From 7fe7295a95bd568711528010d6f15dc21696c489 Mon Sep 17 00:00:00 2001
From: algobytewise <algobytewise@gmail.com>
Date: Sun, 28 Feb 2021 22:34:53 +0530
Subject: [PATCH 1/8] Add algorithm for N-body simulation

---
 other/n-body_simulation.py | 234 +++++++++++++++++++++++++++++++++++++
 1 file changed, 234 insertions(+)
 create mode 100644 other/n-body_simulation.py

diff --git a/other/n-body_simulation.py b/other/n-body_simulation.py
new file mode 100644
index 000000000000..e66af89d57db
--- /dev/null
+++ b/other/n-body_simulation.py
@@ -0,0 +1,234 @@
+"""
+In physics and astronomy, a gravitational N-body simulation is a simulation of a
+dynamical system of particles under the influence of gravity. The system
+consists of a number of bodies, each of which exerts a gravitational force on all
+other bodies. These forces are calculated using Newton's law of universal
+gravitation. The Euler method is used at each time-step to calculate the change in
+velocity and position brought about by these forces. Softening is used to prevent
+numerical divergences when a particle comes too close to another (and the force
+goes to infinity).
+(Description adapted from https://en.wikipedia.org/wiki/N-body_simulation )
+(See also http://www.shodor.org/refdesk/Resources/Algorithms/EulersMethod/ )
+"""
+
+
+from __future__ import annotations
+
+import random
+
+from matplotlib import animation  # type: ignore
+from matplotlib import pyplot as plt
+
+
+class Body:
+    def __init__(
+        self: Body,
+        position_x: float,
+        position_y: float,
+        velocity_x: float,
+        velocity_y: float,
+        mass: float = 1,
+        size: float = 1,
+        color: str = "blue",
+    ) -> None:
+        """
+        The parameters "size" & "color" are not relevant for the simulation itself,
+        they are only used for plotting.
+        """
+        self.position_x = position_x
+        self.position_y = position_y
+        self.velocity_x = velocity_x
+        self.velocity_y = velocity_y
+        self.mass = mass
+        self.size = size
+        self.color = color
+
+    def update_velocity(
+        self: Body, force_x: float, force_y: float, delta_time: float
+    ) -> None:
+        """
+        Euler algorithm for velocity
+        """
+        self.velocity_x += force_x * delta_time
+        self.velocity_y += force_y * delta_time
+
+    def update_position(self: Body, delta_time: float) -> None:
+        """
+        Euler algorithm for position
+        """
+        self.position_x += self.velocity_x * delta_time
+        self.position_y += self.velocity_y * delta_time
+
+
+class BodySystem:
+    """
+    This class is used to hold the bodies, the gravitation constant, the time
+    factor and the softening factor. The time factor is used to control the speed
+    of the simulation. The softening factor is used for softening, a numerical
+    trick for N-body simulations to prevent numerical divergences when two bodies
+    get too close to each other.
+    """
+
+    def __init__(
+        self: BodySystem,
+        bodies: list[Body],
+        gravitation_constant: float = 1,
+        time_factor: float = 1,
+        softening_factor: float = 0,
+    ) -> None:
+        self.bodies = bodies
+        self.gravitation_constant = gravitation_constant
+        self.time_factor = time_factor
+        self.softening_factor = softening_factor
+
+    def update_system(self: BodySystem, delta_time: float) -> None:
+        """
+        For each body, loop through all other bodies to calculate the total
+        force they exert on it. Use that force to update the body's velocity.
+        """
+        for body1 in self.bodies:
+            force_x = 0.0
+            force_y = 0.0
+            for body2 in self.bodies:
+                if body1 != body2:
+                    dif_x = body2.position_x - body1.position_x
+                    dif_y = body2.position_y - body1.position_y
+
+                    # Calculation of the distance using Pythagoras's theorem
+                    # Extra factor due to the softening technique
+                    distance = (dif_x ** 2 + dif_y ** 2 + self.softening_factor) ** (
+                        1 / 2
+                    )
+
+                    # Newton's law of universal gravitation.
+                    force_x += (
+                        self.gravitation_constant * body2.mass * dif_x / distance ** 3
+                    )
+                    force_y += (
+                        self.gravitation_constant * body2.mass * dif_y / distance ** 3
+                    )
+
+            # Update the body's velocity once all the force components have been added
+            body1.update_velocity(force_x, force_y, delta_time * self.time_factor)
+
+        # Update the positions only after all the velocities have been updated
+        for body in self.bodies:
+            body.update_position(delta_time * self.time_factor)
+
+
+def plot(
+    title: str,
+    bodySystem: BodySystem,
+    x_start: float = -1,
+    x_end: float = 1,
+    y_start: float = -1,
+    y_end: float = 1,
+) -> None:
+    INTERVAL = 20  # Frame rate of the animation
+
+    fig = plt.figure()
+    fig.canvas.set_window_title(title)
+
+    # Set section to be plotted
+    ax = plt.axes(xlim=(x_start, x_end), ylim=(y_start, y_end))
+
+    # Each body is drawn as a patch by the plt-function
+    patches = []
+    for body in bodySystem.bodies:
+        patches.append(
+            plt.Circle((body.position_x, body.position_y), body.size, fc=body.color)
+        )
+
+    # Function called once at the start of the animation
+    def init() -> list[patches.Circle]:  # type: ignore
+        axes = plt.gca()
+        axes.set_aspect("equal")
+
+        for patch in patches:
+            ax.add_patch(patch)
+        return patches
+
+    # Function called at each step of the animation
+    def animate(i: int) -> list[patches.Circle]:  # type: ignore
+        # Update the positions of the bodies
+        bodySystem.update_system(INTERVAL)
+
+        # Update the positions of the patches
+        for patch, body in zip(patches, bodySystem.bodies):
+            patch.center = (body.position_x, body.position_y)
+        return patches
+
+    anim = animation.FuncAnimation(  # noqa: F841
+        fig, animate, init_func=init, interval=INTERVAL, blit=True
+    )
+
+    plt.show()
+
+
+if __name__ == "__main__":
+    # Example 1: figure-8 solution to the 3-body-problem
+    position_x = 0.9700436
+    position_y = -0.24308753
+    velocity_x = 0.466203685
+    velocity_y = 0.43236573
+
+    bodies1 = [
+        Body(position_x, position_y, velocity_x, velocity_y, size=0.2, color="red"),
+        Body(-position_x, -position_y, velocity_x, velocity_y, size=0.2, color="green"),
+        Body(0, 0, -2 * velocity_x, -2 * velocity_y, size=0.2, color="blue"),
+    ]
+    body_system1 = BodySystem(bodies1, time_factor=0.003)
+    plot("Figure-8 solution to the 3-body-problem", body_system1, -2, 2, -2, 2)
+
+    # Example 2: Moon's orbit around the earth
+    moon_mass = 7.3476e22
+    earth_mass = 5.972e24
+    velocity_dif = 1022
+    earth_moon_distance = 384399000
+    gravitation_constant = 6.674e-11
+
+    # Calculation of the respective velocities so that total impulse is zero,
+    # i.e. the two bodies together don't move
+    moon_velocity = earth_mass * velocity_dif / (earth_mass + moon_mass)
+    earth_velocity = moon_velocity - velocity_dif
+
+    moon = Body(-earth_moon_distance, 0, 0, moon_velocity, moon_mass, 10000000, "grey")
+    earth = Body(0, 0, 0, earth_velocity, earth_mass, 50000000, "blue")
+    body_system2 = BodySystem([earth, moon], gravitation_constant, time_factor=1000)
+    plot(
+        "Moon's orbit around the earth",
+        body_system2,
+        -430000000,
+        430000000,
+        -430000000,
+        430000000,
+    )
+
+    # Example 3: Random system with many bodies
+    bodies = []
+    for i in range(10):
+        velocity_x = random.uniform(-0.5, 0.5)
+        velocity_y = random.uniform(-0.5, 0.5)
+
+        # Bodies are created pairwise with opposite velocities so that the
+        # total impulse remains zero
+        bodies.append(
+            Body(
+                random.uniform(-0.5, 0.5),
+                random.uniform(-0.5, 0.5),
+                velocity_x,
+                velocity_y,
+                size=0.05,
+            )
+        )
+        bodies.append(
+            Body(
+                random.uniform(-0.5, 0.5),
+                random.uniform(-0.5, 0.5),
+                -velocity_x,
+                -velocity_y,
+                size=0.05,
+            )
+        )
+    body_system3 = BodySystem(bodies, 0.01, 0.01, 0.1)
+    plot("Random system with many bodies", body_system3, -1.5, 1.5, -1.5, 1.5)

