 # Magic Squares - FAQs

(Please note that Questions are shown in italics and Answers in normal text.)

1. Graphic Magic Squares

I have observed that some squares exhibit another dimension of beauty when expressed in a certain manner.

If instead of placing the numbers (1 - n**2) in the cells of a matrix, we place a 'dot', say, in the center of each cell and then start from the dot that is associated with '1' and draw a straight line to the dot associated with '2' and then draw a straight line to the dot associated with '3' and so on to the dot associated with 'n**2', and finally draw a straight line to the dot associated with '1', we may have an interesting and sometimes beautiful graphic pattern.

Has this quality been observed before and if so does this 'class' of magic squares, exhibiting this quality, have a name ?

You will find a similar idea in a book called "The Wonders Of Magic Squares" by Jim Moran, as referenced in my own book, "Magic Squares".

Jim uses them as an aid in trying to ascertain whether two magic squares are identical or not, ignoring rotations and reflections.

I borrowed a copy via the public library system, here in England, but here are the bibliographical details, should you try to obtain a copy:

Moran, J. (1982) "The Wonders Of Magic Squares", New York, New York: Random House, Inc.

2. What does "straight away" mean?

Page 8 of your book on Magic Squares explains the simple formula for a 4 x 4. Step 1 of the formula tells you to write the first 12 number in "straight away", yet I have found no reference to the meaning of "straight away." Can you help with this?

What I should have said is something like this:

"The first (n squared - n) numbers in the magic square may be written in their usual positions, as defined by the basic Magic Square, where n is the number of cells on each side of the Magic Square."

3. How do you know what order to put the numbers in?

Do you just guess?

For most methods of creating Magic Squares, you do need to memorise a "basic" Magic Square first, so you are never just guessing.

Some methods mean that you can write in most of the numbers before you even start, which can be very fast and look very impressive, subject to a slight loss of aesthetic appeal, in my own opinion.

However, if you are willing to learn the basic Magic Square, and it's only 16 numbers for a 4 x 4 one when all is said and done, then you can complete the Magic Square in whatever order the spectator asks for. This is the basis for one half of The Amazing Magic Square & Master Memory Demonstration that I perform, originally marketed by the late Orville Meyer.

4. What formula does your calculator use to figure out the placement for numbers for the 4 x 4 Magic Squares?

I have seen formulae for Magic Squares with odd numbers of cells, but not for an even number of cells.

The formula that I use on my 4 x 4 Magic Squares comes from the booklet "The Amazing Magic Square & Master Memory Demonstration" by the late Orville Meyer.

The "basic" 4 x 4 Magic Square is the one shown at the top left of the above-mentioned page.

To make it add up to the chosen number, which must be greater than 34 since that is the "magic total" for a "basic" 4 x 4 Magic Square:

1. First subtract the 34 from the required total.
2. Divide the result of Step 1 by 4.
3. Add the integer part (i.e. ignore any remainder) of the result of Step 2 to all cells in the "basic" Magic Square.
4. In addition, add the remainder (i.e. ignore the integers before the decimal point) of the result from Step 2 to the four cells with the highest numbers.

As an example, suppose that a 4 x 4 Magic Square is asked for with a required total of 51. The result of Step 1 would be 17 (i.e. 51 - 34). The result of Step 2 would be 4 remainder 1.

The result of Step 3, therefore, would be that 4 (i.e. the integer part of the result of 17 divided by 4) is added to all 16 cells in the "basic" Magic Square.

Step 4 would result in an additional 1 being added to the cells in the "basic" Magic Square that contain 13, 14, 15 and 16; this is because the result of Step 2 gives a value of 4, remainder 1.

The above formula will work in a similar manner for any "doubly-even" Magic Square (i.e. one where the number of cells on each side of the Magic Square is divisible by both two and four without a remainder).

The above information is taken from my book on Magic Squares.

5. What is the most effective presentation?

Is it as a mathematical marvel or as a feat of memory?

I have always found that people find the mathematical aspects of a 4 x 4 Magic Square fascinating. This also applies to people who have seen or are already familiar with 3 x 3 Magic Squares, which are relatively well known. Some people have even suggested that I have memorised all of the different 4 x 4 Magic Squares that add up to 34 to 100. This would be an even better memory feat than I do already!

Although many magicians may say that the mathematical angle is only so-so, I do not believe that many magicians have ever realised how many combinations of four squares do actually add up to the chosen number - you will find the complete analysis in my book, " Magic Squares". If this fact were made more apparent, as I always do, then I think that the mathematical aspect of this effect would be even stronger.

However, it must be said that I usually perform Orville Meyer's "Amazing Magic Square & Master Memory Demonstration", which is, as the title suggested, a combination of a 4 x 4 Magic Square that adds up to a freely chosen number and a memory demonstration of 16 named objects. This either appeals to those who are impressed by the mathematics, or to those who are impressed by the memory demonstration, or, if you are lucky, to those who are impressed by both elements. I have found, however, that it can be a bit on the slow side as a stage / stand-up presentation, but, there again, I have usually performed it as a "party piece".

Occasionally, I have done just the Magic Square bit, but blindfolded, and this too can be very strong.

