class 8 NCERT bridge course Answers Activity 1W1.5: Pattern Observation

 Activity W1.5: Pattern Observation 

In this activity, students explore, identify and generalise patterns using physical movements. 

They, then, connect it to number patterns. 

Step 1: Students may be asked to perform a simple body movement sequence without explaining the pattern. 

💡 Example 1: Clap, Clap, Clap, Clap...

Q1: What do you notice about the movement?
ANSWER

 The same action (clap) is repeated again and again without changing. It’s a simple, repeating pattern.

Q2: Can you predict what comes next? Why?
ANSWER

The next movement will be a clap — the pattern never changes, so it will always be a clap.

Q3: If I stop at the 7th movement, what should the 8th movement be?
ANSWER

The 8th movement will also be a clap, because the same action is repeated.

 Step 2: Change the movements to 

💡 Example 2: Clap, Clap, Jump, Clap, Clap, Jump...

Possible questions could be— 

Q1: How is this different from movements in Example 1?

ANSWER
 In this pattern, there are two claps followed by a jump, so the actions change. It’s not just repeating the same movement like in Example 1.

Q2: Can you describe the rule?
ANSWER

 The rule is: After every two claps, there is one jump. The pattern repeats this sequence: Clap, Clap, Jump.

Q3: If the first jump is at 3, the second jump is at 6; then at what number do we get the third jump?
ANSWER

The jumps happen every 3rd move. So the third jump will be at 9.

 Many such different body movements can be thought of 

Questions followed by discussions should be done. 

Step 3: Connecting to numbers 

💡 Example 3: Clap, clap clap, clap clap clap, clap clap clap, clap, … 

We may write the corresponding sequence of numbers as 1, 2, 3, 4 … 

Q: What sequence of numbers can we assign to Example 2?

ANSWER 

If Clap = 1, 2 and Jump = 0, the number sequence is: 1, 2, 0, 1, 2, 0, 1, 2, 0...

 Students may be given a number sequence, such as 1, 3, 5, 7,… and may be asked to assign corresponding body movements that justify this pattern. 

We may ask students to assign their own numbers and create a sequence of numbers.

 This is an odd number sequence. Students could choose a movement like:

• Jump for odd numbers (1, 3, 5, 7…)

• Clap for even numbers (if extended to 2, 4, 6, 8...)
  In this case, the pattern only shows odd numbers, so maybe only jumping is used.

 Step 4: Students may think of many such movements and their corresponding number patterns. 

Examples of Movements and Corresponding Number Patterns

Example 1:

Movement Pattern:
Tap, Tap, Snap, Tap, Tap, Snap...

Number Pattern:
1, 2, 0, 1, 2, 0, 1, 2, 0, ...

Explanation:

• Tap is represented by 1, 2.

• Snap is represented by 0.

• The pattern repeats every 3 moves.

Example 2:

Movement Pattern:
Jump, Clap, Jump, Clap, Jump, Clap...

Number Pattern:
0, 1, 0, 1, 0, 1, ...

Explanation:

• Jump = 0

• Clap = 1

• Alternates between the two actions.

Example 3:

Movement Pattern:
Clap, Jump, Spin, Clap, Jump, Spin...

Number Pattern:
1, 2, 3, 1, 2, 3, ...

Explanation:

• Clap = 1

• Jump = 2

• Spin = 3

• Repeats every 3 steps.

Example 4:

Movement Pattern:
Step forward, Step backward, Step forward, Step backward...

Number Pattern:
1, -1, 1, -1, 1, -1, ...

Explanation:

• Step forward = 1

• Step backward = -1

• Alternates like a simple plus-minus pattern.

Example 5:

Movement Pattern:
Clap, Clap, Jump, Jump, Clap, Clap, Jump, Jump...

Number Pattern:
1, 1, 0, 0, 1, 1, 0, 0, ...

Explanation:

• Clap = 1

• Jump = 0

• Two claps, two jumps, repeating.

Example 6:

Movement Pattern:
Touch head, Touch shoulders, Touch knees, Touch toes...

Number Pattern:
1, 2, 3, 4, 1, 2, 3, 4, ...

Explanation:

• Each action is numbered from 1 to 4 in a cycle.

• Helps connect actions with counting sequences.

Students to invent their own movement patterns like:
👉 spin, stomp, wave
👉 blink, clap, nod
and match them to any number sequence want!

Reflections on the Activity

 Discussion may be held on questions, such as: 

Q: How do patterns help us make predictions?

ANSWER:

Patterns show regularity and repetition, so once we recognize the rule, we can guess what comes next without seeing the full sequence.