From 579adfca81a6b10ba19a2f89f6daffd498bee4f7 Mon Sep 17 00:00:00 2001
From: algobytewise <algobytewise@gmail.com>
Date: Mon, 1 Mar 2021 12:51:11 +0530
Subject: [PATCH 2/8] doctest & adjusted time_delta

---
 other/n-body_simulation.py | 40 ++++++++++++++++++++++++++------------
 1 file changed, 28 insertions(+), 12 deletions(-)

diff --git a/other/n-body_simulation.py b/other/n-body_simulation.py
index e66af89d57db..579febb4231f 100644
--- a/other/n-body_simulation.py
+++ b/other/n-body_simulation.py
@@ -27,8 +27,8 @@ def __init__(
         position_y: float,
         velocity_x: float,
         velocity_y: float,
-        mass: float = 1,
-        size: float = 1,
+        mass: float = 1.0,
+        size: float = 1.0,
         color: str = "blue",
     ) -> None:
         """
@@ -48,6 +48,11 @@ def update_velocity(
     ) -> None:
         """
         Euler algorithm for velocity
+
+        >>> body = Body(0.,0.,0.,0.)
+        >>> body.update_velocity(1.,0.,1.)
+        >>> body.velocity_x
+        1.0
         """
         self.velocity_x += force_x * delta_time
         self.velocity_y += force_y * delta_time
@@ -55,6 +60,11 @@ def update_velocity(
     def update_position(self: Body, delta_time: float) -> None:
         """
         Euler algorithm for position
+
+        >>> body = Body(0.,0.,1.,0.)
+        >>> body.update_position(1.)
+        >>> body.position_x
+        1.0
         """
         self.position_x += self.velocity_x * delta_time
         self.position_y += self.velocity_y * delta_time
@@ -72,9 +82,9 @@ class BodySystem:
     def __init__(
         self: BodySystem,
         bodies: list[Body],
-        gravitation_constant: float = 1,
-        time_factor: float = 1,
-        softening_factor: float = 0,
+        gravitation_constant: float = 1.0,
+        time_factor: float = 1.0,
+        softening_factor: float = 0.0,
     ) -> None:
         self.bodies = bodies
         self.gravitation_constant = gravitation_constant
@@ -85,6 +95,11 @@ def update_system(self: BodySystem, delta_time: float) -> None:
         """
         For each body, loop through all other bodies to calculate the total
         force they exert on it. Use that force to update the body's velocity.
+
+        >>> body_system = BodySystem([Body(0,0,0,0), Body(10,0,0,0)])
+        >>> body_system.update_system(1)
+        >>> body_system.bodies[0].position_x
+        0.01
         """
         for body1 in self.bodies:
             force_x = 0.0
@@ -118,13 +133,14 @@ def update_system(self: BodySystem, delta_time: float) -> None:
 
 def plot(
     title: str,
-    bodySystem: BodySystem,
+    body_system: BodySystem,
     x_start: float = -1,
     x_end: float = 1,
     y_start: float = -1,
     y_end: float = 1,
 ) -> None:
     INTERVAL = 20  # Frame rate of the animation
+    DELTA_TIME = INTERVAL / 1000  # Time between time steps in seconds
 
     fig = plt.figure()
     fig.canvas.set_window_title(title)
@@ -134,7 +150,7 @@ def plot(
 
     # Each body is drawn as a patch by the plt-function
     patches = []
-    for body in bodySystem.bodies:
+    for body in body_system.bodies:
         patches.append(
             plt.Circle((body.position_x, body.position_y), body.size, fc=body.color)
         )
@@ -151,10 +167,10 @@ def init() -> list[patches.Circle]:  # type: ignore
     # Function called at each step of the animation
     def animate(i: int) -> list[patches.Circle]:  # type: ignore
         # Update the positions of the bodies
-        bodySystem.update_system(INTERVAL)
+        body_system.update_system(DELTA_TIME)
 
         # Update the positions of the patches
-        for patch, body in zip(patches, bodySystem.bodies):
+        for patch, body in zip(patches, body_system.bodies):
             patch.center = (body.position_x, body.position_y)
         return patches
 
@@ -177,7 +193,7 @@ def animate(i: int) -> list[patches.Circle]:  # type: ignore
         Body(-position_x, -position_y, velocity_x, velocity_y, size=0.2, color="green"),
         Body(0, 0, -2 * velocity_x, -2 * velocity_y, size=0.2, color="blue"),
     ]
-    body_system1 = BodySystem(bodies1, time_factor=0.003)
+    body_system1 = BodySystem(bodies1, time_factor=3)
     plot("Figure-8 solution to the 3-body-problem", body_system1, -2, 2, -2, 2)
 
     # Example 2: Moon's orbit around the earth
@@ -194,7 +210,7 @@ def animate(i: int) -> list[patches.Circle]:  # type: ignore
 
     moon = Body(-earth_moon_distance, 0, 0, moon_velocity, moon_mass, 10000000, "grey")
     earth = Body(0, 0, 0, earth_velocity, earth_mass, 50000000, "blue")
-    body_system2 = BodySystem([earth, moon], gravitation_constant, time_factor=1000)
+    body_system2 = BodySystem([earth, moon], gravitation_constant, time_factor=1000000)
     plot(
         "Moon's orbit around the earth",
         body_system2,
@@ -230,5 +246,5 @@ def animate(i: int) -> list[patches.Circle]:  # type: ignore
                 size=0.05,
             )
         )
-    body_system3 = BodySystem(bodies, 0.01, 0.01, 0.1)
+    body_system3 = BodySystem(bodies, 0.01, 10, 0.1)
     plot("Random system with many bodies", body_system3, -1.5, 1.5, -1.5, 1.5)

From fd96cad7d0c440865c35c995445164f4f80d266b Mon Sep 17 00:00:00 2001
From: algobytewise <algobytewise@gmail.com>
Date: Mon, 1 Mar 2021 12:54:53 +0530
Subject: [PATCH 3/8] Rename n-body_simulation.py to n_body_simulation.py

---
 other/{n-body_simulation.py => n_body_simulation.py} | 0
 1 file changed, 0 insertions(+), 0 deletions(-)
 rename other/{n-body_simulation.py => n_body_simulation.py} (100%)

diff --git a/other/n-body_simulation.py b/other/n_body_simulation.py
similarity index 100%
rename from other/n-body_simulation.py
rename to other/n_body_simulation.py

From e7809db79a884535fe1e4760fcf6d117d7bfc444 Mon Sep 17 00:00:00 2001
From: algobytewise <algobytewise@gmail.com>
Date: Mon, 1 Mar 2021 13:06:19 +0530
Subject: [PATCH 4/8] reupload


From d14de5a0810d8ea46ca91e51e26081b81a0b183d Mon Sep 17 00:00:00 2001
From: algobytewise <algobytewise@gmail.com>
Date: Sun, 28 Mar 2021 10:24:15 +0530
Subject: [PATCH 5/8] update n_body_simulation.py

---
 other/n_body_simulation.py | 18 +++++++++++++++---
 1 file changed, 15 insertions(+), 3 deletions(-)

diff --git a/other/n_body_simulation.py b/other/n_body_simulation.py
index 579febb4231f..b7aaaee62b9b 100644
--- a/other/n_body_simulation.py
+++ b/other/n_body_simulation.py
@@ -139,6 +139,11 @@ def plot(
     y_start: float = -1,
     y_end: float = 1,
 ) -> None:
+    """
+    Utility function to plot how the given body-system evolves over time.
+    No doctest provided since this function does not have a return value.
+    """
+
     INTERVAL = 20  # Frame rate of the animation
     DELTA_TIME = INTERVAL / 1000  # Time between time steps in seconds
 
@@ -156,7 +161,7 @@ def plot(
         )
 
     # Function called once at the start of the animation
-    def init() -> list[patches.Circle]:  # type: ignore
+    def init() -> list[patches.Circle]:
         axes = plt.gca()
         axes.set_aspect("equal")
 
@@ -165,7 +170,7 @@ def init() -> list[patches.Circle]:  # type: ignore
         return patches
 
     # Function called at each step of the animation
-    def animate(i: int) -> list[patches.Circle]:  # type: ignore
+    def update(frame: int) -> list[patches.Circle]:
         # Update the positions of the bodies
         body_system.update_system(DELTA_TIME)
 
