Werner Miller has developed some great new approaches for magic squares, and has released them in two eBooks: E-Z Square 1 and E-Z Square 2.
E-Z Square 1
E-Z Square 1 (also available in German) focuses on a 5 by 5 magic square feat in which spectators call out various numbers which are written in the diagonal of the grid. The performer is then able to quickly place numbers in the remaining squares to give the same total in every row, column, and diagonal.There are two methods taught, as well as a bonus. In the first method, once the spectators give the numbers for the diagonal, you fill out the remaining squares quickly, and in what appears to be in random order (although, there is an actual method to the madness). Naturally, there is math involved, but this is where Werner Miller's true genius shows through; the only math required is simple addition and subtraction and is applied consistently through the entire process.
Between the speed of filling in the remaining numbers, the size of the board, and the ways in which the totals can be achieved, this first method alone can be very impressive for an audience. Once you've mastered this, you're ready to move on to the second method.
With the second method, your audience members choose the order in which you fill out the remaining squares. The process is made more baffling, but is still the same as the first method. The addition here is that you need to memorize the pattern in a new way. The new approach is simple, and could even be made simpler with a little work, depending on personal preferences.
The bonus included is another handling for the first method. The bonus approach involves filling out five squares somewhat similar to the right-hand square pattern used in the Knight's Tour (if you add a center square, that is). As a matter of fact, those already familiar with the basics of the Knight's Tour will find that knowledge quite helpful.
E-Z Square 2
E-Z Square 2 (also available in German) turns to the more popularly-performed 4 by 4 magic square. Instead of just the rows, columns, and diagonals, as in the 5 by 5 version, there are 28 patterns that make the magic total in this version (there are actually more, but only 28 are taught).First, simple patterns and some adaptations are shown. With this approach, there is the limitation that you can't generate a magic square for an odd total. At first, this seems like a fatal flaw. However, Werner Miller teaches some approaches that compensate for this, and a little creativity will yield other presentations that can overcome this obstacle.
The first presentation for this approach involves asking for a year special to the spectator, such as when they were born or married. These are placed together near the center of the square, another larger number is requested, and the resulting 4 by 4 square not only gives the total in 28 different ways, but also includes the given year!
There are some very helpful tips in the booklet, such as avoiding negative numbers and duplicates. The variations discussed include starting with other squares, including diagonals. There's also a bonus routine that involves using starting numbers chosen by the audience, and concluding with a prediction of the total!
Final Thoughts
Easily, the best thing about both of these methods is the simplicity of learning the systems. The instruction is clear and understandable, and you don't have to learn the mathematical reasons as to why this work, unless you find you want or need to do so.Also, I like the fact that the resulting square hides the method well. I discussed the problems with some magic square methods in my post on Bill Fritz' free eBook Magic Squares for the Mathematically Challenged. As a matter of fact, many aspects of the magic square that you learn in that book could easily be applied to Werner Miller's approaches.
If you're just starting out with magic squares, these are great places to start. Not only are the basic methods of both quite simple, but they can get more impressive as your skills develop. I highly recommend both E-Z Square 1 and E-Z Square 2 for those interested in magic squares of all levels.
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