Q: Where do we see patterns like this in real life?

ANSWER:

 Patterns are everywhere!

• Days of the week (Monday, Tuesday...)

• Traffic lights (Red, Yellow, Green)

• Music beats and dance steps

• Shapes in tiles or floor designs

• Plant growth (leaf arrangement)

• Numbers like even/odd, multiplication tables.

Participation of Special Children- ADAPTATION 

 Instead of requiring physical movement (for example, clapping, jumping), allow students with mobility disabilities to use gestures, verbal cues, or assistive devices.

 Provide alternative options, such as— 

• Hand tapping or finger snapping instead of clapping. 
• Nodding, blinking, or pointing instead of jumping.
•  Using small objects (counters, flashcards, or digital tools) to represent movements. 
• Pair students with physical disabilities with a peer buddy who can perform the movements on their behalf while they identify, predict, and describe the pattern

class 8 NCERT bridge course Answers Activity 1W1.4: Recreational Puzzles

 Activity W1.4: Recreational Puzzle

Bridge Programme for Grade 8

 Students may be asked to play this game either individually or in pairs. 

1. Locate the following mathematical terms in the above grid. 

2. Encircle them in the grid. 

3. These could be found vertically, horizontally or diagonally. 

4. Time may be allotted for doing this. 

5. Marks may be decided accordingly. 

Words are: 

Circle, 

Octagon,

 Square, 

Parallelogram, 

Star, 

Hexagon, 

 Quadrilateral, 

Triangle, 

Kite and 

Rectangle.

class 8 NCERT bridge course Answers Activity 1W1.3: Recreational Puzzles

 Activity W1.3: Recreational Puzzles 

Students may be asked to play this game either individually or in pairs. 

They may be motivated to disclose their strategy before the class. 

Fill in the missing numbers

 1. The missing values are the whole numbers between 1 and 9. 

2. Each number is used only once.

3. Each row is an arithmetic equation. 

4. Each column is an arithmetic equation. 

5. Remember that multiplication and division are performed before addition and subtraction

Solve the 2nd Column

The equation is:  4 − A +3=2    

-A = 2-7

A = 5

Automatically the 2nd Row solved 

2 x 5 + 8 = 18

Solve the 1st column

A + 2 + B = 15

Solve the 3rd column

C + 8 - D = 0

C - D = -8

The numbers C and D must differ by -8. Possible pairs (from 1-9 without repeats):

• $\mathrm{<br>}$

• $A=2,C=10$ (from 1-9 without repeats so not possible)

• $\mathrm{A=}\mathrm{<span\; class="base"><span\; class="strut"></span><span\; class="mpunct">0,C</span><span\; class="mrel">=8 \; </span><span\; style="font-family:\; "Times\; New\; Roman";\; font-size:\; medium;">(from\; 1-9\; without\; repeats\; so\; not\; possible)</span></span>}$

• So only possible answer is C=1,D=9

Solve the 1st Row

A - 4 - C = 1

A - C = 1 +4

A - C = 5

The numbers $\mathrm{A\; and\; C\; must\; differ\; by\; 5.\; Possible\; pairs\; (from\; 1-9\; without\; repeats):}$

• 
  $A = 6, B = 1$

Solve the 3rd Row

B - 3 X D = -20

D = 9

B - 3 x 9 = - 20

B - 27 = -20

B = -20 + 27

B = 7

Solution for the puzzle

class 8 NCERT bridge course Answers Activity 1W1.2: Dice Math Challenge

 Activity W1.2 Dice Math Challenge 

 Material Required:

• Two large dice (these can be made using cubical cardboard boxes)

Instructions for the Teacher:

• Divide the class into two teams. You can choose creative team names (e.g., “The Math Wizards” and “The Number Ninjas”).
• Draw a line to split the blackboard into two sections and write the team names on each side.
• Each student will take turns throwing the dice for their team.

Steps to Play:

• A student from Team A throws the dice and announces the number that appears on the top.

• The team will multiply the number by itself 3 times (i.e., calculate $number \times number \times number$) and the result will be written under Team A's column on the board.

• A student from Team B takes their turn, following the same steps.

• After each throw, both teams add their new result to their team’s total score on the board.

• Continue until each team has had the set number of turns (e.g., 10 or 12 turns).

• The team whose total score is closest to the target number (500 or 1000, as decided) at the end wins!

