Magic Squares 2024-2025 PDF

 Magic Squares has fascinated man since the ancient times. Magic squares have a long history, the earliest record of magic squares is from China in 2200 BC. and is called "Lo-Shu". The first known example of a magic square ( 3 x 3 ) is said to have been found on the back of a turtle by Chinese...

 Magic Squares has fascinated man since the ancient times. Magic squares have a long history, the earliest record of magic squares is from China in 2200 BC. and is called "Lo-Shu". The first known example of a magic square ( 3 x 3 ) is said to have been found on the back of a turtle by Chinese Emperor Yu in 2200 B.C. The Ancient people believed that any person who can create Magic Squares possesses Magical Powers. 9 4 2 3 5 7 8 16 The First magic square to appear in the Western world was in the year 1514 depicted in a copperplate engraving by the German artist- mathematician Albrecht Dürer. Dürer's Melancholia I (1514) includes a 4 x 4 magic Square whose Magic sum is 34  Magic squares were used by Arab astrologers in the 9th century to help work out horoscopes. The work of the Greek mathematician Moschopoulos in 1300 A.D. help to spread knowledge about magic squares. In this present time , we utilize Magic Squares for Fun and Recreation and for mental Health and Exercises. Mental Health and Non-Academic Benefits of Mathematics “Mathematics is food for the brain” – stated by Dr. Arthur Benjamin, He knew exactly what he was talking about. After all, math is all about finding patterns and making connections. Math requires abstract and concrete thinking, which leads to the development of the brain’s muscles. Studying math is actually a springboard to increasing your overall intelligence, and with regular practice, you get better at various academic pursuits. How is this possible? Well, math allows you to see connections and develop neural pathways that strengthen your brain. Mathematics enhances your analytical and problem-solving skills, creates the basis for systemic thinking, improves the skills required to arrive at logical conclusions, expands the mind to handle unfamiliar tasks with ease and confidence, learns  DEFINITION A Magic Square is a square grid filled with Distinct Integers such that the SUM of each Row, Column and Diagonal is equal. The sum is called the magic constant or magic Number of the magic square. The Mathematical study of magic squares typically deals with its construction, classification, and enumeration. The Magic Squares are Generally Classified into 3 Classifications : 1. ODD if n is odd, ( 3,5,7,9,… squares on each side ) 2. EVENLY EVEN (also referred to as "doubly even") if n = 4k (e.g. 4, 8, 12, and so on), 3. ODDLY EVEN (also known as "singly even") if n = 4k + 2 (e.g. 6, 10, 14, and so on). Each classification Requires Different Techniques and Processes in its Construction of the Magic Squares  ODD , EVENLY EVEN , SINGLY EVEN MAGIC SQUARES SINGLY EVEN N=6,10,14,18 ODD EVENLY EVEN N=3,5,7,9 N=4,8,12,16 CONSTRUCTING A MAGIC SQUARE 3X3 ;5X5;7X7  Rules in Constructing Odd Magic Squares For Odd Magic Squares - Always start in the upper Middle Sq hoose a Starting Number and a Common Difference ( Numbers can be Negative or Positive ) ADD the Common Difference using Upward Diagonal Directio Transpose the numbers outside the Square Grid : Outside Above- Transpose to Bottom Right outside -Leftmost Outside Diagonal -transpose below the number f the Diagonal Slot is already Filled- up with number, rite the next number below the Reference number  Column1 Column2 Column3 Starting Point Row 1 Row 2 𝐶 2 𝑅2 Row 3 𝐶 3 𝑅3  1. For Odd Magic Squares - Always start in the upper Middle Squa BS BS 2. Choose a Starting Number and a Common Difference ( Numbers can be Negative or Positive ) 1 8 6 Starting Number : Common Difference : 1 LS 3. ADD the Common Difference using Upward Diagonal Direction 4. Transpose the 3 5 7 LS numbers outside the Square grid: Upper-Lower/ Right - Left 5. If the Diagonal Slot is already Filled- up, write the 4 9 2 next number below the Reference number. If the Number is located Diagonal Outside the Square Grid write it below the Refrence Number.  