Magic Squares

The literature of magic is replete with descriptions of magic squares but few magicians have ever made a success of them on stage. Magic squares, like memory feats and oddities such as the Knight's Tour, take exceptional presentations to render them entertaining. David's stage performance of the Magic Square is without doubt the best you will ever see. It transcends an audience's fear of mathematics and offers up a clear and dynamic demonstration of the performer's mastery over numbers. It begins with the total involvement of the spectators and ends with an unexpected applause-pulling finish. It takes work, lots of it, but perhaps not in the way you would think.

David has always been fascinated with numbers despite the fact he was not particularly adept at mathematics at school. The dry formulas for algebra and calculus could never compete with his interest in the almost intuitive computations involved in constructing a perfect Magic Square. He has spent many hours moving numbers around numerous grids until finally settling on the best formulas for building Magic Squares. But he has spent even longer devising the presentations that made the Magic Square worthy of inclusion in his professional act.

In a typical stage performance of the Magic Square he begins by getting two men to assist him from the audience. David explains that he is about to show them an experiment in numerology. "Do you know what your lucky number is?" he asks the first man, Robert. He doesn't so David offers to show him.

"What is your birth date?" It's the 16th of May 1958, and David writes it on a large white board, translating the date into a series of numbers: 1+6 + 5+1+9 + 5 + 8. He calls the numbers out and totals them, "One plus six is seven, plus five is twelve, plus one is thirteen, plus nine is twenty-two, plus five is twenty-seven, plus eight is thirty-five." He pauses during the addition to make sure that everyone is following. Finally he writes the number 35 on the board.

"Numerologists don't stop there. They continue to simplify until they arrive at a single number. So they add three plus five to arrive at eight. Therefore Robert your lucky number is eight. Did you know that?" Robert says he didn't but he's not sure that he ought to either.

David then turns to the second man, Joe, "Do you know your lucky number? Again, the answer is "No." David pauses thoughtfully, looks him up and down and then says, "You..." He hesitates, gazing at Joe as if pondering the rest of the sentence. "You..." he repeats, staring hard at the volunteer and keeping him in suspense. "You..." he says for the final time, "look like a very nice man!" Joe relaxes and the audience laughs. It's just a gag. Or is it?

After a pause, David adds, "Well, you look like you might be a nine to me. When is your birthday?" "11th of April 1956," says Joe. Once again David translates the date into numbers, writing them on the board and totalling them. Strangely, they add up to the number nine. Maybe there's something to this numerology after all.

David removes the board from the easel to reveal another board on which is a 4 x 4 grid. "I am going to try to create something that, for well over two thousand years, mathematicians have called a magic square," He pauses for a moment then writes the number 98 above the grid. "In a few moments this will be an important number. Let's see if it works."

He turns to the board once again and proceeds to write furiously, entering various numbers into the cells of the grid. There seems to be no pattern to it and it's all over surprisingly quickly. When David moves aside the audience see that sixteen numbers now fill the board. He has constructed a magic square.

David asks Robert to choose one of the four vertical columns on the board. He selects the second from the left. David picks up a large card and places it over the board. It has a cutout and acts as a mask so that only the chosen column is highlighted. Deliberately and

16

18

59

5

60

4

17

17

S

57

20

IS

19

19

2

SS

slowly David adds up the four numbers in the column. They total 98, the same number written on top of the board. The audience applauds.

"But you're probably wondering what would have happened if he had chosen a different column?" says David. The cardboard mask is placed over another column. The numbers are totalled and again they add up to 98. In fact, as David demonstrates, every column in the Magic Square totals 98!

Not only that but so do all the horizontal columns! David uses the mask to mark them out and the audience checks as he adds up the numbers. Again and again they add up to 98!

But it's not just the vertical and horizontal columns. David brings out an assortment of cardboard masks, each constructed to highlight different combinations of four numbers. It doesn't seem to matter which four numbers he chooses, they all add up to a magical 98. "In fact," says David, "there are at least thirty two different ways of arriving at this total!"

Finally, a mask with four cutouts, one at each corner, is produced and placed over the grid. It's slightly offset and as yet you can't see which four numbers it is meant to highlight. Turning to the first volunteer, David says, "Remind me, Robert, what was your birthday?" Robert says, "It's the 16th May, 1958." David writes the date in the centre of the mask. Then, slowly, he slides the mask across the Magic Square. Four numbers are now revealed through the corner cutouts. They are 16, 5, 19 and 58. Robert's birth date! The audience is absolutely astounded.

"And finally," says David, "If you add those numbers together, 16 plus 5 plus 19 plus 58, you'll find that they too add up to 98." They do and suddenly you realise why 98 was so very important.

Revelations: The major problem with any magic square was that it never had a finish. In the usual performance the performer drew up the square, totalled the numbers and showed, again and again that they all added up to the same number. It's an impressive demonstration but after a few columns have been totalled the audience understands the idea and the rest is mere repetition. David solved this problem by introducing the totally unexpected appearance of the spectator's birth date in the four corners. It is a magnificent finish and utterly baffling. It seems incomprehensible that David could have devised the square by any mathematical means.