                            

SL NO	TEAM A	CUBE	TEAM B	CUBE
1	5	125	2	8
2	1	1	6	216
3	6	216	4	64
4	4	64	3	27
5	2	8	1	1
6	5	125	2	8
7	6	216	5	125
8	3	27	3	27
9	3	27	5	125
10	4	64	5	125
11	1	1	6	216
12	2	8	1	1
13	4	64	2	8
14	3	27	1	1
15	4	64	3	27
	TOTAL	1037	TOTAL	979

TEAM A IS THE WINNER

Based on the above activity, 

Some reflective questions may be discussed, such as: 

1. Number Observation:
   Look at the scores written on the board. Try to find numbers other than seen on the board that are square/cube numbers. 

👉 Question: Can you name a square or cube number that wasn’t written on the board during the game?

ANSWER:
Square numbers (Example: $4, 9, 16, 25, 36, 49, 64, 81, 100, \dots$)

Cube numbers (Example: $1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, \dots$)

2. Cube Building Challenge:

 Students may be given some number of unit cubes say, 8, 10, 25, 27, 30, 50, 64. 

They may be asked to make a bigger cube out of these given number of cubes. 

They may be asked, for which number of cubes they could make a bigger cube and which they could not. Ask them to explain.

You are given unit cubes: 8, 10, 25, 27, 30, 50, 64.

A perfect cube means the total number of cubes can be arranged into a shape where length = width = height (all sides equal). The total must be $n^3$ — a cube number.

Let’s check each one:

Number of Cubes	Can Form a Perfect Cube?	Reason
8	✅ Yes	$2\times 2\times 2=\mathrm{8<span\; class="katex"><br></span> }$$2^3$.
10	❌ No	10 is not a cube number. $2^3 = 8$ and$3^3 = 27$. 10 falls between.
25	❌ No	25 is not a cube number. $2^3 = 8$, $3^3 = 27$. 25 is not in the list.
27	✅ Yes	$3 \times 3 \times 3 = 27$. It is $3^3$.
30	❌ No	30 is not a cube number. $3^3 = 27$ and $4^3=\mathrm{64.\; 30\; falls\; between.}\mathrm{<br>}\mathrm{<br>}\mathrm{<br>}$
50	❌ No	50 is not a cube number. $3^3 = 27$ and $4^3 = 64$. 50 falls between.
64	✅ Yes	$4\times 4\times 4=\mathrm{64.\; It\; is }$$4^3$.

ANSWER: 

You can make a perfect cube with: 8, 27, and 64. these numbers can form a larger cube.

ANSWER:

 You cannot make a perfect cube with: 10, 25, 30, 50 — these numbers cannot form a larger cube because these numbers are not cube numbers.

 3. Rubik’s Cube Mystery:

Look at a Rubik’s Cube. Guess:
👉 How many small unit cubes make up a complete Rubik’s Cube?

💡 Hint: A Rubik's Cube is usually a cube with equal sides — try imagining how many little cubes fit along each edge and then multiply!

A rubik cube may be shown to students to guess how many small unit cubes have been used to make it.

ANSWER: 

A standard Rubik’s Cube is a 3×3×3 cube.

$$3\times 3\times 3=27$$So, 27 small unit cubes are used to make one complete Rubik's Cube! 

??? Extra Challenge:
If you could design your own cube puzzle, how many small cubes would you choose, and why?

Inclusion of Special Children:

• Pair special children with supportive peer buddies for guidance.

• Encourage them to take active roles: throwing the dice, multiplying the numbers, or writing the scores on the board.

• This teamwork helps build confidence and fosters participation.

class 8 NCERT bridge course Answers Activity 1W1.1: Recreational Puzzle

 Bridge Programme Content: Detailed Activity 

Activities for Week1

Activity W1.1: Recreational Puzzle 

Material Required: 

A calendar of any month, coloured sketch pens and sheet of paper

May 2025

A calendar may be given to students in pairs. 

They have to observe the calendar keenly and write their observations in the notebook.

Let the students do on their own. 

Teacher can suggest question but should not reveal the conclusions before taking responses of students’ observations. 

Teacher may ask students to do the following: 

Draw boxes around square numbers (number obtained when a number is multiplied to itself two times. 9 = 3 × 3 is a square number) and colour them with a single colour. 

Draw circles around cube numbers (number obtained when a number is multiplied to itself three times. 8 = 2 × 2 × 2 is a cube number.) and colour them with a new colour. 

Draw triangle around prime numbers. O Students may be asked to choose any 2 by 2 number square grid from the calendar they have. 

Instructions for Students

1. Observing the Calendar:

Students are given a calendar (in pairs). For example, May 2025 starts on Thursday and has 31 days.

They are asked to observe patterns in the calendar and note down their own observations.
Examples of what students might observe:

• The month starts on a Thursday.

• All weeks have 7 days.

