the letter B, and the part of the cylinder that has the number 3 has the
letter C. Remember that the movements of the cylinder are half turns. If we have the cylinder assembled
and we move the numbers 2 and 3, the cylinder looks as seen in the following image that we have below.

Figure 6

Front view of the cylinder disassembled with a single movement

Source: self made.

In the previous image, we realize that the numbers 2 and 3 move when the cylinder moves. It is as if we
were writing the numbers, and the letters B and C did not move even though the cylinder moved. It is as
if the letters were shadows. Then, the letters A, B, C, and D will never move if the cylinder moves; the
letters will never move, and the numbers will move when the cylinder moves, as seen in the previous
image.

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What we are going to do is to have the cylinder assembled as shown in the following image below.

Figure 7

Front view of assembled cylinder

Source: self made.

Then, we make a table and place zero movement in the table. We also place part of cylinder 1, and as
the part of cylinder 1 is in the letter A, we place the letter A in the table. In the table where it says 'part
of cylinder 2,' we place B because the 2 is in the letter B. In the part of cylinder 3, we place the letter C
because the 3 is in the letter C. In the part of cylinder 4, we put the letter D because 4 is in the letter D.
Then, we put the combination in the table as follows: 1 is in A, 2 in B, 3 in C, 4 in D. The combination
would be ABCD. Next, in the table, we put the words 'I return,' and as there are zero movements, we put
'assembled' because the cylinder is assembled.

Now we are going to disassemble the cylinder with a single movement, and we realize that the cylinder
has 2 options of movements that are to move the two parts of the cylinder that is below and the two
parts of the cylinder that is on the left side, remember that the part of the cylinder that is blue does not
move, then we choose a range of movements so that the movement of the bottom part of the cylinder
moves first than the left part of the cylinder, then we disassemble the cylinder with a single movement
of the bottom part of the cylinder, and the cylinder is as shown in the following image below.

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Figure 8

Front view of the cylinder disassembled with a single movement

Source: self made.

And in the table, in movement 1, we place half a turn down as the part of cylinder 1 is in the letter A. We
place in the table the letter A for the part of cylinder 1, in the table where it says part of cylinder 2, we
place C because the 2 is in the letter C. For the part of cylinder 3, we place the letter B because the 3 is
in the letter B, and for the part of cylinder 4, we place the letter D because the 4 is in the letter D. Then,
we place in the table "combination": as the 1 is in A, the 2 in C, the 3 in B, and the 4 in D, the combination
would be ACBD; with 1 in A, 2 in C, 3 in B, and 4 in D, the combination would be ACBD.

Then, in the table, we put the word ‘I return’. As there is a movement and the cylinder is disassembled
with a half turn down, it's as if we were recording a video disarming the cylinder and then we reverse
the video. In the reversed video, the cylinder is disassembled and is assembled with a half turn down,
so we put a half turn down. We do the same with movement one, a half turn to the left. We also do the
same with movements two and three.

Remember that the total number of combinations of the cylinder is 3 factorial, which is equal to 6. As
we disassemble the cylinder with 3 movements and the 3 movements gave us 6 different total
combinations of the cylinder, it is no longer necessary to disassemble the cylinder with 4 movements,
because the combinations of the 4 movements will be repeated with the combinations with fewer
movements. The same will happen with the combinations of 5 movements, 6 movements, and so on.
We will disassemble the cylinder, and the unnecessary movements are eliminated, so we will only
disassemble the cylinder up to 3 movements. Up to 3 movements, there are 6 different options of
combinations: ABCD, ABDC, ACBD, ACDB, ADBC, and ADCB. In the following tables, the letters (HTD)
stand for half turn down, and the letters (HTL) stand for half turn left. Let's look at the table we are
talking about below, which is divided into 4 parts, M 1 means movement 1, M 2 means movement 2, M
3 means movement 3.

