For some reason almost two years went by at LOADSTAR before I had another program (on LOADSTAR #72). It might have been LOADSTAR 128 taking up my time; it might have been that I was learning how to program better and use the programming tools we were publishing. Jeff Jones and Scott Resh had come on board and Jeff was teaching himself assembly language, which Scott was a wizard at. I remained the BASIC programmer.

I just looked at the program and it asks if you want to generate some 5x5 magic squares or read a tutorial that explains how they're made. The Read It below gives you an algorithm for making 5x5 squares, but it's completely different from the one used by the program. I don't know why I did it that way, but you get two algorithms for the price of one. The

*staircase method*(below) is probably the easiest to remember and dazzle your friends with. Ask them to make a 5x5 square where every row, column and diagonal adds up to the same number and it's doubtful they'll be able to do it. But memorize the sequence below and you can come up with as many as you want while they watch.
Here is the Read It from 1990.

I remember the first time I heard of Magic Squares
and how they’re made. It was in a child’s puzzle book and the problem was to
take the numbers from 1 through 9 and place them on a Tic-Tac-Toe grid so that
each possible Tic-Tac-Toe added up to the same number, 15. Three rows, three
columns and two diagonals.

It wasn’t easy. Mainly I remembered the
method that the book gave for solving the problem, and for many years I had a
smug feeling that I knew everything about Magic Squares. This is the method I
learned.

(1) Enter the numbers in order.

**1 2 3**

**4 5 6**

**7 8 9**

(2) Swap the corners.

**9 2 7**

**4 5 6**

**3 8 1**

(3) Then move every number clockwise around
the center number, 5.

**4 9 2**

**3 5 7**

**8 1 6**

Done! I didn’t even think about higher order
Magic Squares (4x4, 5x5, 6x6, etc.) until I picked up a book by Jim Moran
called THE WONDERS OF MAGIC SQUARES. It seemed impossibly complex at first, but
it soon began to sink in and before long I was saying “Aha!” on practically
every page.

Magic Squares, and how to make them, have
been studied by mathematicians and puzzlers for more than 2000 years. They
wanted “algorithms”, or step-by-step methods, for constructing them. Moran’s
book lists fifteen or so methods, one of which I adapted for the Run It program
for this feature.

The earliest mention of Magic Squares was in
China and India, where they were often carved on stone houses, or on pendants
and amulets. “Magic” is a very good name for these squares, at least in a
historical sense.

Benjamin Franklin was a Magic Square freak,
and often bragged that he could construct 8x8’s as fast as he could write the
64 numbers. Obviously, he knew an algorithm or two.

Generally, Magic Squares are divided into two
categories, odd-order and even-order. I’m going to talk about odd-order squares
(3x3, 5x5, 7x7, 9x9, etc.) only, because their algorithms will work for any
size, while even-order squares require different algorithms, depending on their
size.

Each size of square has a constant, which is
the number that all rows, columns and diagonals add up to. The formula for this
constant is

Constant = (n * (n*n+1))/2

One Magic Square is actually four if you
think about it, since you can simply rotate the whole square three times. But
there are also many other ways to completely rearrange the numbers to form
totally new squares. Would you believe that there are an estimated 275 MILLION
(!) possible 5x5 squares?

The algorithm used in the Run It program is a
fairly sophisticated method, developed by Jim Moran. It’s based on an early
method that involves constructing two squares that aren’t magical, then adding
them together to make a square that is. Run the program to see this method explained.

Most of the other methods involve what is
known as the “staircase method”. This method will create one Magic Square,
which can be changed into other squares by a little manipulation, such as rotation.

Here are the rules for the “staircase method”
for a 5x5.

(1) Place the first number, 1, in the middle
box of the first row.

(2) Place each succeeding number in the box
diagonally upward to the right, unless this box is occupied by a previously
placed number.

(3) When the box is “blocked”, place the next
number in the box directly below the last number placed.

(4) Keep going till the square is filled.

The first question is, “Where do I put 2?
Upwards to the right of 1 is off the grid!” This is what you do. Whenever a box
you want to go to is off the grid, imagine that there’s another grid right next
to the main one. In the rules above, 2 would be placed in the 5th row, 4th
column of that imaginary grid. Instead, place the number 2 in the 5th row, 4th
column of your main grid. If it’s blocked, then follow rule (3).

This means that 2 goes in row 5, column 4; 3
goes in row 4, column 5; 4 goes in row 3, column 1, etc. Here is the final grid
constructed by this method.

**17 24 1 8 15**

**23 5 7 14 16**

**4 6 13 20 22**

**10 12 19 21 3**

**11 18 25 2 9**

The first blocked move came when you tried to
place the 6. The 1 was in the way. So 6 goes below 5. Study this till you see
the pattern.

There are several variations on the “staircase”
method. Here are the rules for one variation.

(1) Place the first number, 1, in the box to
the right of the center box.

(2) Fill in the boxes by advancing diagonally
upward and to the right. (Same as the other method.)

(3) In case of a blocked move, place the number in
the box TWO PLACES TO THE RIGHT of the last number placed.

A blocked move will occur every five numbers.
The resulting square will look like this.

**3 16 9 22 15**

**20 8 21 14 2**

**7 25 13 1 19**

**24 12 5 18 6**

**11 4 17 10 23**

Notice that in both of the squares we’ve
made, there is a pattern formed by numbers that add up to 26. Number 1 is
diametrically OPPOSITE to 25. 2 is OPPOSITE to 24. 3 is OPPOSITE to 23, and so
on.

Another way of looking at this is that a
Magic Square is “balanced”. If we were to place weights corresponding to the
numbers in each box and attach a chain to the middle of the middle box, we
could lift the whole square and it would balance about that point. This is only
true for squares made with the “staircase” method.

There are a couple of quick ways to make
variations on a square, once you’ve made it (besides rotation).

(1) Swap Column 1 with Column 5. Then swap
Row 1 with Row 5.

(2) Swap Rows 1 and 2 with Rows 4 and 5. Then
swap Columns 1 and 2 with Columns 4 and 5.

There are several more algorithms for constructing
Magic Squares which I don’t have the room for, including variations on the algorithm
used in the Run It program.

I heartily recommend Jim Moran’s book, THE
WONDERS OF MAGIC SQUARES, published by Vintage Books, a subsidiary of Random
House, New York, 1981. It contains more than you’ll ever want to know about
Magic Squares.

Maurice Kraitchik’s MATHEMATICAL RECREATIONS,
published by Dover Books, is also a great source for Magic Square lore.

Back to 2015. I think a program that generates thousands of magic squares is pretty neat. It shows what a computer is good at if programmed correctly. But I can imagine you wondering if I will ever get to any really useful programs in this project.

The answer is no. I never programmed anything you might call useful. But please stick with me and I think you will find some very interesting stuff coming up. I was a puzzle magazine nut in those days and the C-64 was the perfect platform for turning pencil puzzles into interactive, ego-less, record-keeping tools for solving, or in some cases, generating puzzles. Really! It gets better.

In fact, tomorrow night's program could be considered useful if you are like me and take a half dozen pills every morning and night. Tune in and drop out.