Magic Squares of Order 4n
Here we will generalize the method used to generate fourth-order magic squares to generate squares of order 4n. That is, squares for which the number of cells on a side is a multiple of 4.
We can use almost the same process as we used to generate a fourth-order magic square to create any 4n
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We proceed as we did in the 4
![]() Now we begin in the lower right-hand corner and work our way back using the numbers 1,4,5,8,10,11,... and 64. We put these number in the cells which originally had the diagonal lines starting with 1 in the lower right-hand corner. Our finished product looks like this:
![]() The magic number is 260 = 8(82 + 1)/2. You can check all the rows, columns and diagonals to see that they each have a sum of 260.
Again, we could get new squares by carefully rearranging the rows and columns of the one above. As before, we must take care to make sure that the numbers that are associated together in all rows and columns stay together. A random exchange of rows or columns will not necessarily result in a new magic square. The rows and columns may sum to 260, but the diagonals may not. Below are two rearrangements of our original 8
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A 16
![]() We again start in the upper left-hand corner and fill in the blank squares as before -- except now we use the integers from 1 to 256 (162). When we finish in the lower right-hand corner, we then proceed backwards filling in the squares that were not used before. The following square is complete with the colored squares being the ones in which diagonal lines had appeared. Notice that the rows, columns and diagonals sum to the magic number 16(162 + 1)/2 = 2056.
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The reader can now construct a 12
![]() Use the numbers from 1 to 144. Your magic number will be 12(122 + 1)/2 = 870. Have fun!
There is another method, similar to the above, that can be used to generate 4n
First we partition the n
![]() We can then proceed in one of two ways. Method 1. Number the cells 1 through n2 (in our case, it would be 1 through 144). Leave the numbers as they are in the five squares.
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Then start at the bottom as we did in our first way of generating 4n
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If one is fast at calculations, one could compute the values that goes in each cell. If we let r represent the row number and c represent the column number for each cell in the square. Then the values that would go in the cells in the square blocks would be (r - 1)*n + c and the values that go in the rectangular blocks would be (n2 + 1)-[(r - 1)*n + c]. For example, in our 12 Method 2. If we enter the numbers in every cell in order, we get
![]() We then remove the rectangular blocks, B2, B4, B6, B8. Now we take B2 and rotate it 180º and place it where B8 was and rotate the original block B8 180º and put it where B2 was. We then do the same with blocks B4 and B6. We rotate each 180º and put it where the other was and we again get the square
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| Magic Squares -- Introduction | Magic Squares of Order 3 | 4th Order Magic Squares | Magic Squares of Odd orders | Magic Squares of Even orders--4n+2 | Magic Squares which are not Normal | Ben Franklin's Magic Square | Magic Squares within Magic Squares | |