@@ -175,7 +180,7 @@ def animate(i: int) -> list[patches.Circle]:  # type: ignore
         return patches
 
     anim = animation.FuncAnimation(  # noqa: F841
-        fig, animate, init_func=init, interval=INTERVAL, blit=True
+        fig, update, init_func=init, interval=INTERVAL, blit=True
     )
 
     plt.show()
@@ -183,6 +188,9 @@ def animate(i: int) -> list[patches.Circle]:  # type: ignore
 
 if __name__ == "__main__":
     # Example 1: figure-8 solution to the 3-body-problem
+    # This example can be seen as a test of the implementation: given the right
+    # initial conditions, the bodies should move in a figure-8.
+    # (initial conditions taken from http://www.artcompsci.org/vol_1/v1_web/node56.html)
     position_x = 0.9700436
     position_y = -0.24308753
     velocity_x = 0.466203685
@@ -197,6 +205,10 @@ def animate(i: int) -> list[patches.Circle]:  # type: ignore
     plot("Figure-8 solution to the 3-body-problem", body_system1, -2, 2, -2, 2)
 
     # Example 2: Moon's orbit around the earth
+    # This example can be seen as a test of the implementation: given the right
+    # initial conditions, the moon should orbit around the earth as it actually does.
+    # (mass, velocity and distance taken from https://en.wikipedia.org/wiki/Earth
+    # and https://en.wikipedia.org/wiki/Moon)
     moon_mass = 7.3476e22
     earth_mass = 5.972e24
     velocity_dif = 1022

From 8b5aa24d7c744fc195ccf5a34f92f8e1aa47cc69 Mon Sep 17 00:00:00 2001
From: github-actions <${GITHUB_ACTOR}@users.noreply.github.com>
Date: Sun, 28 Mar 2021 04:55:13 +0000
Subject: [PATCH 6/8] updating DIRECTORY.md

---
 DIRECTORY.md | 10 ++++++++++
 1 file changed, 10 insertions(+)

diff --git a/DIRECTORY.md b/DIRECTORY.md
index d487b39490ed..fc764e14f564 100644
--- a/DIRECTORY.md
+++ b/DIRECTORY.md
@@ -29,7 +29,11 @@
   * [Binary Count Setbits](https://github.com/TheAlgorithms/Python/blob/master/bit_manipulation/binary_count_setbits.py)
   * [Binary Count Trailing Zeros](https://github.com/TheAlgorithms/Python/blob/master/bit_manipulation/binary_count_trailing_zeros.py)
   * [Binary Or Operator](https://github.com/TheAlgorithms/Python/blob/master/bit_manipulation/binary_or_operator.py)
+  * [Binary Shifts](https://github.com/TheAlgorithms/Python/blob/master/bit_manipulation/binary_shifts.py)
+  * [Binary Twos Complement](https://github.com/TheAlgorithms/Python/blob/master/bit_manipulation/binary_twos_complement.py)
   * [Binary Xor Operator](https://github.com/TheAlgorithms/Python/blob/master/bit_manipulation/binary_xor_operator.py)
+  * [Count Number Of One Bits](https://github.com/TheAlgorithms/Python/blob/master/bit_manipulation/count_number_of_one_bits.py)
+  * [Reverse Bits](https://github.com/TheAlgorithms/Python/blob/master/bit_manipulation/reverse_bits.py)
   * [Single Bit Manipulation Operations](https://github.com/TheAlgorithms/Python/blob/master/bit_manipulation/single_bit_manipulation_operations.py)
 
 ## Blockchain
@@ -122,6 +126,7 @@
     * [Fenwick Tree](https://github.com/TheAlgorithms/Python/blob/master/data_structures/binary_tree/fenwick_tree.py)
     * [Lazy Segment Tree](https://github.com/TheAlgorithms/Python/blob/master/data_structures/binary_tree/lazy_segment_tree.py)
     * [Lowest Common Ancestor](https://github.com/TheAlgorithms/Python/blob/master/data_structures/binary_tree/lowest_common_ancestor.py)
+    * [Merge Two Binary Trees](https://github.com/TheAlgorithms/Python/blob/master/data_structures/binary_tree/merge_two_binary_trees.py)
     * [Non Recursive Segment Tree](https://github.com/TheAlgorithms/Python/blob/master/data_structures/binary_tree/non_recursive_segment_tree.py)
     * [Number Of Possible Binary Trees](https://github.com/TheAlgorithms/Python/blob/master/data_structures/binary_tree/number_of_possible_binary_trees.py)
     * [Red Black Tree](https://github.com/TheAlgorithms/Python/blob/master/data_structures/binary_tree/red_black_tree.py)
@@ -274,6 +279,8 @@
 
 ## Graphics
   * [Bezier Curve](https://github.com/TheAlgorithms/Python/blob/master/graphics/bezier_curve.py)
+  * [Koch Snowflake](https://github.com/TheAlgorithms/Python/blob/master/graphics/koch_snowflake.py)
+  * [Mandelbrot](https://github.com/TheAlgorithms/Python/blob/master/graphics/mandelbrot.py)
   * [Vector3 For 2D Rendering](https://github.com/TheAlgorithms/Python/blob/master/graphics/vector3_for_2d_rendering.py)
 
 ## Graphs
@@ -531,6 +538,7 @@
   * [Markov Chain](https://github.com/TheAlgorithms/Python/blob/master/other/markov_chain.py)
   * [Max Sum Sliding Window](https://github.com/TheAlgorithms/Python/blob/master/other/max_sum_sliding_window.py)
   * [Median Of Two Arrays](https://github.com/TheAlgorithms/Python/blob/master/other/median_of_two_arrays.py)
+  * [N Body Simulation](https://github.com/TheAlgorithms/Python/blob/master/other/n_body_simulation.py)
   * [Nested Brackets](https://github.com/TheAlgorithms/Python/blob/master/other/nested_brackets.py)
   * [Palindrome](https://github.com/TheAlgorithms/Python/blob/master/other/palindrome.py)
   * [Password Generator](https://github.com/TheAlgorithms/Python/blob/master/other/password_generator.py)
@@ -840,6 +848,7 @@
   * [Merge Insertion Sort](https://github.com/TheAlgorithms/Python/blob/master/sorts/merge_insertion_sort.py)
   * [Merge Sort](https://github.com/TheAlgorithms/Python/blob/master/sorts/merge_sort.py)
   * [Natural Sort](https://github.com/TheAlgorithms/Python/blob/master/sorts/natural_sort.py)
+  * [Odd Even Sort](https://github.com/TheAlgorithms/Python/blob/master/sorts/odd_even_sort.py)
   * [Odd Even Transposition Parallel](https://github.com/TheAlgorithms/Python/blob/master/sorts/odd_even_transposition_parallel.py)
   * [Odd Even Transposition Single Threaded](https://github.com/TheAlgorithms/Python/blob/master/sorts/odd_even_transposition_single_threaded.py)
   * [Pancake Sort](https://github.com/TheAlgorithms/Python/blob/master/sorts/pancake_sort.py)
@@ -856,6 +865,7 @@
   * [Recursive Quick Sort](https://github.com/TheAlgorithms/Python/blob/master/sorts/recursive_quick_sort.py)
   * [Selection Sort](https://github.com/TheAlgorithms/Python/blob/master/sorts/selection_sort.py)
   * [Shell Sort](https://github.com/TheAlgorithms/Python/blob/master/sorts/shell_sort.py)
+  * [Slowsort](https://github.com/TheAlgorithms/Python/blob/master/sorts/slowsort.py)
   * [Stooge Sort](https://github.com/TheAlgorithms/Python/blob/master/sorts/stooge_sort.py)
   * [Strand Sort](https://github.com/TheAlgorithms/Python/blob/master/sorts/strand_sort.py)
   * [Tim Sort](https://github.com/TheAlgorithms/Python/blob/master/sorts/tim_sort.py)

From 9809124d0ced342711eb4e1e44034db3b8d1f080 Mon Sep 17 00:00:00 2001
From: algobytewise <algobytewise@gmail.com>
Date: Sun, 28 Mar 2021 10:27:52 +0530
Subject: [PATCH 7/8] update DIRECTORY.md

---
 DIRECTORY.md | 61 +++++++++++++++++++++++++++++-----------------------
 1 file changed, 34 insertions(+), 27 deletions(-)

diff --git a/DIRECTORY.md b/DIRECTORY.md
index fc764e14f564..42a6c49c735f 100644
--- a/DIRECTORY.md
+++ b/DIRECTORY.md
@@ -46,6 +46,7 @@
 
 ## Cellular Automata
   * [Conways Game Of Life](https://github.com/TheAlgorithms/Python/blob/master/cellular_automata/conways_game_of_life.py)
+  * [Game Of Life](https://github.com/TheAlgorithms/Python/blob/master/cellular_automata/game_of_life.py)
   * [One Dimensional](https://github.com/TheAlgorithms/Python/blob/master/cellular_automata/one_dimensional.py)
 