• The 1st, 8th, 15th, 22nd, and 29th are all Thursdays.

• The numbers in the same column increase by 7.

2. Marking Specific Numbers:

Using coloured pens, students mark special types of numbers:

Square Numbers (draw boxes around and colour with one colour):
→ 1, 4, 9, 16, 25

Cube Numbers (draw circles around and colour with a second colour):
→ 1, 8, 27

Prime Numbers (draw triangles around):
→ 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31

3. Choose any 2x2 number square from the calendar
(For example: Choose the block with 8, 9, 15, 16)

They should try to answer the following questions: 

What is the sum of the numbers in the diagonals of the grid? 

What is the sum of numbers in the diagonals?

• Diagonal 1 (8 + 16) = 24

• Diagonal 2 (9 + 15) = 24

 What magic do you observe? 

 Observation / Magic:

• The sum of both diagonals is the same in a 2x2 grid.

Based on the above activity, some reflective questions may be discussed, such as: 

Reflective Questions & Sample Student Responses

Q1: How do you differentiate between square and cube numbers?

• A square number is a number multiplied by itself once (e.g., 4 = 2 × 2).

• A cube number is a number multiplied by itself twice more (e.g., 8 = 2 × 2 × 2).

Q2: How do you identify prime numbers?

• Prime numbers have exactly two factors: 1 and itself.

• They are not divisible by any other number.

Q3: Why is the sum of diagonals always same in a 2x2 grid?

• Because the opposite corners in a 2x2 square are symmetrical and balance each other.

• The sums (top-left + bottom-right) and (top-right + bottom-left) are always equal due to arithmetic symmetry.

Extension of the Activity 

 Teacher can extend this activity by suggesting students to take a 3 by 3 grid. 

Extension Activity (3x3 Grid)

Take a 3x3 grid (e.g., from 8 to 16):

Why is the sum of diagonals always same for any 2 by 2 grid? 

They may find— 

1. The sum of diagonal numbers. 

Sum of diagonals:

• Diagonal 1 (8 + 16 + 24) = 48

• Diagonal 2 (10 + 16 + 22) = 48

2. The mean of all numbers. 

Mean of all numbers in the grid:
Sum of all numbers: 8+9+10+15+16+17+22+23+24 = 144
Mean = 144 ÷ 9 = 16

Special Observations:

• The center number (16) is the mean.

• Diagonals again have equal sums.

• There is a symmetry in the calendar numbers.

They may be asked to see what special thing they observed. 

Teacher may encourage students to explore more such patterns in the calendar

Encouragement for Exploration

Students can be encouraged to:

• Try different-sized grids (like 4x4).

• Observe symmetry across weeks.

• Explore patterns in odd/even numbers.

• Make their own puzzles from the calendar.

Circumference of the circle

Today we all use the formulas 2πr or πd to calculate the circumference of a circle. Can you believe it if I tell you that this is a Tamil invention??? Look below and you will definitely be surprised.

The formula for calculating the circumference of a circle is set in the Kanakathikaram, Kakkai Padinium etc.

 Crow Patinium

******************************

Song :-

"Vitamore seven, add four to the circle

Sattena Irati Chain

Thikaipana Churtthane"

Setting :-

(Perimeter - P, Diameter - V, Radius - Aa)

= (Vitamore seven) = V/7 

= (Add four to the circle) = V+4V/7

= (Double chain) = 2[V + 4V/7]

(Thikaipana Churtthane)

= 2[V + 4V/7

= 2[11V/7]

= 2x11V/7

= 22/7 x V

According to the present, if 22/7 = π and V=d (diameter)...

Circumference of the circle = 22/7 x V

= πd

(d = 2r) Circumference = 2πr

So today  It is a matter of pride for Tamils ​​that a Tamil has calculated the formulas used to find the circumference of a circle without using π.....

✍ Tamil Forum

Circumference of the circle 

இன்று நாம் அனைவரும் வட்டத்தின் சுற்றளவை கணிப்பதற்கு 2πr அல்லது πd எனும் சூத்திரங்களை பாவிக்கின்றோம். இது தமிழரின் கண்டுபிடிப்பு என்று சொன்னால் உங்களால் நம்ப முடியுமா??? கீழே பாருங்கள் நிச்சயம் ஆச்சரியப்படுவீர்கள்.

வட்டத்தின் சுற்றளவை கணிப்பதற்கு கணக்கதிகாரம், காக்கை பாடினியம் போன்றவற்றில் சூத்திரம் அமைக்கப்பட்டுள்ளது.