1. For Odd Magic Squares - Always start in the upper Middle Squa BS BS 2. Choose a Starting Number and a Common Difference ( Numbers can be Negative or Positive ) 8 22 18 Starting Number : Common Difference : 8 LS 3. ADD the Common Difference using Upward Diagonal Direction 4. Transpose the 12 16 20 LS numbers outside the Squares: Upper-Lower/ Right - Left 5. If the Diagonal Slot is already Filled- up, write the 14 24 10 next number below the Reference number  1. For Odd Magic Squares - Always start in the upper Middle Squa BS BS 2. Choose a Starting Number and a Common Difference ( Numbers can be Negative or Positive ) -10 4 -10 0 LS Starting Number : 3. ADD the Common Difference using Upward Diagonal Direction CD: 4. Transpose the -6 -2 2 LS numbers outside the Squares: Upper-Lower/ Right - Left 5. If the Diagonal Slot is already Filled- up, write the -4 6 -8 next number below the Reference number  BS 3 BS BS BS SN: CD: 2 35 49 3 17 31 LS 47 11 15 29 33 LS 9 13 27 41 45 LS 21 25 39 43 7 LS 23 37 51 5 19  RULES IN EVENLY EVEN MAGIC SQUARES : 4 x 4 ; 8 x 8 ; 12 x 12 1. Draw Diagonal Lines hoose your Starting Number and a Common Differenc ways start in the FIRST SQUARE UPPER CORNER : C1R pply Left to Right Direction- write only on the squares with onal lines and write imaginary numbers on squares without diagona 5. Reverse the Direction – right to left- starting from the last square in the lower corner ( C4R4 ), write numbers only on the squares without diagonals, start with the starting number , right to left direction moving upward until you reach  Sn: 1 1 15 14 4 2 3 16 13 CD: 1 7 9 5 8 12 6 11 10 8 10 11 5 9 12 7 6 13 3 2 16 14 15 4 1  Sn: 3 3 31 29 9 CD: 2 25 13 15 19 17 21 23 11 27 7 5 33  Sn: 10 10 -18 -16 4 CD: -2 -12 0 -2 -6 -4 -8 -10 2 -14 6 8 -20  1 63 62 4 5 59 58 8 Sn: 1 56 10 11 53 52 14 15 49 CD: 1 48 18 19 45 44 22 23 41 25 39 38 28 29 35 34 32 33 31 30 36 37 27 26 40 24 42 43 21 20 46 47 17 16 50 51 13 12 54 55 9 57 7 6 60 61 3 2 64 MS=260 ANOTHER METHOD FOR 4 x4  Sn: 1 1 12 7 11 8 2 5 10 3 4 6 9  Rules in Singly Even Magic Squares : 6 x 6 ; 10 x 10 ; Divide the Squares into a 3 x 3 or 5 x 5 Magic Squares Choose your Starting Number and Common Differenc 3. Apply the Rules in Odd Magic Squares olve the 3 x3 or 5 x 5 using the Quadrants: II,III, I, IV Interchange Cells : C1R1, C2R2 , C1R3 with C1R4, C2R5, C1R6 respectively Construct a 6 x 6 Magic Square 1. Divide the Squares into a 3 x3 Q II QI 2. Choose your Starting Number and Common Difference Sn: 1 CD: 1 3. Apply the Rules in Odd Magic Q III Q IV Squares Solve each 3 x 3 By quadrants: II,IV, I,III Construct a 6 x 6 Magic Square 1. Divide the Squares into a 3 x3 2. Choose your Starting Number and Common Difference Sn: 1 CD: 1 3. Apply the Rules in Odd Magic Squares 4. Solve each 3 x 3 5. Interchange Cells : C1R1, By C2R2 quadrants: , C1R3 with C1R4, C2R5, C1R6 respectively II,III,I,IV  RULES IN ODD MAGIC SQUARES. Numbers must be an Arithmetic Sequence have a Common Difference 2. Always start in the upper Middle 3. Apply Diagonal Direction 4. Transpose: Upper-Lower/ Right -Left 5. If the Diagonal Slot is already Filled, write the next below the Reference number SUDOKU