 ## Ciphers
@@ -267,6 +268,11 @@
   * Tests
     * [Test Send File](https://github.com/TheAlgorithms/Python/blob/master/file_transfer/tests/test_send_file.py)
 
+## Fractals
+  * [Koch Snowflake](https://github.com/TheAlgorithms/Python/blob/master/fractals/koch_snowflake.py)
+  * [Mandelbrot](https://github.com/TheAlgorithms/Python/blob/master/fractals/mandelbrot.py)
+  * [Sierpinski Triangle](https://github.com/TheAlgorithms/Python/blob/master/fractals/sierpinski_triangle.py)
+
 ## Fuzzy Logic
   * [Fuzzy Operations](https://github.com/TheAlgorithms/Python/blob/master/fuzzy_logic/fuzzy_operations.py)
 
@@ -279,8 +285,6 @@
 
 ## Graphics
   * [Bezier Curve](https://github.com/TheAlgorithms/Python/blob/master/graphics/bezier_curve.py)
-  * [Koch Snowflake](https://github.com/TheAlgorithms/Python/blob/master/graphics/koch_snowflake.py)
-  * [Mandelbrot](https://github.com/TheAlgorithms/Python/blob/master/graphics/mandelbrot.py)
   * [Vector3 For 2D Rendering](https://github.com/TheAlgorithms/Python/blob/master/graphics/vector3_for_2d_rendering.py)
 
 ## Graphs
@@ -319,6 +323,7 @@
   * [Kahns Algorithm Long](https://github.com/TheAlgorithms/Python/blob/master/graphs/kahns_algorithm_long.py)
   * [Kahns Algorithm Topo](https://github.com/TheAlgorithms/Python/blob/master/graphs/kahns_algorithm_topo.py)
   * [Karger](https://github.com/TheAlgorithms/Python/blob/master/graphs/karger.py)
+  * [Markov Chain](https://github.com/TheAlgorithms/Python/blob/master/graphs/markov_chain.py)
   * [Minimum Spanning Tree Boruvka](https://github.com/TheAlgorithms/Python/blob/master/graphs/minimum_spanning_tree_boruvka.py)
   * [Minimum Spanning Tree Kruskal](https://github.com/TheAlgorithms/Python/blob/master/graphs/minimum_spanning_tree_kruskal.py)
   * [Minimum Spanning Tree Kruskal2](https://github.com/TheAlgorithms/Python/blob/master/graphs/minimum_spanning_tree_kruskal2.py)
@@ -404,6 +409,8 @@
   * [Basic Maths](https://github.com/TheAlgorithms/Python/blob/master/maths/basic_maths.py)
   * [Binary Exp Mod](https://github.com/TheAlgorithms/Python/blob/master/maths/binary_exp_mod.py)
   * [Binary Exponentiation](https://github.com/TheAlgorithms/Python/blob/master/maths/binary_exponentiation.py)
+  * [Binary Exponentiation 2](https://github.com/TheAlgorithms/Python/blob/master/maths/binary_exponentiation_2.py)
+  * [Binary Exponentiation 3](https://github.com/TheAlgorithms/Python/blob/master/maths/binary_exponentiation_3.py)
   * [Binomial Coefficient](https://github.com/TheAlgorithms/Python/blob/master/maths/binomial_coefficient.py)
   * [Binomial Distribution](https://github.com/TheAlgorithms/Python/blob/master/maths/binomial_distribution.py)
   * [Bisection](https://github.com/TheAlgorithms/Python/blob/master/maths/bisection.py)
@@ -414,8 +421,9 @@
   * [Decimal Isolate](https://github.com/TheAlgorithms/Python/blob/master/maths/decimal_isolate.py)
   * [Entropy](https://github.com/TheAlgorithms/Python/blob/master/maths/entropy.py)
   * [Euclidean Distance](https://github.com/TheAlgorithms/Python/blob/master/maths/euclidean_distance.py)
+  * [Euclidean Gcd](https://github.com/TheAlgorithms/Python/blob/master/maths/euclidean_gcd.py)
+  * [Euler Method](https://github.com/TheAlgorithms/Python/blob/master/maths/euler_method.py)
   * [Eulers Totient](https://github.com/TheAlgorithms/Python/blob/master/maths/eulers_totient.py)
-  * [Explicit Euler](https://github.com/TheAlgorithms/Python/blob/master/maths/explicit_euler.py)
   * [Extended Euclidean Algorithm](https://github.com/TheAlgorithms/Python/blob/master/maths/extended_euclidean_algorithm.py)
   * [Factorial Iterative](https://github.com/TheAlgorithms/Python/blob/master/maths/factorial_iterative.py)
   * [Factorial Python](https://github.com/TheAlgorithms/Python/blob/master/maths/factorial_python.py)
@@ -432,7 +440,9 @@
   * [Gamma](https://github.com/TheAlgorithms/Python/blob/master/maths/gamma.py)
   * [Gaussian](https://github.com/TheAlgorithms/Python/blob/master/maths/gaussian.py)
   * [Greatest Common Divisor](https://github.com/TheAlgorithms/Python/blob/master/maths/greatest_common_divisor.py)
+  * [Greedy Coin Change](https://github.com/TheAlgorithms/Python/blob/master/maths/greedy_coin_change.py)
   * [Hardy Ramanujanalgo](https://github.com/TheAlgorithms/Python/blob/master/maths/hardy_ramanujanalgo.py)
+  * [Integration By Simpson Approx](https://github.com/TheAlgorithms/Python/blob/master/maths/integration_by_simpson_approx.py)
   * [Is Square Free](https://github.com/TheAlgorithms/Python/blob/master/maths/is_square_free.py)
   * [Jaccard Similarity](https://github.com/TheAlgorithms/Python/blob/master/maths/jaccard_similarity.py)
   * [Kadanes](https://github.com/TheAlgorithms/Python/blob/master/maths/kadanes.py)
@@ -440,11 +450,14 @@
   * [Krishnamurthy Number](https://github.com/TheAlgorithms/Python/blob/master/maths/krishnamurthy_number.py)
   * [Kth Lexicographic Permutation](https://github.com/TheAlgorithms/Python/blob/master/maths/kth_lexicographic_permutation.py)
   * [Largest Of Very Large Numbers](https://github.com/TheAlgorithms/Python/blob/master/maths/largest_of_very_large_numbers.py)
+  * [Largest Subarray Sum](https://github.com/TheAlgorithms/Python/blob/master/maths/largest_subarray_sum.py)
   * [Least Common Multiple](https://github.com/TheAlgorithms/Python/blob/master/maths/least_common_multiple.py)
   * [Line Length](https://github.com/TheAlgorithms/Python/blob/master/maths/line_length.py)
   * [Lucas Lehmer Primality Test](https://github.com/TheAlgorithms/Python/blob/master/maths/lucas_lehmer_primality_test.py)
   * [Lucas Series](https://github.com/TheAlgorithms/Python/blob/master/maths/lucas_series.py)
   * [Matrix Exponentiation](https://github.com/TheAlgorithms/Python/blob/master/maths/matrix_exponentiation.py)
+  * [Max Sum Sliding Window](https://github.com/TheAlgorithms/Python/blob/master/maths/max_sum_sliding_window.py)
+  * [Median Of Two Arrays](https://github.com/TheAlgorithms/Python/blob/master/maths/median_of_two_arrays.py)
   * [Miller Rabin](https://github.com/TheAlgorithms/Python/blob/master/maths/miller_rabin.py)
   * [Mobius Function](https://github.com/TheAlgorithms/Python/blob/master/maths/mobius_function.py)
   * [Modular Exponential](https://github.com/TheAlgorithms/Python/blob/master/maths/modular_exponential.py)
@@ -463,6 +476,7 @@
   * [Prime Factors](https://github.com/TheAlgorithms/Python/blob/master/maths/prime_factors.py)
   * [Prime Numbers](https://github.com/TheAlgorithms/Python/blob/master/maths/prime_numbers.py)
   * [Prime Sieve Eratosthenes](https://github.com/TheAlgorithms/Python/blob/master/maths/prime_sieve_eratosthenes.py)
+  * [Primelib](https://github.com/TheAlgorithms/Python/blob/master/maths/primelib.py)
   * [Pythagoras](https://github.com/TheAlgorithms/Python/blob/master/maths/pythagoras.py)
   * [Qr Decomposition](https://github.com/TheAlgorithms/Python/blob/master/maths/qr_decomposition.py)
   * [Quadratic Equations Complex Numbers](https://github.com/TheAlgorithms/Python/blob/master/maths/quadratic_equations_complex_numbers.py)
@@ -472,6 +486,8 @@
   * [Runge Kutta](https://github.com/TheAlgorithms/Python/blob/master/maths/runge_kutta.py)
   * [Segmented Sieve](https://github.com/TheAlgorithms/Python/blob/master/maths/segmented_sieve.py)
   * Series
+    * [Arithmetic Mean](https://github.com/TheAlgorithms/Python/blob/master/maths/series/arithmetic_mean.py)
+    * [Geometric Mean](https://github.com/TheAlgorithms/Python/blob/master/maths/series/geometric_mean.py)
     * [Geometric Series](https://github.com/TheAlgorithms/Python/blob/master/maths/series/geometric_series.py)
     * [Harmonic Series](https://github.com/TheAlgorithms/Python/blob/master/maths/series/harmonic_series.py)
     * [P Series](https://github.com/TheAlgorithms/Python/blob/master/maths/series/p_series.py)
@@ -485,6 +501,9 @@
   * [Sum Of Geometric Progression](https://github.com/TheAlgorithms/Python/blob/master/maths/sum_of_geometric_progression.py)
   * [Test Prime Check](https://github.com/TheAlgorithms/Python/blob/master/maths/test_prime_check.py)
   * [Trapezoidal Rule](https://github.com/TheAlgorithms/Python/blob/master/maths/trapezoidal_rule.py)
+  * [Triplet Sum](https://github.com/TheAlgorithms/Python/blob/master/maths/triplet_sum.py)
+  * [Two Pointer](https://github.com/TheAlgorithms/Python/blob/master/maths/two_pointer.py)
+  * [Two Sum](https://github.com/TheAlgorithms/Python/blob/master/maths/two_sum.py)
   * [Ugly Numbers](https://github.com/TheAlgorithms/Python/blob/master/maths/ugly_numbers.py)
   * [Volume](https://github.com/TheAlgorithms/Python/blob/master/maths/volume.py)
   * [Zellers Congruence](https://github.com/TheAlgorithms/Python/blob/master/maths/zellers_congruence.py)
@@ -514,43 +533,23 @@
 
