www.MAZES.com *
Magic Squares Index *
Properties of a Perfect Square *
Almost Perfect Magic Trick *
Perfect Magic Square Trick for Even Sums *
Odd Perfect Squares? *
Usage Agreement & Donations Deeply Appreciated
Magic Square Magic Trick Secrets Revealed
All of the directions for performing this magic square magic trick are
based on memorizing one or both of these first sample magic squares.
You can perform the Almost-Perfect Magic Square Trick with just the 30 square
but in order to perform the Perfect-Magic-Square Trick, you need the 32 also.
You don't need to memorize these particular magic squares.
You could print them on the barrel of the magic marker you'll use doing the trick.
Click Reload to see different sample magic squares, 64 in all.
Perfect Magic Square
with Magic Sum = 30... using all of the numbers from zero through fifteen.
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Perfect Magic Square
with Magic Sum = 32... using all numbers from zero thru sixteen (except 8).
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Did you notice that you can turn the perfect 30 square into a perfect 32 square
by simply adding one to the each of the eight biggest values in the 30 square?
(Just add one to every value greater than 8.)
Properties of a Perfect Magic Square
(with examples from the Perfect Magic 30 Square above)
- The numbers in each row add up to the magic sum: (11+6+13+0=30)
- The numbers in each column add up to the same sum: (11+12+2+5=30)
- The numbers in both diagonals add up to that sum: (11+1+4+14=30)
- The four corner numbers add up to the same value: (11+0+5+14=30)
- The four inside numbers add up to the magic sum: (1+10+15+4=30)
- The four numbers in each corner add up to the sum: (11+6+12+1=30)
- Every 2x2 block of numbers add up to the same sum: (6+13+1+10=30)
- Even if you wrap the magic square around a cylindrical tube,
every 2x2 number block will add up to that sum: (7+12+9+2=30)
- Each of the six minor diagonals add up to the sum: (13+7+2+8=30)
- If you cut the row of numbers off the bottom and paste it onto the top,
the square is still perfect
- If you cut the column off the left side and paste it onto the right side,
the square is still perfect
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You may have seen a magician ask an audience member for a number, then proceed to create a magic square on the spot, a square in which every row, every column, and both
diagonals added up to the audience member's chosen number. It's likely that the magician pointed out, or let an audience member discover, other special qualities, such as the
four corners adding up to the sum, the four numbers in any corner having the same sum, and a few other features. What the magician probably doesn't point out to you is the fact
that the four middle squares on the left or right side of the square probably do not add up to the magic sum. That's because he's probably using a variation of the trick that I
describe here:
Almost-Perfect Magic Square
with
Magic Sum larger than 30
Subtract the given values from the magic sum in the four squares shown here.
| 11 | 6 | Sum-17 | 0 | | Sum-18 | 1 | 10 | 7 | | 2 | Sum-15 | 4 | 9 | | 5 | 8 | 3 | Sum-16 |
If you add up just the numbers shown here,
and ignore the 'Sum' variable,
every column, row, diagonal, and selected boxes add up to zero?
That's why this method works.
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Almost-Perfect Magic Square
with
Magic Sum less than 30
Subtract the given values from the magic sum in the four squares shown here.
| 11 | 6 | 13 | Sum-30 | | 12 | Sum-29 | 10 | 7 | | Sum-28 | 15 | 4 | 9 | | 5 | 8 | Sum-27 | 14 |
One or more of these calculated numbers will be negative.
There is no way to create a magic square
with magic sum less than 30
without using at least one negative number.
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In other words, all you have to do is memorize the perfect square at the top of this page, and you can use this magic trick to make any almost-perfect magic square.
If you're making a magic square with a sum greater than 30, just add the difference between your magic sum and 30 to the four biggest numbers in the perfect square above (for
example, to make a 35 square, add 5 to the largest number in each column; and if you're making a magic square with a sum less than 30, just subtract the difference
between 30 and your magic sum from the four smallest numbers in the above square (for example, to make a 26 square, subtract 4 from the smallest number in each column).
Here is a Magic Square that combines both of these methods into one square. Since you might want to print this particular magic square onto the barrel of your magic marker
pen (you could print it onto a label, then paste the label around the barrel of the pen), I have printed it smaller and summarized the directions:
You could sell magic markers with this printed on the barrel:
| 11 | 6 | 13+d | 0-d | | 12+d | 1-d | 10 | 7 | | 2-d | 15+d | 4 | 9 | | 5 | 8 | 3-d | 14+d |
| The Almost-Perfect Magic Square Magic Trick ...
- calculate: d = magicsum - 30 (difference between desired sum and 30)
- desired sum > 30? add the difference to the four large numbers
- desired sum < 30? subtract the difference from the four small #s
copyright 2009 by John & David @ www.MAZES.com/magic-squares
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If you sell pens with this square and directions printed on the barrel,
please send us at least 25% of the gross selling price as our fair share.