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Table 1

First process to make God algorithm table, part 1

M
1

M
2

M
3

part
of

cylinder
1

part
of

cylinder
2

part
of

cylinder
3

part
of

cylinder
4

combination I
return

Zero
movements

A B C D ABCD assembled
puzzle

H
T
D

A C B D ACBD HTD

H
TL

A B D C ABDC HTL

Source: self made.

Table 2

First process to make God algorithm table, part 2

HTD HTD A B C D ABCD HTD HTD
HTD HTL A D B C ADBC HTL HTD
HTL HTD A C D B ACDB HTD HTL
HTL HTL A B C D ABCD HTL HTL

Source: self made.

Table 3

First process to make God algorithm table, part 3

HTD HTD HTD A C B D AC
BD

HTD HTD HTD

HTD HTD HTL A B D C AB
DC

HTL HTD HTD

HTD HTL
HTD

A D C B AD
CB

HTD HTL HTD

HTD HTL HTL A C B D ACBD HTL HTL HTD

Source: self made.

Table 4

First process to make God algorithm table, part 4

HTL HTD HTD A B D C ABDC HTD HTD HTL
HTL HTD HTL A D C B ADCB HTL HTD HTL
HTL HTL HTD A C B D ACBD HTD HTL HTL
HTL HTL HTL A B D C ABDC HTL HTL HTL

Source: self made.

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Now we organize alphabetically the combinations of the table, and we only copy the parts of the table
where the combinations and the movements that are returned. Then, where it says "I return," we replace
it with the phrase "is assembled with." We place a new column in the table called "combination number,"
and there we count the number of combinations of the cylinder.

The combination ABDC, assembled with 1 movement that is a half turn left, is also assembled with 3
movements: half turn left, half turn down, half turn down. It is also assembled with 3 other movements:
half turn down, half turn down, half turn left. So, God's algorithm looks for the least number of possible
movements, the perfect or optimal number of movements to assemble the cylinder. Then, the
combination can be assembled with a movement that is a half turn left, or with 3 movements, and God's
algorithm chooses 1 movement. Then, for the 3 movements, we place "eliminated," and we do the same
with all the combinations of the following table. In the combination ADCB, there are 2 options of
assembling with 3 movements: half turn down, half turn left, half turn down, and another option with 3
movements: half turn left, half turn down, half turn left. As the two options are 3 movements, it is the
same number of movements that this combination has. This means that God's algorithm can be
assembled in 2 possible ways with 3 movements. God's algorithm is not unique; it seeks to assemble
the combination with the least number of movements possible. If there are several options with the
same number of movements to be assembled, then any of these options can be used.

In the following tables, the letters (HTD) stand for half turn down, and the letters (HTL) stand for half
turn left. We are listing the combinations that are not eliminated, and it has to give us 6 combinations,
which is equal to the number of total combinations that the cylinder has: 3 factorial, equal to 6.
Everything that has been explained is observed in the following table, which is divided into 2 parts.

Table 5

Second process to make God algorithm table, part 1 of the table.

Combination is assembled with Combination number
ABCD assembled puzzle 1
ABCD HTD HTD Eliminated
ABCD HTL HTL Eliminated
ABDC HTL 2
ABDC HTL HTL HTL Eliminated
ABDC HTL HTD HTD Eliminated
ABDC HTD HTD HTL Eliminated

Source: self made.

Table 6

Second process to make God algorithm table, part 2 of the table

ACBD HTD 3
ACBD HTD HTD HTD Eliminated
ACBD HTD HTL HTL Eliminated
ACBD HTL HTL HTD Eliminated
ACDB HTD HTL 4
ADBC HTL HTD 5
ADCB HTD HTL HTD 6
ADCB HTL HTD HTL 6

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Source: self made.

Now we remove the eliminated combinations from the previous table, and the table will be called God's
Algorithm. We replace 'assembled' with 'assembled with God's Algorithm,' and it appears as follows.

Table 7

Third and final process to make God's algorithm table

God's Algorithm
Combinations It is assembled with God's algorithm Combination number

ABCD assembled puzzle 1
ABDC HTL 2
ACBD HTD 3
ACDB HTD HTL 4
ADBC HTL HTD 5
ADCB HTD HTL HTD 6
ADCB HTL HTD HTL 6

Source: self made.