காக்கை பாடினியம்

****************************

பாடல் :-

"விட்டமோர் ஏழு செய்து

திகைவர நான்கு சேர்த்து

சட்டென இரட்டி செயின்

திகைப்பன சுற்றுத்தானே"

நிறுவல் :-

(பரிதி - ப, விட்டம் - வி, ஆரை - ஆ என வைத்தால்)

= (விட்டமோர் ஏழு செய்து) = வி/7 

= (திகைவர நான்கு சேர்த்து) = வி+4வி/7

= (சட்டென இரட்டி செயின்) = 2[வி + 4வி/7]

(திகைப்பன சுற்றுத்தானே)

= 2[வி + 4வி/7

= 2[11வி/7]

= 2x11வி/7

= 22/7 x வி

தற்காலத்தின் படி 22/7 = π எனவும் வி=d (விட்டம்) எனவும் கொண்டால்...

வட்டத்தின் சுற்றளவு = 22/7 x வி

= πd

(d = 2r ஆக) சுற்றளவு = 2πr

ஆக இன்று வட்டத்தின் சுற்றளவு காண பயன்படுத்தப்படும் சூத்திரங்களை தமிழன் π பயன்படுத்தாமலே கணித்துள்ளான் என்பது தமிழருக்கு பெருமையே.....

✍ தமிழ் கருத்துக்களம்

 MATHEMATICAL GARDEN IDEAS

An illustration of a mathematical garden that visually represents the spiral root concept using plants and flowers. 

A mathematical garden illustrating the spiral root concept using plants and flowers. 

Creating a spiral root concept in a math garden using plants and flowers sounds fascinating! 

1. Center Point: Start with a central point in your garden.

2. Spiral Layout: Arrange plants and flowers in a spiral pattern, expanding outward from the center.

3. Square Root Spiral: Each segment of the spiral can represent a square root value, such as √2, √3, √4, and so on

Spiral Root Math Garden Layout

1. Central Focal Point (🌼):

   • Start with a vibrant flower or plant at the center of your garden. This represents the origin point (0).

2. Spiral Pathway:

   • Create a path that spirals outward from the center, resembling the shape of a square root spiral.

   • Each turn of the spiral represents an incremental square root value (√1, √2, √3, etc.).

3. Plant Arrangement:

   • First Loop (√1):

     • Plant low-growing flowers like pansies or marigolds close to the center.

   • Second Loop (√2):

     • Use medium-height plants like lavender or sage.

   • Third Loop (√3):

     • Incorporate taller plants such as roses or dahlias.

   • Fourth Loop (√4):

     • Include even taller elements like sunflowers or ornamental grasses.

4. Visualizing Square Roots:

   • The increasing distance from the center symbolizes the increasing value of square roots.

   • The spiral's expansion represents the non-linear growth of square root functions.

5. Pathway Materials:

   • Define the spiral walkway with stones, bricks, or mulch.

   • Consider using different colors or textures for each loop to enhance the visual distinction between square root values.

Metaphorical Elements:

• Growth and Learning:

  • The spiral symbolizes the journey of learning and discovery in mathematics.

  • As one walks the spiral path, they're metaphorically delving deeper into mathematical concepts.

• Connection to Nature:

  • By integrating mathematical concepts with natural elements, you create a space where abstract ideas become tangible.

Additional Enhancements:

• Educational Markers:

  • Place small signs along the path indicating the square root values and perhaps interesting facts or equations.

• Golden Ratio Spiral:

  • For an added mathematical twist, design the spiral based on the Fibonacci sequence or the golden ratio, which occurs frequently in nature.

• Color Gradation:

  • Choose plants whose flower colors transition gradually, creating a gradient effect that enhances the spiral design.

Example Layout Idea:

• Start with white flowers at the center.

• Gradually transition to yellow, then orange, red, purple, and finally blue as you move outward.

• This color shift not only adds visual appeal but can represent the spectrum of light or other mathematical gradients.

Implementing the Spiral:

1. Planning:

   • Use a rope or hose to lay out the spiral shape on the ground before planting.

   • Mark the path with stakes to ensure accuracy.

2. Calculating Distances:

   • Use the spiral equation $r=a\sqrt{\theta }r\; =\; a\; \backslash sqrt\{\backslash theta\}$ where:

     • $rr$ is the radius,

     • $aa$ is a constant that controls the distance between loops,

     • $\theta \backslash theta$ is the angle in radians.

   • This will help you space the loops appropriately according to square root values.

3. Soil Preparation:

   • Ensure the soil is well-prepared with compost and suitable for the plants you've chosen.

   • Consider soil testing to balance pH levels.