 ## Other
   * [Activity Selection](https://github.com/TheAlgorithms/Python/blob/master/other/activity_selection.py)
-  * [Anagrams](https://github.com/TheAlgorithms/Python/blob/master/other/anagrams.py)
-  * [Autocomplete Using Trie](https://github.com/TheAlgorithms/Python/blob/master/other/autocomplete_using_trie.py)
-  * [Binary Exponentiation](https://github.com/TheAlgorithms/Python/blob/master/other/binary_exponentiation.py)
-  * [Binary Exponentiation 2](https://github.com/TheAlgorithms/Python/blob/master/other/binary_exponentiation_2.py)
   * [Davis–Putnam–Logemann–Loveland](https://github.com/TheAlgorithms/Python/blob/master/other/davis–putnam–logemann–loveland.py)
-  * [Detecting English Programmatically](https://github.com/TheAlgorithms/Python/blob/master/other/detecting_english_programmatically.py)
   * [Dijkstra Bankers Algorithm](https://github.com/TheAlgorithms/Python/blob/master/other/dijkstra_bankers_algorithm.py)
   * [Doomsday](https://github.com/TheAlgorithms/Python/blob/master/other/doomsday.py)
-  * [Euclidean Gcd](https://github.com/TheAlgorithms/Python/blob/master/other/euclidean_gcd.py)
   * [Fischer Yates Shuffle](https://github.com/TheAlgorithms/Python/blob/master/other/fischer_yates_shuffle.py)
-  * [Frequency Finder](https://github.com/TheAlgorithms/Python/blob/master/other/frequency_finder.py)
-  * [Game Of Life](https://github.com/TheAlgorithms/Python/blob/master/other/game_of_life.py)
   * [Gauss Easter](https://github.com/TheAlgorithms/Python/blob/master/other/gauss_easter.py)
+  * [Graham Scan](https://github.com/TheAlgorithms/Python/blob/master/other/graham_scan.py)
   * [Greedy](https://github.com/TheAlgorithms/Python/blob/master/other/greedy.py)
-  * [Integeration By Simpson Approx](https://github.com/TheAlgorithms/Python/blob/master/other/integeration_by_simpson_approx.py)
-  * [Largest Subarray Sum](https://github.com/TheAlgorithms/Python/blob/master/other/largest_subarray_sum.py)
   * [Least Recently Used](https://github.com/TheAlgorithms/Python/blob/master/other/least_recently_used.py)
   * [Lfu Cache](https://github.com/TheAlgorithms/Python/blob/master/other/lfu_cache.py)
   * [Linear Congruential Generator](https://github.com/TheAlgorithms/Python/blob/master/other/linear_congruential_generator.py)
   * [Lru Cache](https://github.com/TheAlgorithms/Python/blob/master/other/lru_cache.py)
   * [Magicdiamondpattern](https://github.com/TheAlgorithms/Python/blob/master/other/magicdiamondpattern.py)
-  * [Markov Chain](https://github.com/TheAlgorithms/Python/blob/master/other/markov_chain.py)
-  * [Max Sum Sliding Window](https://github.com/TheAlgorithms/Python/blob/master/other/max_sum_sliding_window.py)
-  * [Median Of Two Arrays](https://github.com/TheAlgorithms/Python/blob/master/other/median_of_two_arrays.py)
-  * [N Body Simulation](https://github.com/TheAlgorithms/Python/blob/master/other/n_body_simulation.py)
   * [Nested Brackets](https://github.com/TheAlgorithms/Python/blob/master/other/nested_brackets.py)
-  * [Palindrome](https://github.com/TheAlgorithms/Python/blob/master/other/palindrome.py)
   * [Password Generator](https://github.com/TheAlgorithms/Python/blob/master/other/password_generator.py)
-  * [Primelib](https://github.com/TheAlgorithms/Python/blob/master/other/primelib.py)
   * [Scoring Algorithm](https://github.com/TheAlgorithms/Python/blob/master/other/scoring_algorithm.py)
   * [Sdes](https://github.com/TheAlgorithms/Python/blob/master/other/sdes.py)
-  * [Sierpinski Triangle](https://github.com/TheAlgorithms/Python/blob/master/other/sierpinski_triangle.py)
   * [Tower Of Hanoi](https://github.com/TheAlgorithms/Python/blob/master/other/tower_of_hanoi.py)
-  * [Triplet Sum](https://github.com/TheAlgorithms/Python/blob/master/other/triplet_sum.py)
-  * [Two Pointer](https://github.com/TheAlgorithms/Python/blob/master/other/two_pointer.py)
-  * [Two Sum](https://github.com/TheAlgorithms/Python/blob/master/other/two_sum.py)
-  * [Word Patterns](https://github.com/TheAlgorithms/Python/blob/master/other/word_patterns.py)
 
 ## Project Euler
   * Problem 001
@@ -760,6 +759,8 @@
     * [Sol1](https://github.com/TheAlgorithms/Python/blob/master/project_euler/problem_102/sol1.py)
   * Problem 107
     * [Sol1](https://github.com/TheAlgorithms/Python/blob/master/project_euler/problem_107/sol1.py)
+  * Problem 109
+    * [Sol1](https://github.com/TheAlgorithms/Python/blob/master/project_euler/problem_109/sol1.py)
   * Problem 112
     * [Sol1](https://github.com/TheAlgorithms/Python/blob/master/project_euler/problem_112/sol1.py)
   * Problem 113
@@ -768,6 +769,8 @@
     * [Sol1](https://github.com/TheAlgorithms/Python/blob/master/project_euler/problem_119/sol1.py)
   * Problem 120
     * [Sol1](https://github.com/TheAlgorithms/Python/blob/master/project_euler/problem_120/sol1.py)
+  * Problem 121
+    * [Sol1](https://github.com/TheAlgorithms/Python/blob/master/project_euler/problem_121/sol1.py)
   * Problem 123
     * [Sol1](https://github.com/TheAlgorithms/Python/blob/master/project_euler/problem_123/sol1.py)
   * Problem 125
@@ -814,6 +817,7 @@
 
 ## Searches
   * [Binary Search](https://github.com/TheAlgorithms/Python/blob/master/searches/binary_search.py)
+  * [Binary Tree Traversal](https://github.com/TheAlgorithms/Python/blob/master/searches/binary_tree_traversal.py)
   * [Double Linear Search](https://github.com/TheAlgorithms/Python/blob/master/searches/double_linear_search.py)
   * [Double Linear Search Recursion](https://github.com/TheAlgorithms/Python/blob/master/searches/double_linear_search_recursion.py)
   * [Fibonacci Search](https://github.com/TheAlgorithms/Python/blob/master/searches/fibonacci_search.py)
@@ -876,11 +880,15 @@
 