The previous table is arranged in alphabetical order for easier combinations search. The more
intelligent we are and the more we strive, the better we will become at solving more complex problems.
In other words, intelligence and effort are directly proportional to becoming better at solving more
complex problems. It can also be said that the more intelligence and effort we apply, the better
algorithms we will create with the fewest possible steps to solve that problem. However, let's remember
that God's Algorithm represents the limit, and there is no algorithm that can solve the problem in fewer
steps than God's Algorithm. This is why a problem of type P may have a shortcut for solving it in
polynomial time, but if it's very complex, we might not have the necessary intelligence to solve it. We
might either solve it with an algorithm that requires a significant number of steps or with an algorithm
in non-polynomial time.

WE CAN MAKE GOD´S ALGORITHM FOR THE CYLINDER DIVIDED IN 4 PARTS IN ANOTHER WAY, WITH
INTELLIGENCE

For example, if the cylinder is assembled, it is obvious that if we repeat the same movement 2 times,
such as half turn down, half turn down, the cylinder is going to return to zero movements. Then, we
remove the combinations that have the movements that are repeated in a row 2 times or more. Thus,
we make the table of God's algorithm faster. However, in a cylinder with more divisions or more
complexity, the obvious can become more and more complex, to infinity, to the point where we might
not have the sufficient intelligence to realize it. For that reason, the table of God's algorithm is created
step by step in this investigation, in case you don't realize the obvious.

In the following table below, we observe the number of combinations with respect to the number of
movements of the cylinder with 4 divisions, which has to give 6 total combinations.

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Table 8

Incomplete table of god's algorithm, cylinder with 4 divisions

Number of movements to arm with God's algorithm 0 1 2 3 Total combinations
Number of combinations with respect to the number of
moves to assemble with God's algorithm of the cylinder
divided into 4 parts

1 2 2 1 6

Source: self made.

The above table is an incomplete table of God's algorithm of the cylinder divided into 4 parts because
it tells how many combinations it has with respect to the number of movements to assemble with God's
algorithm. The same happens with other puzzles, like, for example, the Rubik's Cube. On Wikipedia it
says that the maximum number of face turns needed to assemble any combination of the Rubik's cube
is 20 moves, I'm not sure if it is with God's algorithm, but assuming it is with God's algorithm the
incomplete table of the algorithm of God would be as shown below. The table is divided in 2.

Let's see the first part of the table below.

Table 9

Incomplete table of god's algorithm, Rubik's cube

Number of movements to arm
with God's algorithm

Number of combinations with respect to the number of movements to be
assembled with God's algorithm

0 1
1 18
2 243
3 3,240
4 43,239
5 574,908
6 7,618,438
7 100,803,036
8 1,332,343,288
9 17,596,479,795

10 232,248,063,316

Source: self made.

Now let's look at the second part of the table below.

Table 10

Incomplete table of god's algorithm, Rubik's cube

11 3,063,288,809,012
12 40,374,425,656,248
13 531,653,418,284,628
14 6,989,320,578,825,358
15 91,365,146,187,124,313
16 about 1,100,000,000,000,000,000
17 about 12,000,000,000,000,000,000
18 about 29,000,000,000,000,000,000
19 about 1,500,000,000,000,000,000

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20 about 490,000,000

Source: own elaboration and modified with data from Davidson, Morley; Dethridge, John; Kociemba,
Herbert; Rokicki, Tomas. «God's Number is 20»

Exercise number 1

Everything we did above is to assemble the cylinder with God's algorithm, let's assume we have the
cylinder disassembled as shown in the following image below.

Figure 8

Front view of disassembled cylinder with 2 movements

Source: self made.

In the previous image, we observe that the combination of the cylinder is ACDB. In the following table,
the letters (HTD) stand for half turn down, and the letters (HTL) stand for half turn left. So, we search
for the combination in the God's Algorithm table, and it will be easy to find because the table has the
combinations in alphabetical order.