Bringing It All Together:

By integrating the spiral root concept into your garden, you create a living embodiment of mathematical principles. This not only makes for a beautiful garden but also a conversation starter and an educational experience.

Let’s Visualize the Concept Further:

Imagine walking through your garden starting at the center:

• Step 1: You see delicate white pansies symbolizing the beginning.

• Step 2: As you follow the spiral, yellow marigolds appear, marking the transition to the next square root value.

• Step 3: Further along, red roses emerge, taller and more prominent.

• Step 4: Nearing the outer loops, majestic sunflowers tower above, representing larger square root values.

Final Thoughts:

Incorporating the spiral root concept blends the beauty of nature with the elegance of mathematics. It's a creative way to explore mathematical ideas while enjoying the pleasures of gardening.

The garden features a spiral pattern with roots intertwining in.

                       🌼

                      /   \

                     🌻     🌸

                    /         \

                   🌷           🌺

                  /               \

                 🌹-----------------🌿

                  \               /

                   🌱           🌵

                    \         /

                     🍄     🌴

                      \     /

                       🌳

The spiral pattern follows a Fibonacci sequence, blending mathematical beauty with nature. 

4 4 4 4 4 4 4 4 .....so what is special about these ‘four’...?????

01. ‘Four’ people speak like ‘four’.

02. If good things happen to ‘four’ people, nothing is wrong.

03. ‘Four’ does not earn money, so they should study....????

04. Only ‘Four’ can understand the world if they go around the city.

05. He who knows ‘four’., He who understands ‘four’.

06. ‘Four’ should listen carefully to the words of ‘four’.

Why is this ‘four’ so special....

In the Sangha literature, in the books of the eight-digit number system, four comes along with the name. Some four., forty and eight are four hundred., and in the universe, ‘four’ comes as four thousand.

Naladiyar., Nanmanikadigai., Inna forty., Iniyavai Narpathu.

AgaNaanooru.., PuraNaanooru., Nalaira diviya prabantham....

“Milk., Talithenum., Bagum., Dalum, these ‘four’ mixed together I will give you”

A song of God’s greeting in the ‘Nal’vazhi Righteousness book of Auvaiyar.

Nalum., Both are the words... In this ‘Nal’ is.. Naladiyar....

“Becoming in love and shedding tears

He who recites it, he who rises to the path of virtue

‘The four Vedas’ is the true meaning

Nathan Namam Namachivayeva”

‘Nanmarai’.... means ‘four’ of the Vedas.

The foremost of the Nayanmars who spread the Saivism are ‘four’.

Appar., Sambandar., Sundarar., Manickam. We call them the four.

Out of the ten (nine) incarnations of Mahavishnu, only ‘four’ incarnations took human form (in pregnancy).

Veda Vyasa, who divided the Vedas into four, entrusted the responsibility of spreading them to ‘four’ Rishis. They were

Ruk ═ Bailar., Yajur ═ Jaimini., Sama ═ Vaisampayana., Atharvana ═ Sumandhu.

Dasharatha had ‘four’ children.

The ‘four’ purusharthas....

They are Dharma., Artha., Kama., Moksha.

The stages that a person has to go through from birth to death

are also ‘four’

They are ~ Brahmacharya., Grihasthashrama., Vanaprastha., Sannyasa.

The sons of Brahma are ‘four’. Sanagar, Sanadhan, Sanandhan, Sanatkumar.

Brahma has ‘four’ heads. Brahma is called Chaturmukha.

Adi Shankara established ‘four’ monasteries in the ‘four’ directions/corners of India and appointed ‘four’ disciples.

Fire has ‘four’ forms namely Gampira, Yamala, Mahati, Panchami.

The directions are ‘four.’

The Kumbh Mela is held in ‘four’ places namely Haridwar, Allahabad (Prayagai, Triveni Sangamam), Nashik, and Ujjain.

The four types of armies are: Ratha, Gaja, Duraga, Padati (chariot, elephant, horse, infantry...).

Aham Brahmasmi, Tattvam Asi, Prajnanam Brahma, Ayamatma Brahma.... The four great words mentioned in the Upanishads.

The four that cannot be conquered

“Sleep, dream, woman, Kama Agni Indana Kara Pana”

Dreaming does not satisfy sleep,

Women do not satisfy desire,

Firewood does not satisfy fire,

Drinking does not satisfy a drunkard.

“Na Swapnena Jayet Nidram Na Kamena Jayet Striyaha

Na Indanena Jayet Agni Na Pane Na Kara Pana Kara Pana”

This is what Vidura Neethi Hidopadesam

“Fire cannot be extinguished by wood.

Rivers cannot fill the ocean.

