Pan Square
Producing a perfect magic square is like filling in an easy Sudoku puzzle
Don’t understand me wrong. Structuring and enumeration of magic squares needs a scientific (mathematical) approach. But producing perfect magic squares is like filling in an easy Sudoku puzzle. There is even an important link between the magic square and the Sudoku. See for example the 5x5 panmagic square.
In a pure 5x5 (pan)magic square you fill in the digits 1 up to 25 or 0 up to 24 (= easier to calculate with). Each digit can be split up in a column coordinate and a row coordinate.
5x column coordinate + 1x row coordinate = Digit
0 0 0 0 0 0 1 2 3 4 10 0 1 2 3 4
1 1 1 1 1 0 1 2 3 4 10 5 6 7 8 9
2 2 2 2 2 0 1 2 3 4 10 10 11 12 13 14
3 3 3 3 3 0 1 2 3 4 10 15 16 17 18 19
4 4 4 4 4 0 1 2 3 4 10 20 21 22 23 24
10 10 10 10 10
The magic sum is the (odd) size of the magic square multiplied by the middle digit: 5 x 12 = 60. You can also calculate the magic sum by adding the (average) sum of the column coordinates, multiplied by the size of the square, to the (average) sum of the row coordinates: 10 x 5 + 10 = 60. If you ensure that all the column coordinates from 0 up to 4 and all the row coordinates from 0 up to 4 are in each row/column/pandiagonal, than you will produce everywhere the magic sum of 60. The trick is to create two Sudoku’s. The first Sudoku contains the column coordinates and the second Sudoku contains the row coordinates.
Fill in the first row of the 1st Sudoku. There are 24 combinations of digits which lead to unique basic solutions (01234, 01243, 01324, 01342, 01423, 01432, 02134, 02143, 02314, 02341, 02413, 02431, 03124, 03142, 03214, 03241, 03412, 03421, 04123, 04132, 04213, 04232, 04312, 04321). Fill in row two up to five by moving the first row each time 2 places to the left.
1st Sudoku
0 1 2 3 4 0 1 2 3 4 0 1 2 3 4
0 1 2 3 4 0 1 2 3 4 0 1 2 3 4
0 1 2 3 4 0 1 2 3 4 0 1 2 3 4
0 1 2 3 4 0 1 2 3 4 0 1 2 3 4
0 1 2 3 4 0 1 2 3 4 0 1 2 3 4
Adopt as first row of the 2nd Sudoku, the first row of the 1st Sudoku. Fill in row two up to five by moving the first row each time 2 places to the right.
2nd Sudoku
0 1 2 3 4 0 1 2 3 4 0 1 2 3 4
0 1 2 3 4 0 1 2 3 4 0 1 2 3 4
0 1 2 3 4 0 1 2 3 4 0 1 2 3 4
0 1 2 3 4 0 1 2 3 4 0 1 2 3 4
0 1 2 3 4 0 1 2 3 4 0 1 2 3 4
Take a digit from the 1st Sudoku multiplied by 5 and add (1x) the digit from the same cell of the 2nd Sudoku; add 1 to each cell.
5x digit + 1x digit = +1 = panmagic 5x5
0 1 2 3 4 0 1 2 3 4 0 6 12 18 24 1 7 13 19 25
2 3 4 0 1 3 4 0 1 2 13 19 20 1 7 14 20 21 2 8
4 0 1 2 3 1 2 3 4 0 21 2 8 14 15 22 3 9 15 16
1 2 3 4 0 4 0 1 2 3 9 10 16 22 3 10 11 17 23 4
3 4 0 1 2 2 3 4 0 1 17 23 4 5 11 18 24 5 6 12
Notify that the connection between the column coordinates and the row coordinates lead to unique column/row-combinations, so you can find all the digits 0 up to 24 in the square.
1st Sudoku (= column) 2nd Sudoku (= row) column/row-combinations
0 1 2 3 4 0 1 2 3 4 0,0 1,1 2,2 3,3 4,4
2 3 4 0 1 3 4 0 1 2 2,3 3,4 4,0 0,1 1,2
4 0 1 2 3 1 2 3 4 0 4,1 0,2 1,3 4,2 0,3
1 2 3 4 0 4 0 1 2 3 1,4 2,0 3,1 4,2 3,0
3 4 0 1 2 2 3 4 0 1 3,2 4,3 0,4 1,0 2,1
Do you want more explanation and/or learn step by step how to produce perfect magic squares for every order, go to my website www.perfectmagicsquares.com
About the Author
My name is Arie Breedijk. I was born in the Netherlands in 1966. I am married and have two daughters (born in 2001 and 2004). I work for the Dutch Department of Education (initially as an auditor and now as budget administrator). I have no mathematical background. One of my hobby’s is magic square.
Do you want more explanation and/or learn step by step how to produce perfect magic squares for every order, go to my website www.perfectmagicsquares.com
how many square inch of sheet tin are required to make the pan?
a cubic ft.of water weighs about 62.4 lb. What must be the diameter of a cylindrical pail 1 ft high in order for it to hold the water from 25 lb of ice?how many square inches of sheet tin are required to make the pan?
62.4 lb.s of water/cu.Ft.
25 / 62.4 = 0.4 Cu ft of melt water from your ice.
Your circular pail is 1 foot high, so the area of the circular base is .4 Sq.ft.
You need the area and circumference of a circle with that area
to find the total area of metal needed.
The circumference is Pi x D
The area of the circle Is Pi x (D/2) squared.
The total area is the base + the wall (1 Ft. x C)
Now go finish your homework.
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