Table 11

Third and final process to make God's algorithm table

God's Algorithm
combinations It is assembled with God's algorithm Combination number

ABCD assembled puzzle 1
ABDC HTL 2
ACBD HTD 3
ACDB HTD HTL 4
ADBC HTL HTD 5
ADCB HTD HTL HTD 6
ADCB HTL HTD HTL 6

Source: self made.

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We observe that in the table the combination ACDB is assembled with God's algorithm with a half turn
down and a half turn left and we assemble the cylinder with God's algorithm as shown in the following
image.

Graphic 9

Front view of assembled cylinder

Source: self made.

There is no algorithm that solves the problem in fewer steps than God's algorithm; that algorithm is
impossible. For example, the ADCB combination is assembled with God's algorithm, with 2 options of
3 moves, and it is impossible to assemble it with fewer than 3 moves.

With this method, we can assemble puzzles in 3 dimensions with God's algorithm, such as the Rubik's
Cube, the Pyraminx Duo, Rubik's Cube 4x4, etc. No matter how complex the puzzle may seem, it can be
solved using the God's Algorithm with the same method we use to solve the cylinder divided into 4
divisions. So, if the cylinder is divided in half to infinity; this method serves to assemble it with God's
algorithm.

There is no algorithm that solves the problem in fewer steps than God's algorithm; that algorithm is
impossible. Continuing with the article, all the puzzles can be put together with God's algorithm. But
let's imagine that we are putting together a puzzle, for example, a Rubik's Cube. With God's algorithm,
20 moves are the optimal solution to put together that puzzle. Then, we put it together with 20 moves
and record a video with a camera putting it together with 20 moves. Then, we watch the video in reverse,
and obviously, in the video, we see that the puzzle is assembled. Afterward, we disassemble it with 20
moves. Then, we realize that the puzzle can be assembled and disassembled with the number of moves
of God's algorithm, which in this case would be 20 moves.

Now, if the cylinder has 4 divisions, we disassemble it with God's algorithm with 1 movement, to then
assemble it with God's algorithm with 1 movement. If the cylinder has 8 divisions, we disassemble it
with God's algorithm with 2 movements, to then assemble it with God's algorithm with 2 movements. If
the cylinder has 12 divisions, we disassemble it with God's algorithm with 3 movements, to then
assemble it with God's algorithm with 3 movements, and we follow this sequence to infinity.

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ISSN en línea: 2789-3855, marzo, 2025, Volumen VI, Número 1 p 891

The sequence is that the number of divisions of the cylinder increases by 4, and the number of
movements with which the cylinder will be assembled with God's algorithm increases by 1. In other
words, the number of movements with which we are going to assemble the cylinder with God's
algorithm is one-fourth of the number of divisions of the cylinder.

The cylinder divided into 4 parts has 3 factorial of total combinations, and it is going to be disassembled
with God's algorithm with 1 movement. Then, you can move parts 2 and 3 of the cylinder and assemble
it with God's algorithm, or you can disassemble it by moving parts 3 and 4 of the cylinder and then
assemble it with God's algorithm. As there are 2 options of movements to move the cylinder, we make
a formula: 2, which is the number of options of movements, raised to the power of the number of
movements with which it will be assembled. Since it will be assembled with 1 movement (because 1 is
the fourth part of 4), then it would be 2 raised to the power of 1, which is equal to 2. This is the number
of possibilities of combinations with which the cylinder will be assembled using God's algorithm.

Figure 10

Cylinder with 4 divisions

Source: self made.

Following this sequence, the cylinder divided into 8 parts has 7 factorial of total combinations and has
4 options of movements. In the first option of movement, you can move the part of the cylinder with the
numbers 2, 3, 4, and 5. In the second option, the numbers 3, 4, 5, and 6 are moved. In the third option,
the numbers 4, 5, 6, and 7 are moved. And in the fourth option, the numbers 5, 6, 7, and 8 are moved. In
the formula, 4 is placed because it has 4 options of movements, and it is going to be assembled with 2
movements because 2 is the fourth part of 8. The formula would be 4 raised to 2 = 16, which is the
number of possibilities of combinations with which the cylinder will be assembled using the God's
Algorithm. Let's look at the cylinder divided into 8 parts in the following image below.