Yama cannot be conquered even by all living beings.

Men cannot satisfy beauties.”

The ages.... Gratam., Tretam., Dvaparam., Kali are ‘four’

Fear., Math., Naanam., Pasalppu... The ‘four’ types of qualities of a woman

On Shivaratri, ‘Nalu’ Kala Puja is performed.

Those who have learned all the ‘four’ Vedas are called Chaturvedi.

Although the number ‘four’ has special qualities.

Be as good in life as ‘four’ people...

It also has the concept that “Even in death.,

For good.,

For bad., ‘four’ people are needed”

I want to let 'four' people know., 'four' wrote the paragraph., if at least 'four' people read it....💐💐💐

Read it in Tamil also pleasure one

4 4 4 4 4 4 4 .....அப்படி இந்த ‘நாலு’க்கு என்னதாங்க ஸ்பெஷல்....?????

01. ‘நாலு’ பேரு ‘நாலு’ விதமா பேசுவாங்க.

02. ‘நாலு’ பேருக்கு நல்லது நடக்கும்னா எதுவும் தப்பில்ல. 

03. ‘நாலு’ காசு சம்பாதிக்கவாது படிக்கணும்ல....????

04. ‘நாலு’ ஊரு சுற்றினாதான் உலகம் புரியும்.

05. அவரு ‘நாலு’ம் தெரிஞ்சவரு., ‘நாலு’ம் புரிஞ்சவரு. 

06. ‘நாலு’ வார்த்த நறுக்குன்னு நல்லா கேக்கணும்.

ஏன் இந்த ‘நாலு’ மட்டும் இவ்ளோ ஸ்பெஷல்....

சங்க இலக்கியத்தில் பதினெண்கீழ்கணக்கு நூல்களில்., பெயருடன் நான்கு சேர்ந்து வரும். சில நாலு., நாற்பது மற்றும் எட்டுத்தொகையில் நானூறு., பிரபந்தத்தில் நாலாயிரம் என ‘நான்கு’ வரும்.

நாலடியார்., நான்மணிக்கடிகை.,இன்னா நாற்பது., இனியவை நாற்பது

அக நானூறு., புற நானூறு., நாலாயிர திவ்ய பிரபந்தம்....

“பாலும்., தெளிதேனும்., பாகும்., பருப்பும் இவை ‘நாலு’ம் கலந்து உனக்கு நான் தருவேன்”

ஔவையாரின் ‘நால்’வழி நீதி நூலில் கடவுள் வாழ்த்து பாடல்.

நாலும்., இரண்டும் சொல்லுக்குறுதி... இதில் ‘நாலு’ என்பது.. நாலடியார்....

“காதலாகிக் கசிந்து கண்ணீர் மல்கி

ஓதுவார் தமை நன்நெறிக்கு உய்ப்பது

‘வேதம் நான்கினும்’ மெய்ப்பொருளாவது

நாதன் நாமம் நமச்சிவாயவே”

‘நான்மறை’.... என்பது வேதங்கள் ‘நான்கு’.

சைவ நெறியைப் பரப்பிய நாயன்மார்களில் முதன்மையானவர்கள் ‘நான்கு’ பேர்.

அப்பர்., சம்பந்தர்., சுந்தரர்., மாணிக்க வாசகர். இவர்களை நால்வர் என அழைக்கிறோம்.

மஹாவிஷ்ணுவின் பத்து (ஒன்பதில்) அவதாரங்களில் ‘நான்கு’ அவதாரங்களுக்கு மட்டுமே மனிதனாக (கர்பவாசத்தில்) எடுத்ததாகும்.

வேதங்களை நான்காகப் பகுத்த வேத வியாசர்., அவற்றை ‘நாலு’ ரிஷிக்களிடம் பரப்பும் பொறுப்பை ஒப்படைத்தார். அவர்கள்

ருக் ═ பைலர்., யஜூர் ═ ஜைமினி., சாம ═ வைசம்பாயன., அதர்வண ═ சுமந்து.

தசரதனுக்கு ‘நான்கு’ பிள்ளைகள்.

‘நான்கு’ புருஷார்த்தங்கள்....

அவை தர்ம., அர்த்த., காம., மோட்சம்.

மனிதன் பிறப்பு முதல் இறப்பு வரை

கடக்க வேண்டிய நிலைகளும் ‘நான்கு’

அவை ~ பிரம்மசர்யம்., கிருஹஸ்தாச்ரமம்., வானப்ரஸ்தம்., சந்யாசம்.

பிரம்மாவின் மானஸ புத்திரர்கள் ‘நான்கு’ பேர். சநகர்., சநாதனர்., சநந்தனர்., சனத் குமாரர்.

