71-Magic Square-6 (Jumbo magic square)

Click here for the next blog on the magic square.

A) Magic square using sketches of 3x3 on 4x4

🧮 Delving into the Realm of Jumbo Magic Squares  

We've previously examined how to create a Magic Square using a Magic Sketch—an innovative and visual approach that introduces order and clarity to intricate configurations.  

Now, let’s elevate our exploration!  

By broadening and merging different kinds of Magic Sketches, we can develop Magic Squares of significantly larger dimensions. These impressive creations demonstrate numerical balance and showcase the captivating aesthetics of mathematical design on a grander scale.  

These are Jumbo Magic Squares — a harmonious combination of structure, symmetry, and magnitude.  

🔷 Whether a 16×16 square or even larger, the Magic Sketch method enables us to construct them systematically and creatively.  

🔷 Each digit is strategically positioned, creating rows, columns, and diagonals that yield the same magical sum.  

Therefore, if you enjoy numbers and patterns, crafting Jumbo Magic Squares offers a true pleasure — both a mental challenge and a visual enjoyment!

🧠 Creating a 12 × 12 Jumbo Magic Square with 3×3 and 4×4 Magic Sketches  

In the captivating realm of Magic Squares, possibilities are endless. We can construct larger and more complex magic configurations by utilizing smaller, manageable units, such as a 12 × 12 jumbo magic square.

Let’s explore how this can be achieved by combining two familiar magic sketches: 3×3 and 4×4.

🔷 Two Effective Methods to Construct a 12 × 12 Magic Square:  

1️⃣ Method One: Starting with a 4×4 Magic Sketch  

Initiate with the 4 × 4 Magic Sketch — this will serve as your foundational layout.

Now, envision each point in the 4 × 4 base sketch as the beginning point for a 3 × 3 square of houses.

This means each point of the 4 × 4 sketch transforms into a 3 × 3 Magic Square, resulting in:

4  x 3 = 12

(for both rows and columns)  

4×3=12(both rows and columns)  

Therefore, the overall square expands to a 12 × 12 grid of 144 magic numbers!

🔗 Arranging the Squares:  

Commence from point “A” in the first 3 × 3 sketch.

Link it to point “B” of the subsequent 3 × 3 sketch following the pathway from your 4 × 4 base.

Proceed with this meticulous connection for all 16 positions in the 4 × 4 sketch.

This visual linking culminates in a Magic Sketch for 12 × 12, preserving structure and numerical balance.

📐 Refer to the diagram below for a clear depiction of how these connections evolve to finalize the Jumbo Magic Square.

In this section, we can create two varieties of magic squares.  

1) Gradually build up the magic square by incorporating a 3x3 pattern onto a 4x4 template.  

Here, the foundational sketch is a 4x4 magic sketch, and we are using this base to apply the 3x3 magic design. 

2) A consistently ascending magic square with a 3x3 design on a 4x4 template. This foundational sketch is a 4x4 magic template, and we are integrating the 3x3 magic design onto this base.

B) Magic square using sketches of 4x4 on 3x3

🔷 Method 2: Utilizing a 3 × 3 Magic Sketch as the Foundation for Creating a 12 × 12 Jumbo Magic Square  

Similar to piecing together a stunning mosaic, we can develop a Jumbo Magic Square (12 × 12) by employing a different method, beginning with a 3 × 3 Magic Sketch as our base.

🧩 Step-by-Step Assembly:  

Start with the 3 × 3 Magic Sketch — our initial layout.

Next, each point within this 3 × 3 framework will be expanded into a 4 × 4 Magic Square, which contains 16 houses.

Given that:

3 x 4 = 12

(for both rows and columns)

3×4=12(for both rows and columns)

We will construct a complete 12 × 12 Magic Square, which will encompass a total of 144 numbers.

🔗 Linking the Sketch:  

Assuming in your foundational 3 × 3 sketch, you have the first point labeled “A.”

Utilize your 4 × 4 magic sketch to determine the movement:  

From point “B” in your magic sketch, trace a path that connects to the “A” of the subsequent point in the base sketch.