Click reload ... there are 64 different square/direction sets you could sell.
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If you think that this particular magic square might be a hard one to memorize, don't worry, there are 64 different magic squares that you can choose from for this trick.
Just click REFRESH or RELOAD and you'll get a different magic square.
You actually don't need to memorize the perfect square at the top of this page. Instead, print the magic square or squares onto mailing labels and paste them onto the marker
pens that you'll be using when you perform this trick. This way, you can sneak peaks at the grid on the pen while you are amazing your audience.
(There are actually 384 different perfect squares, but only 64 of them can be used for this magic trick. If you would like to see all 384 perfect squares,
click here.
Using this trick more than once with the same audience?
One of the problems with this trick is that if you make more than one magic square for the same audience, they might figure out your trick. This is a good reason to reload
this page to get different magic squares to put on the barrels of your pens. If you have a different magic square on each pen, your observers will be less likely to figure out your trick.
Of course, you might want to explain your trick as a method of selling pens after the show, so if you have multiple pens with many different squares, you might get increased
sales in your gift shop area. If you do decide to reveal this magic trick, you could then turn around and explain that this trick for sale is the one that most magicians use, but
you have an even better trick, then proceed to create a Perfect Magic Square with an even sum as described below, but don't necessarily teach that trick, though you could if you wish.
Yes, there is a trick that you can use to create Perfect Squares that have any even magic sum, and it is also based on memorizing the perfect magic square at the top of
this page (and possibly memorizing the second one, too, or remembering the trick about adding 1 to each of the eight largest numbers).
Perfect Magic Square with Magic Sum
equal to two more than a multiple of four
34, 38, 42, 46, 50, 54, 58, 62, etc.
26, 22, 18, 14, 10, 6, 2, etc.
- Calculate: diff = Sum - 30
(subtract 30 from the desired magic sum to get the difference.)
(If desired sum is less than 30, this result will be negative)
- Calculate: q = diff / 4
(Divide that difference by four.)
- Add that 'q' amount to every value in the perfect 30 square.
(If your desired magic sum is less than 30, the 'q' value will be negative.)
- For example, to make a perfect 34 magic square, add 1 to every value in the 30 square.
To make a perfect 42 square, add 3 to every value.
| 11+q | 6+q | 13+q | 0+q | | 12+q | 1+q | 10+q | 7+q | | 2+q | 15+q | 4+q | 9+q | | 5+q | 8+q | 3+q | 14+q |
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Perfect Magic Square with Magic Sum
equal to an exact multiple of four
36, 40, 44, 48, 52, 56, 60, 64, etc.
28, 24, 20, 16, 12, 8, 4, etc.
- Calculate: diff = Sum - 32
(subtract 32 from the desired magic sum to get the difference.)
(If desired sum is less than 32, this result will be negative)
- Calculate: q = diff / 4
(Divide that difference by four.)
- Add that 'q' amount to every value in the perfect 32 square.
(If your desired magic sum is less than 32, the 'q' value will be negative.)
- For example, to make a perfect 36 magic square, add 1 to every value in the 32 square.
To make a perfect 40 square, add 2 to every value.
| 12+q | 6+q | 14+q | 0+q | | 13+q | 1+q | 11+q | 7+q | | 2+q | 16+q | 4+q | 10+q | | 5+q | 9+q | 3+q | 15+q |
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You could also sell magic markers with this PERFECT method printed on the barrel:
(though I'm guessing that you'd want to sell the Almost-Perfect pens and use this one as an unexplained trick)
Sum = ... 22,26,30,34,38,42 ...
| 11+q | 6+q | 13+q | 0+q | | 12+q | 1+q | 10+q | 7+q | | 2+q | 15+q | 4+q | 9+q | | 5+q | 8+q | 3+q | 14+q |
Perfect 30 Magic Square
| The Perfect Magic Square Magic Trick ...
- calculate: d = magicsum - 30 (left square) or
calculate: d = magicsum - 32 (right square)
(difference between desired sum and base sum)
- calculate: q = d / 4 (divide that difference by 4)
- add q to every value in the left or right square
- New Magic Sum will be equal to original sum plus four times q
copyright 2009 by John & David @ www.MAZES.com/magic-squares
| Sum = ... 24,28,32,36,40,44 ...
| 12+q | 6+q | 14+q | 0+q | | 13+q | 1+q | 11+q | 7+q | | 2+q | 16+q | 4+q | 10+q | | 5+q | 9+q | 3+q | 15+q |
Perfect 32 Magic Square
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If you sell pens with this square and directions printed on the barrel,
please send us at least 25% of the gross selling price as our fair share.
Click reload ... there are 64 different square/direction sets you could sell.
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If you sell pens containing these brief directions, you might want to also give them a printout of this web page with purchase.
(If you sell the 'Almost-Perfect' pen, you might not want to print out the Perfect part of this page.)