பிரம்மாவுக்கு ‘நான்கு’ தலைகள். சதுர்முகன் என்ற பெயர் பிரம்மாவுக்கு உண்டு.

ஆதிசங்கரர் பாரத நாட்டின் ‘நான்கு’ திசைகளிலும்/மூலைகளிலும் ‘நான்கு’ மடங்கள் நிறுவி., ‘நான்கு’ சீடர்களை நியமித்தார்.

அக்னிக்கு கம்பீரா., யமலா., மஹதி., பஞ்சமி என ‘நான்கு’ வடிவங்கள்.

திசைகள் ‘நான்கு.’

ஹரித்வார்., அலஹாபாத் (பிரயாகை, த்ரிவேணி சங்கமம்)., நாசிக்., உஜ்ஜையினி என ‘நான்கு’ இடங்களில் கும்ப மேளா நடைபெறும்.

ரத., கஜ., துரக., பதாதி (தேர்., யானை., குதிரை., காலாட் படைகள்.... என ‘நால்’ வகைப் படைகள்.

அஹம் பிரம்மாஸ்மி., தத்வம் அஸி., பிரக்ஞானம் பிரம்ம., அயமாத்ம ப்ரம்ம.... உபநிஷத்தில் கூறப்படும் ‘நாலு’ மஹா வாக்யங்கள்.

வெல்ல முடியாத ‘நாலு’

“நித்ரா., ஸ்வப்ன., ஸ்த்ரீ., காமஅக்னி இந்தன கரா பாண”

கனவு காண்பது தூக்கத்தை நிறைவு செய்யாது.,

பெண்களுக்கு ஆசை நிறைவு செய்யாது.,

தீயை விறகு நிறைவு செய்யாது.,

குடிகாரனை குடி நிறைவு செய்யாது.

“ந ஸ்வப்னேன ஜயேத் நித்ராம் ந காமேன ஜயேத்ஸ்திரியஹ

ந இந்தனேன ஜயேதக்னீம் ந பானேன கராம் ஜயேத்.” விதுர நீதி

இதையே ஹிதோபதேசம்

“அக்னியை விறகு அணைக்காது.,

சமுத்திரத்தை ஆறுகள் நிறைக்காது.,

யமனை எல்லா உயிர்களும் சேர்ந்தாலும் வெல்ல முடியாது.,

அழகிகளை ஆண்கள் த்ருப்தி செய்ய முடியாது” என சொல்கிறது

யுகங்களும்.... கிரதம்., திரேதம்., துவாபரம்., கலி என ‘நான்கு’

அச்சம்., மடம்., நாணம்., பயிர்ப்பு... பெண்டிரின் ‘நால்’ வகை குணங்கள்

சிவராத்ரியில் ‘நாலு’ கால பூஜை நடக்கும்.

‘நான்கு’ வேதங்களும் கற்றவர்களை சதுர்வேதி என்பார்கள்.

‘நான்கு’ என்ற எண் சிறப்புக்கள் கொண்டிருந்தாலும்., 

‘நாலு’ பேரை போல வாழ்வில் நல்லா இருக்கனும்....

“செத்தாலும்., 

நல்லதுக்கும்.,

கெட்டதுக்கும்., ‘நான்கு’ பேர் வேண்டும்” 

என்ற கருத்தாக்கத்தையும் கொண்டுள்ளது.

ஒரு ‘நாலு’ பேருக்கு தெரியட்டும் என்று., ‘நாலு’ பத்தி எழுதினதை., ஒரு ‘நாலு’ பேராவது படிச்சா சரி....💐💐💐

calendar 2025

What a fascinating set of mathematical properties of the year 2025! You've highlighted some remarkable characteristics, including:

1. 2025 being a square number (45²)

2. Being the product of two squares (9² x 5²)

3. Being the sum of three squares (40² + 20² + 5²)

4. Being the first square year after 1936

5. Being the sum of cubes of single digits from 1 to 9

These properties make 2025 a unique and interesting year indeed! Thank you for sharing this delightful mathematical wonder!

Our next calendrer year 2025 is a mathematical wonder

Interesting 2025

1) 2025, itself is a square 

2) It's a product of two squares, 

Viz. 9² x 5² = 2025

3) It is the sum of 3-squares, 

viz. 40²+ 20²+5²= 2025

4) It's the first square after 1936

5) It's the sum of cubes, of all the single digits, from 1 to 9,

 viz. 1³+2³+3³+4³+5³+6³+7³+8³+9³= 2025.

This is going to be our NEXT YEAR. 😊