The magic square has been around for centuries. There are some very complex magic squares, but I want to show a simple magic square, as more of an introduction. The basics of the magic square is that you allow the participant to freely select numbers from a grid, according to some simple rules. When they are done, you can show them that you predicted the total of their numbers. To start, you make a grid any size 3×3 or bigger and fill it with consecutive numbers. Today I am using a 4×4. The magic square does not work for 2×2, and while a 1 box ‘grid’ technically fits the criteria to be a magic square… 1 box is not impressive.
To simplify, I will show the trick as I explain it. You give them a free choice on the 1st number, and it is a truly free choice. Then, they cannot use any of the numbers on that same row or column. I will gray out the numbers for clarity. In performance you would need to X them out, or put some mark on them.
The 2nd choice can be freely chosen from the remaining numbers. The choices are free, but as the choices continue, the choices are self-limiting. For the example, I am choosing the diagonal numbers but any numbers can be used, as long as you block off the column and row. Using the diagonal shows how much their choice gets limited as they go on. Also, the diagonal will help explain the trick in a minute.
By the time they get to their 4th choice, there is only one number left to choose. You will have a prepared prediction stating that their total will be 34.
The Secret
In the magic square, you put in consecutive numbers, add up the diagonal, and as long as you follow the row/column rule any selected numbers will add up to the diagonal total. It is that simple (for the basic square).
In this example I used 1-16 which gives a diagonal total of 34. I made another box using 40-55 which gives a total of 190. On the simple magic square, any set of consecutive numbers will work e.g. 1-16, 40-55, 100-116… any set of consecutive numbers. I would not make the numbers too large as the math will get cumbersome – keep it simple.
It works on both diagonals, and as long as you follow the rules, any selection works. You can also show them that the 4 corners add up to the same total, and the center 4 squares do too (6, 7, 10, 11). On a prepared simple square like this I only show their total and my prediction, because I want it to appear to be predictive mental magic, not just a math trick. There are more complex squares, and different handlings, that make those revelations very powerful. On this simple square I would avoid drawing too much attention to the math trick side of it.
More Advanced Magic Squares
There is a long math proof to show how this actually works, but to use the trick, it is optional reading. For more information on the math side of magic squares you can check the wiki page, it goes much further in depth than I cared to read. This simple magic square adds up on the diagonals and as long as you follow the row/column rule, their selected numbers work. This is a good simple start to magic squares but they can get much more complex.
Harry Anderson did a magic square with a giant folded piece of blank paper. Each grid square was about the size of a piece of typing paper. He would only see one square at a time. He would make a long funny bit of folding it around and writing in what seemed to be random numbers. When he was done, the diagonals added up to a random number selected by the audience in advance. But then he would show that the 4 corners added up, the center 4 squares, each row and each column, and the 4 corner quadrants all added up to the selected number. (One corner quadrant would be, on the 1-16, the 9, 10, 13, & 14… in the simple magic square it does not add up.) Harry Anderson has a much more complex magic square.
His square is based off Annemann’s classic square, but in Annemann’s version the magician had to do quite a bit of math in his head, while filling in the square. Harry Anderson, developed a system to use basically the same square, but without doing the math on the fly (the math is done in advance). This made it easier to perform, and allowed for the folding of the square. Filling in the grid with only one square visible at a time makes it look quite impossible. Harry’s presentation looked like a giant sudoku on steroids.
I think this simple magic square is a good start. If you have an interest in getting more complex, there are plenty of books with tons of information available.
Further reading:
Harry Anderson and Jon Raucherbaumer – Magic2
Ted Annemann – The Book Without A Name
Martin Gardner has numerous books on math magic
Mathematical Puzzles and Diversions, Mathematical Carnival, Mathematical Magic Show, The Magic Numbers of Dr Matrix (and more)
David Roth – The Magic Square
Harry Lorayne – Reputation Makers
This barely scratches the surface of available literature on the magic square.
Here are a couple simple squares you can print out and use.
1-16=34
40-55=190