 ## Strings
   * [Aho Corasick](https://github.com/TheAlgorithms/Python/blob/master/strings/aho_corasick.py)
+  * [Anagrams](https://github.com/TheAlgorithms/Python/blob/master/strings/anagrams.py)
+  * [Autocomplete Using Trie](https://github.com/TheAlgorithms/Python/blob/master/strings/autocomplete_using_trie.py)
   * [Boyer Moore Search](https://github.com/TheAlgorithms/Python/blob/master/strings/boyer_moore_search.py)
   * [Can String Be Rearranged As Palindrome](https://github.com/TheAlgorithms/Python/blob/master/strings/can_string_be_rearranged_as_palindrome.py)
   * [Capitalize](https://github.com/TheAlgorithms/Python/blob/master/strings/capitalize.py)
   * [Check Anagrams](https://github.com/TheAlgorithms/Python/blob/master/strings/check_anagrams.py)
   * [Check Pangram](https://github.com/TheAlgorithms/Python/blob/master/strings/check_pangram.py)
+  * [Detecting English Programmatically](https://github.com/TheAlgorithms/Python/blob/master/strings/detecting_english_programmatically.py)
+  * [Frequency Finder](https://github.com/TheAlgorithms/Python/blob/master/strings/frequency_finder.py)
   * [Is Palindrome](https://github.com/TheAlgorithms/Python/blob/master/strings/is_palindrome.py)
   * [Jaro Winkler](https://github.com/TheAlgorithms/Python/blob/master/strings/jaro_winkler.py)
   * [Knuth Morris Pratt](https://github.com/TheAlgorithms/Python/blob/master/strings/knuth_morris_pratt.py)
@@ -889,6 +897,7 @@
   * [Manacher](https://github.com/TheAlgorithms/Python/blob/master/strings/manacher.py)
   * [Min Cost String Conversion](https://github.com/TheAlgorithms/Python/blob/master/strings/min_cost_string_conversion.py)
   * [Naive String Search](https://github.com/TheAlgorithms/Python/blob/master/strings/naive_string_search.py)
+  * [Palindrome](https://github.com/TheAlgorithms/Python/blob/master/strings/palindrome.py)
   * [Prefix Function](https://github.com/TheAlgorithms/Python/blob/master/strings/prefix_function.py)
   * [Rabin Karp](https://github.com/TheAlgorithms/Python/blob/master/strings/rabin_karp.py)
   * [Remove Duplicate](https://github.com/TheAlgorithms/Python/blob/master/strings/remove_duplicate.py)
@@ -898,11 +907,9 @@
   * [Swap Case](https://github.com/TheAlgorithms/Python/blob/master/strings/swap_case.py)
   * [Upper](https://github.com/TheAlgorithms/Python/blob/master/strings/upper.py)
   * [Word Occurrence](https://github.com/TheAlgorithms/Python/blob/master/strings/word_occurrence.py)
+  * [Word Patterns](https://github.com/TheAlgorithms/Python/blob/master/strings/word_patterns.py)
   * [Z Function](https://github.com/TheAlgorithms/Python/blob/master/strings/z_function.py)
 
-## Traversals
-  * [Binary Tree Traversals](https://github.com/TheAlgorithms/Python/blob/master/traversals/binary_tree_traversals.py)
-
 ## Web Programming
   * [Co2 Emission](https://github.com/TheAlgorithms/Python/blob/master/web_programming/co2_emission.py)
   * [Covid Stats Via Xpath](https://github.com/TheAlgorithms/Python/blob/master/web_programming/covid_stats_via_xpath.py)

From bb25fdfab3f873e0153a1fb9682593c6ac6f943a Mon Sep 17 00:00:00 2001
From: github-actions <${GITHUB_ACTOR}@users.noreply.github.com>
Date: Sun, 28 Mar 2021 04:58:16 +0000
Subject: [PATCH 8/8] updating DIRECTORY.md

---
 DIRECTORY.md | 61 +++++++++++++++++++++++-----------------------------
 1 file changed, 27 insertions(+), 34 deletions(-)

diff --git a/DIRECTORY.md b/DIRECTORY.md
index 42a6c49c735f..fc764e14f564 100644
--- a/DIRECTORY.md
+++ b/DIRECTORY.md
@@ -46,7 +46,6 @@
 
 ## Cellular Automata
   * [Conways Game Of Life](https://github.com/TheAlgorithms/Python/blob/master/cellular_automata/conways_game_of_life.py)
-  * [Game Of Life](https://github.com/TheAlgorithms/Python/blob/master/cellular_automata/game_of_life.py)
   * [One Dimensional](https://github.com/TheAlgorithms/Python/blob/master/cellular_automata/one_dimensional.py)
 
 ## Ciphers
@@ -268,11 +267,6 @@
   * Tests
     * [Test Send File](https://github.com/TheAlgorithms/Python/blob/master/file_transfer/tests/test_send_file.py)
 
-## Fractals
-  * [Koch Snowflake](https://github.com/TheAlgorithms/Python/blob/master/fractals/koch_snowflake.py)
-  * [Mandelbrot](https://github.com/TheAlgorithms/Python/blob/master/fractals/mandelbrot.py)
-  * [Sierpinski Triangle](https://github.com/TheAlgorithms/Python/blob/master/fractals/sierpinski_triangle.py)
-
 ## Fuzzy Logic
   * [Fuzzy Operations](https://github.com/TheAlgorithms/Python/blob/master/fuzzy_logic/fuzzy_operations.py)
 
@@ -285,6 +279,8 @@
 
 ## Graphics
   * [Bezier Curve](https://github.com/TheAlgorithms/Python/blob/master/graphics/bezier_curve.py)
+  * [Koch Snowflake](https://github.com/TheAlgorithms/Python/blob/master/graphics/koch_snowflake.py)
+  * [Mandelbrot](https://github.com/TheAlgorithms/Python/blob/master/graphics/mandelbrot.py)
   * [Vector3 For 2D Rendering](https://github.com/TheAlgorithms/Python/blob/master/graphics/vector3_for_2d_rendering.py)
 
