A magic square is a square grid of numbers where the sum of the numbers in each row, each column, and both main diagonals are the same. The 49 cell magic square is a special type of magic square that consists of seven rows and seven columns, totaling 49 cells. This particular magic square has a rich history and is known for its uniqueness. In a 49 cell magic square, each cell is filled with a distinct positive integer starting from 1 up to 49. The numbers are arranged in such a way that the sum of each row, column, and diagonal is equal to 175. This specific sum is obtained by dividing the total sum of numbers from 1 to 49, which is 1225, by 7 (the number of rows, columns, and diagonals).
This is an Order-16 pandiagonal pure magic square so uses the consecutive numbers from 1 to 256.
Each of the 16 rows, columns, and diagonals sum to the constant 2056
The E. S. each also sum to 2056 and the H. H. each sum to 2056 x 2.
The magic constant of a normal magic square depends only on n and has the value M n n sup2 1 2 For normal magic squares of order n 3, 4, 5, 6, 7,8, 183 183 183 183, the magic constants are 15, 34, 65, 111, 175, 260, 183 183 183 183. The magic constant of a normal magic square depends only on n and has the value M n n sup2 1 2 For normal magic squares of order n 3, 4, 5, 6, 7,8, 183 183 183 183, the magic constants are 15, 34, 65, 111, 175, 260, 183 183 183 183.
This specific sum is obtained by dividing the total sum of numbers from 1 to 49, which is 1225, by 7 (the number of rows, columns, and diagonals). The 49 cell magic square is often represented as a table, with each cell containing a number. It is a challenge to create such a square as there are many factors and combinations that need to be considered.
Unusual Magic Squares
This pattern, which is a torus drawn in two dimensions may be used as an order-5 pandiagonal magic square generator.
Examples:
Start at number 1, and follow the big circles, to generate the rows of the A. magic square (below).
Start at number 2, and follow the big circles, to generate the columns of the B magic square.
25 different pandiagonal magic squares can be formed this way by starting with each of the 25 numbers on the model.
Another 25 different magic squares can be constructed by forming the rows and columns with the numbers along the spiral lines. See Magic square C, below.
Actually, four magic squares may be constructed by following the radial lines, and another four by following the spiral lines, in either direction around the torus. However, three of these magic squares are just disguised versions of the fourth one, because they are rotations or reflections.
Magic Circles
These two diagrams, between them, illustrate some relationships in this order-4 magic square.
1 6 12 15 A. B.
11 16 2 5 1 + 15 + 4 + 14 -- biggest circle 1 + 15 + 10 + 8 -- 1 of 4 big circles
8 3 13 10 1 + 12 + 13 + 8 -- 1 of 4 medium circles 11 + 2 + 13 + 8 -- 1 of 4 small circles
14 9 7 4 1 + 6 + 16 + 11 -- 1 of 5 small circles
Thanks for the idea to W. S. Andrews, Magic Squares and Cubes, Dover, 1960.
Pandiagonal with Special Numbers.
Prime Number Heterosquares
The Order-3 heterosquare on the left consists of 9 prime numbers. The 3 rows, 3 columns and the 2 main diagonals all sum to different prime numbers. The sum of all 9 cells is also a prime number.
Is this the square with the smallest possible total with eighteen unique primes (including the totals)?
The Square on the right has identical features, but in addition consists of consecutive primes.
Is this the square with the smallest possible total with nine consecutive primes?
These squares designed by Carlos Rivera, Sept. 98. See his Web page on Prime Puzzles & Problems at
http://www.sci.net.mx/~crivera/
Double HH
This is an Order-16 pandiagonal pure magic square so uses the consecutive numbers from 1 to 256.
Each of the 16 rows, columns, and diagonals sum to the constant 2056
The E. S. each also sum to 2056 and the H. H. each sum to 2056 x 2.
Constructed in Sept./98 by E.W. Shineman, Jr. for myself. Thanks Ed.
Update: Sept. 14, 2001
After investigating the Franklin 16x16 squares, I did the same tests on this one. Here are the results of that test.
If there are 16 cells in the pattern, they sum to S. If there are only 4 cells to a pattern, their sum is S/4, and 8 cell patterns produce S/2.
The word All with no qualifier means that the pattern may be started at ANYof the 256 cells of the magic square.
| All rows of 16 cells. All columns of 16 cells. All rows of 8 cells starting on EVEN columns All columns of 8 cells starting on rows 8 & 16 All rows of 4 cells starting on EVEN columns All columns of 4 cells starting on rows 2 & 10 All rows of 2 cells starting on EVEN columns All 16 cell diagonals All 2x2 square arrays Corners of all even squares All 16 cell small patterns (fully symmetrical within a 6x6 or 8x8 square array) All 16 cell midsize patterns (fully symmetrical within a 10 or 12 square array) All 16 cell large patterns (fully symmetrical within a 14 or 16 square array) All horizontal 2-cell segment bent-diagonals All vertical 2-cell segment bent-diagonals, R, L starting on ODD rows All vertical 2-cell segment bent-diagonals, L, R starting on EVEN rows All horizontal 4-cell segment bent-diagonals starting in column 4, 8, 12 and 16 All vertical 4-cell segment bent-diagonals starting in column 2, 6, 10, 14 NO 8-cell segment (regular) bent-diagonals All knight-move horizontal 8-cell segment, bent-diagonals All knight-move vertical 8-cell segment, bent-diagonals |
See more on the Franklin page
Shineman's Magic Diamonds
Constructed by E. W. Shineman, Jr. , treasurer, to commemorate his company's 75 th (Diamond) Anniversary in 1966 . It contains 5 special numbers.