I also produce 4 x 4 Magic Squares that add up to somebody's age for their birthdays, just for a bit of fun, but I usually print these off my computer.

I suppose, in conclusion, that it depends on the circumstances, but I have never found the effect to be unappreciated by any audience, be it a few people round a desk at work, a party or as a formal presentation.

6. Partial Magic Squares

I am often asked if I can complete a Magic Square where only some of the cells are filled in, or whether there is some formula that can be used.

I have, unfortunately, not found a generic formula for this, so it very much depends on the Magic Square with which I am presented.

Therefore, if anybody can provide me with a method or algorithm for completing "partial" Magic Squares, I would be very grateful!

7. 3 x 3 Magic Square Using Specific Numbers

I need a 3 x 3 Magic Square that includes only these nine numbers: 1,2,3,5,6,8,9,10 & 12.

I don't believe that there is a 3 x 3 Magic Square that can be created using the nine numbers that you supplied.

The reason for this is that the sum of these nine numbers (i.e. 56) is not evenly divisible by three.

If you take the "basic" 3 x 3 Magic Square:

8 1 6
3 5 7
4 9 2

then you will see that each row adds up to 15 (obviously, or else it wouldn't be a Magic Square), and the sum of each row adds up to 45. This is clearly the same as the sum of the nine numbers from one to nine inclusive.

Since your nine numbers add up to 56, which, when divided by three, gives 18 remainder 2, you cannot create a true Magic Square using them. The closest you can get, I think, is one where some rows / columns / diagonals add up to 18 and some add up to 19.

Even if the nine numbers did sum to a total that was evenly divisible by three, I still don't think that guarantees that a "true" Magic Square can be created (i.e. one where all three rows, all three columns and both diagonals sum to the same total).

Some of this can be established using pure logic / deductive reasoning.

For example, assume that the nine numbers were 1, 2, 3, 5, 6, 8, 9, 10 and 13.

In this case, the sum of these nine numbers is 57, which is evenly divisible by three, so the "magic total" ought to be 19.

The number that would go into the centre cell cannot be 6, 8, 9, 10 or 13. This is because, if any one of these five numbers were in the centre cell, at least one of the remaining four of these numbers would create a total of 19 or more, and there would still be the third cell in that row / column / diagonal to fill.

The number that would go into the centre cell cannot be 1, 2 or 3. This is because, if any of these three numbers were in the centre cell, at least one of the remaining two of these numbers would create a need for the third number in that row / column / diagonal to be greater than 13.

Therefore, the centre cell must contain the number 5.

Next, let us try to place the number 13.

It cannot go in one of the four corner cells. This is because it would leave the numbers 8, 9 and 10 to be placed in a cell that is not "in line" with the 13 - i.e. one of the cells marked "x" in the following diagram:

13 . .
. 5 x
. x .

However, there are only two cells marked with an "x" and three numbers that need to be placed there.

Therefore, the number 13 must go on a centre cell on one side, as per:

. 13 .
. 5 .
. . .

This means that the number 1 must go opposite the 13, as per:

. 13 .
. 5 .
. 1 .

This leaves two more numbers on the first row that need to add up to 6, making that row's total equal to 19. However, having used the 1 and the 5, no such numbers remain.

Therefore, even though the numbers being used add up to an even multiple of three, no "true" 3 x 3 Magic Square may be formed.

(The above logic is taken from a short section near the beginning of an excellent book called "In Code" by Sarah & David Flannery - you can find details of it at Amazon.co.uk if you are interested.)

8. When Was The First Magic Square Produced?

I was wondering if you knew, or had any idea of, when and how the first Magic Square was produced?

As far as I know, the first recorded Magic Square is the Loh-Shu, which is a Chinese Magic Square dating from approximately 2,800 BC (see my page on Magic Square History for further information).

There are many different methods for creating Magic Squares, one of which is described on my page about Odd Magic Squares.

9. What Is A Total Diagonal Magic Square?

I need to know something (for school). What is a "total diagonal magic square"?

I have not come across the phrase "total diagonal magic square" before, but I am familiar with the term "panmagic squares" (otherwise known as "diabolic squares").

"Basic" Magic Squares must have the same " magic total" on each row, each column and both of the two "corner" diagonals (i.e. top left to bottom right, and top right to bottom left).

For example:
 16 2 3 13 5 11 10 8 9 7 6 12 4 14 15 1

In this Magic Square, you can see that each row adds up to 34, as does each column, as do the two corner diagonals.

"Panmagic" Magic Squares have an additional property that means that the "broken" diagonals also add up to the "magic total".

A "broken" diagonal, using the above Magic Square, might consist of, for example, the numbers 2, 10, 12 and 4. These four numbers clearly do not sum to 34.

However, if you transpose the Magic Square to make it "panmagic" or "diabolic", you might end up with the following Magic Square:
 9 6 3 16 4 15 10 5 14 1 8 11 7 12 13 2

In this instance, not only do all rows, all columns and both corner diagonals still sum to 34, but so do all of the "broken" diagonals:

• 6, 10, 11 and 7
• 3, 5, 14 and 12
• 4, 6, 13 and 11
• 14, 15, 3 and 2

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