 ## Graphs
@@ -323,7 +319,6 @@
   * [Kahns Algorithm Long](https://github.com/TheAlgorithms/Python/blob/master/graphs/kahns_algorithm_long.py)
   * [Kahns Algorithm Topo](https://github.com/TheAlgorithms/Python/blob/master/graphs/kahns_algorithm_topo.py)
   * [Karger](https://github.com/TheAlgorithms/Python/blob/master/graphs/karger.py)
-  * [Markov Chain](https://github.com/TheAlgorithms/Python/blob/master/graphs/markov_chain.py)
   * [Minimum Spanning Tree Boruvka](https://github.com/TheAlgorithms/Python/blob/master/graphs/minimum_spanning_tree_boruvka.py)
   * [Minimum Spanning Tree Kruskal](https://github.com/TheAlgorithms/Python/blob/master/graphs/minimum_spanning_tree_kruskal.py)
   * [Minimum Spanning Tree Kruskal2](https://github.com/TheAlgorithms/Python/blob/master/graphs/minimum_spanning_tree_kruskal2.py)
@@ -409,8 +404,6 @@
   * [Basic Maths](https://github.com/TheAlgorithms/Python/blob/master/maths/basic_maths.py)
   * [Binary Exp Mod](https://github.com/TheAlgorithms/Python/blob/master/maths/binary_exp_mod.py)
   * [Binary Exponentiation](https://github.com/TheAlgorithms/Python/blob/master/maths/binary_exponentiation.py)
-  * [Binary Exponentiation 2](https://github.com/TheAlgorithms/Python/blob/master/maths/binary_exponentiation_2.py)
-  * [Binary Exponentiation 3](https://github.com/TheAlgorithms/Python/blob/master/maths/binary_exponentiation_3.py)
   * [Binomial Coefficient](https://github.com/TheAlgorithms/Python/blob/master/maths/binomial_coefficient.py)
   * [Binomial Distribution](https://github.com/TheAlgorithms/Python/blob/master/maths/binomial_distribution.py)
   * [Bisection](https://github.com/TheAlgorithms/Python/blob/master/maths/bisection.py)
@@ -421,9 +414,8 @@
   * [Decimal Isolate](https://github.com/TheAlgorithms/Python/blob/master/maths/decimal_isolate.py)
   * [Entropy](https://github.com/TheAlgorithms/Python/blob/master/maths/entropy.py)
   * [Euclidean Distance](https://github.com/TheAlgorithms/Python/blob/master/maths/euclidean_distance.py)
-  * [Euclidean Gcd](https://github.com/TheAlgorithms/Python/blob/master/maths/euclidean_gcd.py)
-  * [Euler Method](https://github.com/TheAlgorithms/Python/blob/master/maths/euler_method.py)
   * [Eulers Totient](https://github.com/TheAlgorithms/Python/blob/master/maths/eulers_totient.py)
+  * [Explicit Euler](https://github.com/TheAlgorithms/Python/blob/master/maths/explicit_euler.py)
   * [Extended Euclidean Algorithm](https://github.com/TheAlgorithms/Python/blob/master/maths/extended_euclidean_algorithm.py)
   * [Factorial Iterative](https://github.com/TheAlgorithms/Python/blob/master/maths/factorial_iterative.py)
   * [Factorial Python](https://github.com/TheAlgorithms/Python/blob/master/maths/factorial_python.py)
@@ -440,9 +432,7 @@
   * [Gamma](https://github.com/TheAlgorithms/Python/blob/master/maths/gamma.py)
   * [Gaussian](https://github.com/TheAlgorithms/Python/blob/master/maths/gaussian.py)
   * [Greatest Common Divisor](https://github.com/TheAlgorithms/Python/blob/master/maths/greatest_common_divisor.py)
-  * [Greedy Coin Change](https://github.com/TheAlgorithms/Python/blob/master/maths/greedy_coin_change.py)
   * [Hardy Ramanujanalgo](https://github.com/TheAlgorithms/Python/blob/master/maths/hardy_ramanujanalgo.py)
-  * [Integration By Simpson Approx](https://github.com/TheAlgorithms/Python/blob/master/maths/integration_by_simpson_approx.py)
   * [Is Square Free](https://github.com/TheAlgorithms/Python/blob/master/maths/is_square_free.py)
   * [Jaccard Similarity](https://github.com/TheAlgorithms/Python/blob/master/maths/jaccard_similarity.py)
   * [Kadanes](https://github.com/TheAlgorithms/Python/blob/master/maths/kadanes.py)
@@ -450,14 +440,11 @@
   * [Krishnamurthy Number](https://github.com/TheAlgorithms/Python/blob/master/maths/krishnamurthy_number.py)
   * [Kth Lexicographic Permutation](https://github.com/TheAlgorithms/Python/blob/master/maths/kth_lexicographic_permutation.py)
   * [Largest Of Very Large Numbers](https://github.com/TheAlgorithms/Python/blob/master/maths/largest_of_very_large_numbers.py)
-  * [Largest Subarray Sum](https://github.com/TheAlgorithms/Python/blob/master/maths/largest_subarray_sum.py)
   * [Least Common Multiple](https://github.com/TheAlgorithms/Python/blob/master/maths/least_common_multiple.py)
   * [Line Length](https://github.com/TheAlgorithms/Python/blob/master/maths/line_length.py)
   * [Lucas Lehmer Primality Test](https://github.com/TheAlgorithms/Python/blob/master/maths/lucas_lehmer_primality_test.py)
   * [Lucas Series](https://github.com/TheAlgorithms/Python/blob/master/maths/lucas_series.py)
   * [Matrix Exponentiation](https://github.com/TheAlgorithms/Python/blob/master/maths/matrix_exponentiation.py)
-  * [Max Sum Sliding Window](https://github.com/TheAlgorithms/Python/blob/master/maths/max_sum_sliding_window.py)
-  * [Median Of Two Arrays](https://github.com/TheAlgorithms/Python/blob/master/maths/median_of_two_arrays.py)
   * [Miller Rabin](https://github.com/TheAlgorithms/Python/blob/master/maths/miller_rabin.py)
   * [Mobius Function](https://github.com/TheAlgorithms/Python/blob/master/maths/mobius_function.py)
   * [Modular Exponential](https://github.com/TheAlgorithms/Python/blob/master/maths/modular_exponential.py)
@@ -476,7 +463,6 @@
   * [Prime Factors](https://github.com/TheAlgorithms/Python/blob/master/maths/prime_factors.py)
   * [Prime Numbers](https://github.com/TheAlgorithms/Python/blob/master/maths/prime_numbers.py)
   * [Prime Sieve Eratosthenes](https://github.com/TheAlgorithms/Python/blob/master/maths/prime_sieve_eratosthenes.py)
-  * [Primelib](https://github.com/TheAlgorithms/Python/blob/master/maths/primelib.py)
   * [Pythagoras](https://github.com/TheAlgorithms/Python/blob/master/maths/pythagoras.py)
   * [Qr Decomposition](https://github.com/TheAlgorithms/Python/blob/master/maths/qr_decomposition.py)
   * [Quadratic Equations Complex Numbers](https://github.com/TheAlgorithms/Python/blob/master/maths/quadratic_equations_complex_numbers.py)
@@ -486,8 +472,6 @@
   * [Runge Kutta](https://github.com/TheAlgorithms/Python/blob/master/maths/runge_kutta.py)
   * [Segmented Sieve](https://github.com/TheAlgorithms/Python/blob/master/maths/segmented_sieve.py)
   * Series
-    * [Arithmetic Mean](https://github.com/TheAlgorithms/Python/blob/master/maths/series/arithmetic_mean.py)
-    * [Geometric Mean](https://github.com/TheAlgorithms/Python/blob/master/maths/series/geometric_mean.py)
     * [Geometric Series](https://github.com/TheAlgorithms/Python/blob/master/maths/series/geometric_series.py)
     * [Harmonic Series](https://github.com/TheAlgorithms/Python/blob/master/maths/series/harmonic_series.py)
     * [P Series](https://github.com/TheAlgorithms/Python/blob/master/maths/series/p_series.py)
@@ -501,9 +485,6 @@
   * [Sum Of Geometric Progression](https://github.com/TheAlgorithms/Python/blob/master/maths/sum_of_geometric_progression.py)
   * [Test Prime Check](https://github.com/TheAlgorithms/Python/blob/master/maths/test_prime_check.py)
   * [Trapezoidal Rule](https://github.com/TheAlgorithms/Python/blob/master/maths/trapezoidal_rule.py)
-  * [Triplet Sum](https://github.com/TheAlgorithms/Python/blob/master/maths/triplet_sum.py)
-  * [Two Pointer](https://github.com/TheAlgorithms/Python/blob/master/maths/two_pointer.py)
-  * [Two Sum](https://github.com/TheAlgorithms/Python/blob/master/maths/two_sum.py)
   * [Ugly Numbers](https://github.com/TheAlgorithms/Python/blob/master/maths/ugly_numbers.py)
   * [Volume](https://github.com/TheAlgorithms/Python/blob/master/maths/volume.py)
   * [Zellers Congruence](https://github.com/TheAlgorithms/Python/blob/master/maths/zellers_congruence.py)
@@ -533,23 +514,43 @@
 