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Figure 11

Front view of the cylinder armed with 8 divisions

Source: self made.

The cylinder, divided into 12 parts, has 11 total combinations factorial, and it has 6 options of
movements. In the formula, 6 is placed, and it will be assembled with 3 movements because 3 is the
fourth part of 12. The formula would be 6 raised to the power of 3 = 216, which is the number of
possibilities of combinations with which the cylinder will be assembled with God's algorithm, and so
on, to infinity.

Then, we realize that the number of total combinations of the cylinder is greater than the number of
possibilities of combinations with which the cylinder will be assembled with God's algorithm.

Then, we take the percentage of the number of possibilities of combinations to assemble the cylinder
divided by the number of total combinations of the cylinder. For example, a cylinder divided into 4 parts
has a total number of combinations of 3 factorial, which is equal to 6, and the number of possibilities
of combinations to assemble the cylinder with the algorithm of God is 2. So, 6, which is the number of
total combinations of the cylinder, is 100%. We want to know what percentage the 2 represents out of
6. To find this, we divide 2 by 6, which gives us 0.33. When multiplied by 100, it becomes 33%. In the
following table, we observe that the percentage for the cylinder divided into 8 is 0.317%. We can see
that the percentage decreases as the cylinder has more divisions. This trend continues to lower the
percentage toward infinity. This implies that we can assemble the cylinder with God's algorithm using
the fourth part of the number by which the cylinder is divided. This is because the number of
combinations to assemble the cylinder with God's algorithm will be less than the total number of
combinations by which the cylinder is divided.

The number of combinations with which the cylinder is going to be assembled with God's algorithm is
a gross unit, because God's algorithm is going to eliminate unnecessary combinations of unnecessary
movements, this means that the number of combinations with which it will be assembled with God's
algorithm the cylinder is a number smaller than the one in the following table, and the percentage is
also smaller than the one in the following table, but I am not doing this work to summarize this article
as much as possible and this has already been explained as do it in this same article when we made

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ISSN en línea: 2789-3855, marzo, 2025, Volumen VI, Número 1 p 893

the table of God's algorithm, the important thing is to know that the percentage is going down. The table
is divided into 2 parts, let's see the first part below.

Table 12

Percentage: of the number of possible combinations to assemble the cylinder with God's algorithm
divided by the number of total combinations of the cylinder with half of its even divisions

Row
number

Percentage: of the number of
possible combinations to

assemble the cylinder with
God's algorithm divided by the
number of total combinations
of the cylinder with half of its

even divisions

Its number of
combination

possibilities to
assemble the cylinder
with God's algorithm
with half of its even

divisions

Total number
of

combinations
of the cylinder
with half of its

even
divisions.

Number
of

divisions
of the

cylinder
with half

of its even
divisions

1 33,33333333 % 2 6 4

Table 13

Now let's look at the second part of the table below

2 0,317460317 % 16 5.040 8
3 5,41 x 10−4 % 216 39.916.800 12
4 3,13 x 10−7 % 4.096 1,31 x 1012 16
5 8,22 x 10−11 % 100.000 1,22 x 1017 20
6 1,16 x 10−14 % 2.985.984 2,59 x 1022 24
7 9,68 x 10−19 % 105.413.504 1,09 x 1028 28
8 5,22 x 10−23 % 4.294.967.296 8,22 x 1033 32
9 1,92 x 10−27 % 198.359.290.368 1,03 x 1040 36

10 5,02 x 10−32 % 10.240.000.000.000 2,04 x 1046 40

Source: self made.

ANSWERING THE MILLENIUM QUESTION: IS P EQUAL TO NP?

When we ask: Is P equal to NP? In reality, these 2 questions are being asked, which are the following.

Question number 1, if a non-deterministic Turing machine finds solutions to problems in polynomial
time, does that imply that it is also possible for a deterministic Turing machine to find solutions to those
same problems in polynomial time?