The Hundred Dollar Challenge
Can you make a perfect magic square with an ODD Magic Sum?
It is mathematically impossible to create a Perfect Magic Square with an odd magic sum, so if you wish, you could hold up a hundred dollar bill and offer it to the first person
in the audience who brings you a perfect magic square made with all integers that has a magic sum equal to 31, 33, 35 or 37 (for example), by the end of the night's show.
If you would like to see proof that a Perfect Magic Square cannot be made with an odd magic sum, please see
www.mazes.com/magic-squares/OddPerfectSquares.htm.
If someone comes up after the show with what they claim is a perfect odd square, you should be able to quickly prove that it is not perfect by examining the middle 2x2
box of numbers on each side of the square. This is where the mistakes are usually to be quickly found, but in any case, I guarantee that you will find that the alleged odd perfect
square is not perfect, so this is a safe challenge if you enjoy challenging your audiences. Be sure that your printed information form about your $100 challenge contains the list of
properties from this page, so that you'll be able to point out their mistake when you find it. The other side of the challenge sheet could be an enlarged $100 bill (be sure that it's
a big enough enlargement to not violate the law (we are not allowed to copy United States currency at size that is reasonably close to 100% so be safe and at least double the size).
How to use the Almost-Perfect Marked-Magic-Marker to make Even Perfect Squares
Here is the diagram for the Marked-Magic-Marker again, complete with the directions for the Almost-Perfect Magic Trick.
You could sell magic markers with this printed on the barrel:
| 11 | 6 | 13+d | 0-d | | 12+d | 1-d | 10 | 7 | | 2-d | 15+d | 4 | 9 | | 5 | 8 | 3-d | 14+d |
| The Almost-Perfect Magic Square Magic Trick ...
- calculate: d = magicsum - 30 (difference between desired sum and 30)
- desired sum > 30? add the difference to the four large numbers
- desired sum < 30? subtract the difference from the four small #s
copyright 2009 by John & David @ www.MAZES.com/magic-squares
|
If you sell pens with this square and directions printed on the barrel,
please send us at least 25% of the gross selling price as our fair share.
Click reload ... there are 64 different square/direction sets you could sell.
|
Here is how to use the Almost-Perfect Magic Square diagram to make Perfect Squares:
- Ignore the +d's and the -d's
- Ask someone to pick an even number. If you want to avoid fussing with negative numbers, ask for an even number greater than 28.
If you do not want to have a zero in the finished square, ask for an even number greater than 32.
- Subtract 30 from the desired Magic Sum. If the desired magic sum is less than 30, your temporary result will be negative.
For example: 42-30=12, or 40-30=10, or 20-30=-10
- Divide that temporary result by four. You should end up with either a whole number, or a number ending in .5 (one half).
For example: 12/4=3, or 10/4=2.5, or -10/4=-2.5
- If you ended up with a whole number, add that value to every number in the Magic Square diagram on the Marked Magic Marker.
- If you ended up with positive number ending in .5 (one-half):
- Truncate the number (ignore the .5)
- Add the truncated value to the two smallest numbers in every row (red and violet)
- Add the truncated value PLUS ONE to the two largest numbers in every row (blue and green)
For example, if you were making a 40 perfect square, the top row numbers would be:
11+2+1 = 14, 6+2 = 8, 13+2+1 = 16, 0+2 = 2
- If you ended up with a negative number ending in .5 (one-half):
- Ignore the minus sign and truncate the number
- Subtract the truncated value from the two largest numbers in every row (blue and green)
- Subtract the truncated value AND subtract one from the two smallest numbers in every row (red and violet)
- For example, if you were making a 20 perfect square, the top row numbers would be:
11-2 = 9, 6-2-1 = 3, 13-2 = 11, 0-2-1 = -3
There is a method that is harder than the methods described on this page that can be used to create perfect magic squares that contain three different numbers that are picked by
three different members of your audience. If you want an easy method, use one of the tricks on this page. If you want an interesting trick that very few magicians would even care
to attempt, then come back after we've had time to create that web page. You can see a hint about how this trick works by looking at our Odd Perfect Square page
(www.mazes.com/magic-squares/OddPerfectSquares.htm).
The author puts a great deal of time and effort into creating these web pages. You are welcome to use the information on this page in any
non-profit way that you wish, but if you use our information in any way as part of a profit operation, you are required
to send an appropriate donation as a large token of your appreciation. (Please be as generous as you can.) If you use this method as part
of a regular act, or if you use it on any semi-regular basis, you should make donations on a regular basis.
In any case, if you enjoy what you learn from my web pages, please make a donation. Web pages are our main source of income, and your donations help keep them free for all to view.
Donations may be mailed to:
Arleen Knoderer, John, and David Hrbacek
P O Box 235
Sulphur Springs, AR 72768-0235 USA
Or you can make donations via PayPal by clicking here