 ## Other
   * [Activity Selection](https://github.com/TheAlgorithms/Python/blob/master/other/activity_selection.py)
+  * [Anagrams](https://github.com/TheAlgorithms/Python/blob/master/other/anagrams.py)
+  * [Autocomplete Using Trie](https://github.com/TheAlgorithms/Python/blob/master/other/autocomplete_using_trie.py)
+  * [Binary Exponentiation](https://github.com/TheAlgorithms/Python/blob/master/other/binary_exponentiation.py)
+  * [Binary Exponentiation 2](https://github.com/TheAlgorithms/Python/blob/master/other/binary_exponentiation_2.py)
   * [Davis–Putnam–Logemann–Loveland](https://github.com/TheAlgorithms/Python/blob/master/other/davis–putnam–logemann–loveland.py)
+  * [Detecting English Programmatically](https://github.com/TheAlgorithms/Python/blob/master/other/detecting_english_programmatically.py)
   * [Dijkstra Bankers Algorithm](https://github.com/TheAlgorithms/Python/blob/master/other/dijkstra_bankers_algorithm.py)
   * [Doomsday](https://github.com/TheAlgorithms/Python/blob/master/other/doomsday.py)
+  * [Euclidean Gcd](https://github.com/TheAlgorithms/Python/blob/master/other/euclidean_gcd.py)
   * [Fischer Yates Shuffle](https://github.com/TheAlgorithms/Python/blob/master/other/fischer_yates_shuffle.py)
+  * [Frequency Finder](https://github.com/TheAlgorithms/Python/blob/master/other/frequency_finder.py)
+  * [Game Of Life](https://github.com/TheAlgorithms/Python/blob/master/other/game_of_life.py)
   * [Gauss Easter](https://github.com/TheAlgorithms/Python/blob/master/other/gauss_easter.py)
-  * [Graham Scan](https://github.com/TheAlgorithms/Python/blob/master/other/graham_scan.py)
   * [Greedy](https://github.com/TheAlgorithms/Python/blob/master/other/greedy.py)
+  * [Integeration By Simpson Approx](https://github.com/TheAlgorithms/Python/blob/master/other/integeration_by_simpson_approx.py)
+  * [Largest Subarray Sum](https://github.com/TheAlgorithms/Python/blob/master/other/largest_subarray_sum.py)
   * [Least Recently Used](https://github.com/TheAlgorithms/Python/blob/master/other/least_recently_used.py)
   * [Lfu Cache](https://github.com/TheAlgorithms/Python/blob/master/other/lfu_cache.py)
   * [Linear Congruential Generator](https://github.com/TheAlgorithms/Python/blob/master/other/linear_congruential_generator.py)
   * [Lru Cache](https://github.com/TheAlgorithms/Python/blob/master/other/lru_cache.py)
   * [Magicdiamondpattern](https://github.com/TheAlgorithms/Python/blob/master/other/magicdiamondpattern.py)
+  * [Markov Chain](https://github.com/TheAlgorithms/Python/blob/master/other/markov_chain.py)
+  * [Max Sum Sliding Window](https://github.com/TheAlgorithms/Python/blob/master/other/max_sum_sliding_window.py)
+  * [Median Of Two Arrays](https://github.com/TheAlgorithms/Python/blob/master/other/median_of_two_arrays.py)
+  * [N Body Simulation](https://github.com/TheAlgorithms/Python/blob/master/other/n_body_simulation.py)
   * [Nested Brackets](https://github.com/TheAlgorithms/Python/blob/master/other/nested_brackets.py)
+  * [Palindrome](https://github.com/TheAlgorithms/Python/blob/master/other/palindrome.py)
   * [Password Generator](https://github.com/TheAlgorithms/Python/blob/master/other/password_generator.py)
+  * [Primelib](https://github.com/TheAlgorithms/Python/blob/master/other/primelib.py)
   * [Scoring Algorithm](https://github.com/TheAlgorithms/Python/blob/master/other/scoring_algorithm.py)
   * [Sdes](https://github.com/TheAlgorithms/Python/blob/master/other/sdes.py)
+  * [Sierpinski Triangle](https://github.com/TheAlgorithms/Python/blob/master/other/sierpinski_triangle.py)
   * [Tower Of Hanoi](https://github.com/TheAlgorithms/Python/blob/master/other/tower_of_hanoi.py)
+  * [Triplet Sum](https://github.com/TheAlgorithms/Python/blob/master/other/triplet_sum.py)
+  * [Two Pointer](https://github.com/TheAlgorithms/Python/blob/master/other/two_pointer.py)
+  * [Two Sum](https://github.com/TheAlgorithms/Python/blob/master/other/two_sum.py)
+  * [Word Patterns](https://github.com/TheAlgorithms/Python/blob/master/other/word_patterns.py)
 
 ## Project Euler
   * Problem 001
@@ -759,8 +760,6 @@
     * [Sol1](https://github.com/TheAlgorithms/Python/blob/master/project_euler/problem_102/sol1.py)
   * Problem 107
     * [Sol1](https://github.com/TheAlgorithms/Python/blob/master/project_euler/problem_107/sol1.py)
-  * Problem 109
-    * [Sol1](https://github.com/TheAlgorithms/Python/blob/master/project_euler/problem_109/sol1.py)
   * Problem 112
     * [Sol1](https://github.com/TheAlgorithms/Python/blob/master/project_euler/problem_112/sol1.py)
   * Problem 113
@@ -769,8 +768,6 @@
     * [Sol1](https://github.com/TheAlgorithms/Python/blob/master/project_euler/problem_119/sol1.py)
   * Problem 120
     * [Sol1](https://github.com/TheAlgorithms/Python/blob/master/project_euler/problem_120/sol1.py)
-  * Problem 121
-    * [Sol1](https://github.com/TheAlgorithms/Python/blob/master/project_euler/problem_121/sol1.py)
   * Problem 123
     * [Sol1](https://github.com/TheAlgorithms/Python/blob/master/project_euler/problem_123/sol1.py)
   * Problem 125
@@ -817,7 +814,6 @@
 
 ## Searches
   * [Binary Search](https://github.com/TheAlgorithms/Python/blob/master/searches/binary_search.py)
-  * [Binary Tree Traversal](https://github.com/TheAlgorithms/Python/blob/master/searches/binary_tree_traversal.py)
   * [Double Linear Search](https://github.com/TheAlgorithms/Python/blob/master/searches/double_linear_search.py)
   * [Double Linear Search Recursion](https://github.com/TheAlgorithms/Python/blob/master/searches/double_linear_search_recursion.py)
   * [Fibonacci Search](https://github.com/TheAlgorithms/Python/blob/master/searches/fibonacci_search.py)
@@ -880,15 +876,11 @@
 
 ## Strings
   * [Aho Corasick](https://github.com/TheAlgorithms/Python/blob/master/strings/aho_corasick.py)
-  * [Anagrams](https://github.com/TheAlgorithms/Python/blob/master/strings/anagrams.py)
-  * [Autocomplete Using Trie](https://github.com/TheAlgorithms/Python/blob/master/strings/autocomplete_using_trie.py)
   * [Boyer Moore Search](https://github.com/TheAlgorithms/Python/blob/master/strings/boyer_moore_search.py)
   * [Can String Be Rearranged As Palindrome](https://github.com/TheAlgorithms/Python/blob/master/strings/can_string_be_rearranged_as_palindrome.py)
   * [Capitalize](https://github.com/TheAlgorithms/Python/blob/master/strings/capitalize.py)
   * [Check Anagrams](https://github.com/TheAlgorithms/Python/blob/master/strings/check_anagrams.py)
   * [Check Pangram](https://github.com/TheAlgorithms/Python/blob/master/strings/check_pangram.py)
-  * [Detecting English Programmatically](https://github.com/TheAlgorithms/Python/blob/master/strings/detecting_english_programmatically.py)
-  * [Frequency Finder](https://github.com/TheAlgorithms/Python/blob/master/strings/frequency_finder.py)
   * [Is Palindrome](https://github.com/TheAlgorithms/Python/blob/master/strings/is_palindrome.py)
   * [Jaro Winkler](https://github.com/TheAlgorithms/Python/blob/master/strings/jaro_winkler.py)
   * [Knuth Morris Pratt](https://github.com/TheAlgorithms/Python/blob/master/strings/knuth_morris_pratt.py)
@@ -897,7 +889,6 @@
   * [Manacher](https://github.com/TheAlgorithms/Python/blob/master/strings/manacher.py)
   * [Min Cost String Conversion](https://github.com/TheAlgorithms/Python/blob/master/strings/min_cost_string_conversion.py)
   * [Naive String Search](https://github.com/TheAlgorithms/Python/blob/master/strings/naive_string_search.py)
-  * [Palindrome](https://github.com/TheAlgorithms/Python/blob/master/strings/palindrome.py)
   * [Prefix Function](https://github.com/TheAlgorithms/Python/blob/master/strings/prefix_function.py)
   * [Rabin Karp](https://github.com/TheAlgorithms/Python/blob/master/strings/rabin_karp.py)
   * [Remove Duplicate](https://github.com/TheAlgorithms/Python/blob/master/strings/remove_duplicate.py)
@@ -907,9 +898,11 @@
   * [Swap Case](https://github.com/TheAlgorithms/Python/blob/master/strings/swap_case.py)
   * [Upper](https://github.com/TheAlgorithms/Python/blob/master/strings/upper.py)
   * [Word Occurrence](https://github.com/TheAlgorithms/Python/blob/master/strings/word_occurrence.py)
-  * [Word Patterns](https://github.com/TheAlgorithms/Python/blob/master/strings/word_patterns.py)
   * [Z Function](https://github.com/TheAlgorithms/Python/blob/master/strings/z_function.py)
 
+## Traversals
+  * [Binary Tree Traversals](https://github.com/TheAlgorithms/Python/blob/master/traversals/binary_tree_traversals.py)
+
 ## Web Programming
   * [Co2 Emission](https://github.com/TheAlgorithms/Python/blob/master/web_programming/co2_emission.py)
   * [Covid Stats Via Xpath](https://github.com/TheAlgorithms/Python/blob/master/web_programming/covid_stats_via_xpath.py)