Repeat this connection pattern for all 9 positions of the 3 × 3 foundation.

Doing so ensures that each 4 × 4 block aligns correctly with the design and flow of the original 3 × 3 sketch, resulting in a Magic Sketch for the entire 12 × 12 grid.

📐 A well-drawn diagram will help you visualize how each 4 × 4 square integrates into the larger sketch, ultimately leading to the completion of a Jumbo Magic Square!

🔶 Building a 12 × 12 Jumbo Magic Square Using the “4 × 4” Base Sketch

In our previous method, we used the 4 × 4 Magic Sketch as the base and extended each point to represent a 3 × 3 magic square, leading to a beautifully crafted 12 × 12 Jumbo Magic Square.

Now, let’s continue the explanation with more detail.

---

🔁 Completing the Full Sketch:

• Start with the 4 × 4 Magic Sketch (which contains 16 points).

• Each of these 16 points will be replaced with a 3 × 3 square, each consisting of 9 houses.

• These small 3 × 3 blocks will be filled with numbers carefully so that when combined, they form a fully functional 12 × 12 Magic Square, since 4 × 3 = 12.

👉 Process of Expansion:

• Consider a point “A” in the base 4 × 4 sketch.

• Attach the “B” point (from your smaller 3 × 3 sketch) with the next “A” of the next square — this defines the direction of progression.

• Repeat this connection and placement until all 16 squares are aligned, reaching the last point of the base sketch.

---

🌟 What Do We Get?

Once complete, we achieve a Jumbo Magic Square made up of 144 cells (12 rows × 12 columns) that maintains the magic properties — same row, column, and diagonal sums — thanks to the underlying logic of magic sketches.

---

🧠 Two Unique Possibilities:

Using this idea, we can prepare two distinct types of Magic Squares:

1. Type 1 – 4 × 4 base with each point as a 3 × 3 Magic Square
   Result: 12 × 12 Jumbo Magic Square

   • Focuses on the 4 × 4 movement.

   • Builds complexity on finer 3 × 3 arrangements.

2. Type 2 – 3 × 3 base with each point as a 4 × 4 Magic Square
   Result: Another version of 12 × 12 Jumbo Magic Square

   • Compact path movement of the 3 × 3 base.

   • Detailed 4 × 4 filling at each node.

Each method is equally elegant and exciting to explore, revealing the limitless creative potential of numbers and mathematical patterns.

1) Gradually enhance the magic square by incorporating a 4x4 design onto a 3x3 layout. The foundational sketch is a 3x3 magic design, and we are building the 4x4 magic design on top of this base.

2) Uniformly escalating magic square featuring a 4x4 design on a 3x3 template. The foundation is a 3x3 magic layout, and we are building upon this framework to incorporate the magic design of 4x4.

Click here for the next blog on the magic square.

ANIL SATPUTE

70-11 Basics of Quadratic Equations

Click here for the previous basics.

Facebook: http://www.facebook.com/anil7pute

YouTube: https://www.youtube.com/user/Anil7pute

                  

Some special and critical types of factors:

Please download the following file and study it very carefully so that you will not find any difficulties while solving quadratic equations. Download the following file by clicking on it.

Click here to download the critical-type-of-factors.pdf file.

Just go through this downloaded file and be prepared to solve any problem pertaining to these critical factors.

Some Important Problems related to Quadratic Equations:

Anil Satpute

69-10 Basics of Quadratic Equations

Click here for the previous basics.

Facebook: http://www.facebook.com/anil7pute

YouTube: https://www.youtube.com/user/Anil7pute

Some special and critical types of factors:

Please download the following file and study it very carefully so that you will not find any difficulties while solving quadratic equations. Download the following file by clicking on it.

Click here to download the critical-type-of-factors.pdf file.

Just go through this downloaded file and be prepared to solve any problem pertaining to these critical factors.

Some Important Problems related to Quadratic Equations:

A few more problems on Quadratic Equations related to factors will be discussed in the next Blog.

Click here for the next basics.

Anil Satpute

68-09 Basics of Quadratic Equations

Click here for the previous basics.