75 The anniversary.
18 & 91 1891 The year the company was founded.
206 Net sales in 1966 (millions of dollars).
244 Net earnings (cents per share).
24 combinations of 4 numbers sum to 1966 .
Also constructed by E. W. Shineman, Jr., this in 1990 for his 75 th birthday.
This one contains 11 special numbers.
75 Age on reaching diamond anniversary.
33 (1933) Year graduated from high school.
4-9-15 Date of birth.
1878 Year father was born.
22 Age when graduated from college
86 (1886) Birthyear of Father-in-law & mother-in-law
1885 Year mother was born.
63 & 68 (1963 &1968) Years of career milestones
24 combinations of 4 numbers sum to 1990 .
Order-8 with embedded star
This order-8 magic square is composed of four order-4 pure magic squares. The embedded magic star is index # 16 and is super-magic (the points also sum to the constant 34).
The index numbers of the magic squares are:
upper left # 390 equivalent upper right # 142 the basic solution
lower left # 724 equivalent lower right # 271 equivalent
The equivalent solutions require rotations and/or reflections in order to match the basic solution # shown.
Fr�nicle, assigned these magic square index numbers about 1675, when he published a list of all 880 basic solutions for the order-4 magic square. For more information, see
Benson & Jacoby, New Recreations with Magic Squares, Dover Publ., 1976.
The magic star index numbers were designed and assigned by me and a full description appears at Magic Star Definitions.
Thanks to Arto Juhani Heino who e-mailed me this pattern on Jul. 15/98.
Order-4 square to order-8 star.
This diagram shows some relationships between an order-8B magic star and an order-4 magic square.
Both patterns are basic solutions. The star is index # 57 (Heinz) and the square is index # 666 (Fr�nicle).
Thanks to Arto Juhani Heino for this design.
Franklin 8 x 8 Magic Square
This magic square was constructed by Benjamin Franklin (1706-1790).
It has many interesting properties as illustrated by the following cell patterns.
Because the square is continuous, (wraps around), each pattern is repeated 64 times ( 8 in each direction).
It also has the property that any 4 by 4 square sums to the constant, 2056, as well as some other combinations.
It has many interesting properties as illustrated by the following cell patterns.
However, once constructed, it is a visually pleasing and mathematically intriguing puzzle. The uniqueness of the 49 cell magic square lies in its properties and symmetries. It is a symmetrical square, both vertically and horizontally, as well as diagonally. This symmetry adds to its aesthetic appeal and makes it a fascinating mathematical object to study. The 49 cell magic square has been a subject of interest for mathematicians and puzzle enthusiasts throughout history. Its construction and properties have been extensively studied, and numerous variations and extensions of this square have been explored. Overall, the 49 cell magic square is a visually striking and mathematically stimulating puzzle. Its unique properties and symmetrical design make it a fascinating subject for further exploration and study..
Reviews for "Unveiling the Mathematical Algorithm Behind a 49 Cell Magic Square"
1. John - 1/5 - I found the "49 cell magic square" to be incredibly confusing and unenjoyable. The rules were not clearly explained, and I struggled to understand how the magic square was supposed to work. Additionally, the game lacked any sort of excitement or challenge, as it felt like simply filling in numbers randomly. Overall, I was thoroughly disappointed with this game and would not recommend it.
2. Sarah - 2/5 - While I appreciate the concept behind the "49 cell magic square," I found the execution to be lacking. The game quickly became repetitive and monotonous, as the same patterns seemed to appear over and over again. There was a lack of variety and excitement, making it difficult to stay engaged. I also had trouble with the controls and found them to be clunky and unresponsive. Overall, I was not impressed with this game and would not play it again.
3. Alex - 1/5 - I was extremely disappointed in the "49 cell magic square" game. The graphics were dull and unappealing, and the gameplay was repetitive and uninteresting. There was no clear objective or goal, and I quickly lost interest in trying to complete the magic square. The lack of any sort of challenge or excitement made it feel like a waste of time. I would not recommend this game to anyone looking for a fun and engaging puzzle experience.
4. Emily - 2/5 - I had high hopes for the "49 cell magic square," but unfortunately, it did not live up to my expectations. The game felt overly complicated and confusing, with unclear instructions and rules. I found myself constantly questioning whether I was completing the square correctly, which took away from the overall enjoyment. Additionally, the game lacked any sort of visual appeal, with plain and uninteresting graphics. Overall, I was disappointed with this game and would not recommend it to others.