How to Create a 86x86 Magic Square
with a Magic Sum of 318071
The method described on this page can be used to create
14x14, 18x18 and larger 4N+2
magic squares
Steps:
- Create an overlay 86x86 Magic Square grid consisting
only
of the
numbers 0, 1, 2 and 3
- Create a 43x43 Magic Square (half the size of our
final
86x86 square) that starts with the number 1
- Double the size of that 43x43 Magic Square,
quadrupling every
number, to create a 86x86 magic square that has each number in it
four times.
- Subtract the numbers in the overlay grid from the
quadrupled
numbers in the doubled square to make your new square.
Alternate directions: You may create a 43x43 Magic Square that starts
with zero in step 2 if you add the two squares together in step 4.
Step One: Create a 86x86 Magic
Square
overlay grid
Since we are going
to
create a magic square that contains each number four times, we
need to also create a magic square consisting only of the numbers 0, 1,
2 and 3, and we need to make sure that each 2x2 part of this overlay
grid contains all four numbers.
When I created my 6x6 Magic Square page, I knew that one method of
creating these squares was to simply double the size of the 3x3 magic
square, quadrupling its values, then use brute-force to find an overlay
pattern that satisfied my requirements (every number from 0 to 3
overlaying each identical number in the quadrupled square, in such a
way that every row, column and diagonal added up to 9). Imagine my
surprise when I found over a million different overlay squares. You can
see information about how I brute-forced that solution at www.mazes.com/magic-squares/magic-06.html.
Obviously, I do not want to brute-force a 10x10 or larger 4N+2 magic
square (there must be billions and billions of answers), so I decided
to seek out a method of creating an overlay grid that will work with
all 4N+2 squares larger than 6x6. I decided to use binary
numbers as part of my
search for a universal method. Binary (or base two) numbers, as many
math students know, use the digits 0 and 1 to represent any number. The
binary equivalents for the decimal (or base 10) numbers 0, 1, 2 and 3
are 00, 01, 10, and 11. Since half of each binary number represents
whether or not the 1 is present, and half of each number represents
whether or not the two is present, I decided to create two temporary
overlay grids, one for the value 1, and one for the value 2, so that
each square was independently magic.
Also, I wanted to make it easy for you, my visitor, to use the same
method with pencil and paper, so I wanted to use basically the same
grid twice, just doubling the "one" grid to get the "two" grid, with a
simple rotation to make sure that the sum of the values will be
different.
Create a 86x86 Magic Square of
all 1's
and 0's
Here are the steps to create the first grid (based on the pattern
'11100000000000000000000000000000000000000000001111111111111111111111111111111111111111'):
- Draw a 86x86 grid on your paper.
- Put the number 1 into the first column of the first 3
rows
of your grid
- Put the number 0 into the first column of the next 43
rows
of your grid
- Put the number 1 into the remaining spaces of the
first
column.
- Double-check
your work. You should now have 43 1's and 43 0's in the first column, which I have bolded so you can see it clearly.
As a further double-check, you have used the digits in the pattern '11100000000000000000000000000000000000000000001111111111111111111111111111111111111111')
- For the second column, put the opposite number ... if
there
is a 1 in the first column, put 0 in the second column, and vice-versa.
- Double-check.
The first column and the second column number should add up to 1.
- Copy the first column to the third column, and copy
the
second column to the fourth column.
- Turn the first two columns upside-down and copy them
into
the fifth and sixth columns.
- Copy the fifth and sixth columns to the rest of the
square.
- Double-check.
Each row, each column and both diagonals should have exactly 43 1's in
it.
Create a 86x86 Magic Square of
all 2's
and 0's
Here are the steps to create the second grid:
- Rotate the first grid 90 degrees counter-clockwise
and
- double all the numbers.
- You can see how the bolded left column has become the bolded bottom row after rotation.
Add Magic Square 1 to Magic
Square 2
And, of course, here are the steps to create the third
grid,
which is your overlay
grid:
- Add the two squares together
- You should end up with every 2x2 block containing all
four
digits, 0, 1, 2 and 3.
- (I have bolded alternating 2x2 blocks so that you can see this more easily.)
- and, if you have done the work properly, each row,
each
column, and
both diagonals add up to the same magic sum.
- Lay this grid aside for a
few minutes. You won't need it until after you've finished your
quadrupled grid
Step Two: Create a 43x43 Magic
Square
Next, you need to create a magic square that is half the
size
as your desired square. Since this half-sized square is odd (43x43), it
is easy to create. Just follow the method you see described at the
bottom of every odd square page at www.mazes.com/magic-squares.
Here is one such square, in which I started with the number 1 in the
middle of the top row, then moved one square up one and two squares to
the right after
each move, except when the next move would be blocked, in which case I
dropped down one for the next number.
I deliberately used a different magic square than the
one you
learned in school, just because I enjoy being different.
Now that you have an odd square, we are ready for the
next step
Step Three: Double the
size
of the 43x43 magic
square
and quadruple all the numbers
Draw a 86x86 grid, then, for each number in the 43x43 magic square:
- Multiply the number times four (quadruple it), and
- Write that number four times, into a 2x2 area of the
new
grid that corresponds to the position of the original number in the
43x43 square. (I have bolded alternating 2x2 blocks so you can see how the 43x43 square has expanded into a 86x86 square.)
Step Four: Put the Quadrupled and
the
Overlay grids together
This last part should be simple.
- If the smallest number in the quadrupled grid is 4,
use
subtraction: Quadrupled minus Overlay = Magic Square
The lowest number in the new Magic Square will be one, just like in the original 43x43 magic square.
- If the smallest number in the quadrupled grid is 0,
use
addition: Quadrupled plus Overlay = Magic Square
The lowest number in the new Magic Square will be zero, just like in the original.
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