Answer, if it is possible for a deterministic Turing machine to find solutions to those same problems in
polynomial time, because the god algorithm or the optimal algorithm can be used to solve these same
problems, for example if we limit the number of moves to assemble with the God's algorithm the
cylinder for example 20 movements and if the cylinder has 80 divisions, from 80 divisions forward the
cylinder will be assembled with 20 movements with God's algorithm, it does not matter if the cylinder
has 1 million, 10 million, 1 trillion and infinities of divisions, the cylinder will always be assembled with
20 movements with God's algorithm and it would only take 20 movements or 20 seconds to assemble
the cylinder, regardless of whether the cylinder becomes infinitely more complex, it will always be
assembled with 20 movements. Let's assume that 1 movement is equal to 1 second.

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In other words, if a non-deterministic Turing machine finds solutions to problems in polynomial time, a
deterministic Turing machine will also find solutions to the same problems in polynomial time with the
optimal algorithm or with the god algorithm.

Question number 2, if it is possible to quickly "verify" solutions to problems, does that imply that it is
also possible to "find" the answers just as quickly?

Answer, in some cases it will be possible to verify the solutions to the problems in polynomial time
because what is similar to the hand of a clock will almost turn around in a minute or so, but it will not
be possible to find the solutions to the problems in polynomial time because for example if the cylinder
has 400 million divisions it must be assembled with 100 million movements with God's algorithm.
Remember that it is a quarter of the number of total divisions of the cylinder, that is why it will take 100
million seconds to assemble it because let's assume that a movement is equal to a second and the
more divisions the cylinder has, more time it will take to assemble it with God's algorithm. Let's
remember that the more divisions the cylinder has, the problem becomes more complex because the
cylinder has more options. movements, let us also remember that there is no algorithm that solves the
problem in fewer steps than God's algorithm, that algorithm is impossible. It does not matter if the
Turing machine is deterministic or non-deterministic, a non-deterministic Turing machine will not be
able to solve the problem in polynomial time because it will take 100 million seconds, assuming that 1
second is equal to 1 movement.

In other cases it will be possible to verify the solutions to the problems in polynomial time, and it will
also be possible to find the solutions to the problems in polynomial time because for example if we
limit the number of movements to assemble the cylinder with God's algorithm, for example 20
movements and if the cylinder has 80 divisions, from 80 divisions forward the cylinder will be
assembled with 20 movements with God's algorithm, it does not matter if the cylinder has 1 million, 10
million, 1 billion and infinity of divisions, the cylinder will always be assembled with 20 movements with
God's algorithm and it would only take 20 movements or 20 seconds to assemble the cylinder,
regardless of whether the cylinder becomes infinitely more complex, it will always be assembled with
20 movements.

The more divisions the cylinder has, the more complex the algorithm will become because the cylinder
will have more possibilities for movements. Let us remember that the cylinder will be assembled with
20 movements with God's algorithm and will be constant and in Big O notation, it is represented like
this O(1), the cylinder can have infinite divisions, the cylinder will always be assembled with 20
movements with God's algorithm.

Answering these 2 questions we also answered: Is P equal to NP? And we also answered: Is P equal to
NP-complete?

CONCLUSIONS

The results of this research support the idea that the complexities of the problems increase to infinity
because the more divisions the cylinder has, the more complex it will become, if we ask: Is P equal to
NP? As explained above, we are actually asking 2 questions, the answers to these 2 questions are: If a
non-deterministic Turing machine finds solutions to problems in polynomial time, that implies that it is
also possible for a deterministic Turing machine to find solutions to those same problems in polynomial
time with God's algorithm, by answering the other question it is possible to quickly verify the solutions
of the problems, but in some cases this implies that it is also possible to find the answers in polynomial
time with God's algorithm, but in In other cases, the solutions to the problems are quickly checked, but
it is not possible to find the answers in polynomial time with God's algorithm. There is no algorithm that
solves the problem in fewer steps than God's algorithm; that algorithm is impossible.

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