Facebook: http://www.facebook.com/anil7pute

YouTube: https://www.youtube.com/user/Anil7pute

Some special and critical types of factors:

Please download the following file and study it very carefully so that you will not find any difficulties while solving quadratic equations. Download the following file by clicking on it.

Click here to download the critical-type-of-factors.pdf file.

Just go through this downloaded file and be prepared to solve any problem pertaining to these critical factors.

Some Important Problems related to Quadratic Equations:

A few more problems on Quadratic Equations related to factors will be discussed in the next Blog.

Click here for the next basics.

Anil Satpute

67-08 Basics of Quadratic Equations

Click here for the previous basics.

Facebook: http://www.facebook.com/anil7pute

YouTube: https://www.youtube.com/user/Anil7pute

Some Important Problems related to Quadratic Equations:

 
a) If the sum of the root of the quadratic equation [1/(x + m)] + [1/(x + n)] = 1/p is 0, then show that the product of the roots is -[(m ² + n ²)/2]

Now by using the above method, we can easily solve the following examples.

A few more problems on Quadratic Equations related to factors will be discussed in the next Blog.

Click here for the next basics.

Anil Satpute

66-07 Basics of Quadratic Equations

Click here for the previous basics.

Some Important Problems related to Quadratic Equations:

a) If one root of the quadratic equation k x ² - 5 x + 2 = 0 is 4 times the other, then find " k "
Solution:

b) If the roots of the quadratic equation  x ² +  m x + n = 0 differ by 3, then prove that m ² =  4 n + 9
Solution:

c) If one root of the quadratic equation p x ² +  x + p = 0 is the square of the other, then find  prove that 1 + p q ( p + q - 3 ) = 0
Solution:

Hence Proved

A few more problems on Quadratic Equations related to factors will be discussed in the next Blog.

Click here for the next basics.

Anil Satpute

65-06 Basics of Quadratic Equations

Click here for the previous basics.

Now we will see a few quadratic equations using the formula method.

a) Solve the quadratic equation  5 x ² - 2 x - 9 = 0.

Solution:

       1) Comparing  5 x ² - 2 x - 9 = 0 with a x ² + b x + c = 0 we get,

            a = 5, b = - 2, c = - 9

b) Solve the quadratic equation  3 x ² - 4 x - 3 = 0.

Solution:

       1) Comparing  3 x ² - 4 x - 3 = 0  with a x ² + b x + c = 0 we get,

            a = 3, b = - 4, c = - 3

Now we will see some rules about the roots of the quadratic equation.

       1) Sum of the roots (α, β) of the quadratic equation a x ² + b x + c = 0,

              α  +  β  =  - b / a  and   α β  =  c / a

              α  +  β  =  - Coefficient of x / Coefficient of x ²  and   α β  =  Constant / Coefficient of x ^2.

       2) If α and β are the roots of the quadratic equation then the quadratic equation will be 

            x ² - (Sum of the roots)  x + (Product of the roots) = 0,

            x ² - (α  +  β)  x + (α β) = 0,

^{Now we will see some important problems in Quadratic equations and the roots of the Quadratic Equations}

       ^{a) If the sum of the roots of the quadratic equation is 3 and the sum of their cubes is 63, then find the equation.}

         ^Solution:

          1)  Here, α + β =  3 and  α ³ + β ³ =  63

          2)  We know that , α ³ + β ³ =  (α + β)³ - 3 α β (α + β)

                                   so ,     63      =  (3)³ - 3 α β (3)

                                               63      =  27 - 9 α β       Dividing by 9 through out we get

                                                7       =  3  - α β 

                                                 α β   =  3  - 7 

                                                 α β   =  - 4

          3) We know that  the quadratic equation will be 

                       x ² - (α  +  β)  x + (α β) = 0 

                       x ² - (3)  x + (- 4) = 0 

                       x ² - 3 x - 4 = 0 

          4)  Answer: So the required quadratic equation will be x ² - 3 x - 4 = 0.

A few more problems on Quadratic Equations related to factors will be discussed in the next Blog.

Click here for the next basics.

Anil Satpute