David, as ever, has devised a number of approaches to the Magic Square and several will be detailed here. But let's start with the stage presentation you have just read. It begins with two volunteers being chosen to help with the routine. What the audience don't know is that one of those volunteers, Joe, was approached before the show and asked if he wouldn't mind his birth date being used in one of the routines. That's how David knew he was a "nine" when his birth date was reduced to a single digit.

David explained to him that some people are a bit sensitive about information of that type and he didn't want to embarrass anyone during the performance. It sounded a logical

enough excuse. David didn't ask him for his birth date but he did ask him to write the date down in full, just in case he has to show it around later, on a slip of paper and then keep it in his pocket. You won't be surprised to learn that this allowed David to gain access to the information. But you might be surprised as to why Joe was asked to write the date in full. It ensures that when he reveals his date of birth on stage he will say, "11th of April 1956," rather than reducing it to a series of numbers, 11/4/1956. It's a small point but helps further conceal the surprise finish.

The Magic Square can be genuinely calculated during the performance but obtaining the birth date ahead of time enables David to create an aesthetically pleasing arrangement of numbers, one that has no duplicates. The ingenious but relatively simple method to construct this square is dealt with later. For now, let us continue with the presentation and add one detail about discovering the volunteer's birth date.

Instead of indulging in clipboards and centre tears or any of the other devices, David sometimes just asks for the information. How he asks and whom he asks is important for this bold ruse to work. For instance, when he starred in his Swedish television series, he called his producer and asked him if he had any friends coming to the show. "I need a man to volunteer for one routine. It's nothing important, it's to do with numbers and he won't have to calculate anything. I just need a friendly looking guy." The producer said he had quite a few people coming along to the show and promised to find someone suitable. Two or three days later David was speaking to the producer again, about something completely different, and as an aside asked if he had managed to locate anyone willing to volunteer for the number experiment. The producer said he had just the chap. "Could you find out his birth date?" asked David, "but if you do, do it discreetly, I don't want him to know about it." A short while later the producer came back with the required information, having got it from the volunteer's wife or some other source. For him it was a minor detail in a busy production process. For David it was central to the routine and would simplify its working.

The secret of making these inquiries is to keep them low key. Take them step-by-step, first find the volunteer and then the birth date. It seems so casual and unimportant that people freely give information that David might otherwise have to obtain in the short and often hectic period prior to the show when time is at a premium. Getting this cooperation from complete strangers is a tremendous skill and requires empathy, understanding and just the right amount of psychological manoeuvring for it to work but it has saved David lots of time and energy in preparing his performances.

Getting the right volunteer onstage during the show also requires some thought. It would be embarrassing if the arranged volunteer rushed onto the stage. His eagerness to participate should not be obvious. To prepare the studio audience David usually talks with them before the show and openly tells them that he will be doing something using numbers and only wants people to volunteer who don't mind revealing their exact date of birth. It seems a gentlemanly warning but prepares the audience for the fact that some people will readily put up their hand when called for. Naturally David picks out the volunteer he spoke to earlier. He has, on occasions, set up more than one volunteer and yet only used one of them during a performance. This leaves one, or more, of them thinking, "Well, I guess he didn't need me after all," and even further from solving the mystery.

Let's recap the details of the performance. With the two men on stage, David turns to the first, Robert, and asks whether he knows anything about numerology. He asks for his birth date, 16th May 1958, and demonstrates how the numbers can be added together and reduced to a single digit, the number 8. This is Robert's lucky number.

There is no magic at this stage. But it introduces the subject of personal lucky numbers and accustoms the audience to the fact that this routine depends on adding various numbers together. It's a gentle introduction and quite amusing. Then David takes the matter a stage further and suggests that by just looking at the second volunteer, Joe, he can divine his lucky number, something that at this stage even Joe doesn't know. It lends credibility to David's talk about lucky numbers. Later it will help confuse the audience as to the method, an additional detail that will draw even those who know how to construct magic squares away from any solution.

With a little humour David announces that Joe's lucky number is nine. But is it? The only way the audience can find out is to add up the numbers in Joe's birth date. In other words David has given the audience a reason to be interested in numbers, they are harbingers of secrets and seem to have some kind of relationship with their owners.

All the figure work for the lucky numbers is done on the board resting on the easel. For the next stage of the routine the board is removed and the Magic Square grid revealed. David appears to concentrate and then writes the number 98 above the grid. 98 is the number David arrived at when he mentally totalled Robert's birth date as follows: 16 + 5 + 19 + 58 = 98.

At this stage there is no explanation for the number 98. It serves only to intrigue the audience as to what part it might play in the plot. A little more concentration and David starts writing numbers in the Magic Square. He calculates these numbers according to a formula explained later.

A little more concentration and David starts writing numbers in the Magic Square, putting them into the cells apparently at random. In fact he is careful to put the figures in so that no rows are completed until the last possible moment. That way the audience can't add up the numbers until he is ready. Not only does this build the tension but it prevents anyone, particularly those who might have come across a magic square before, getting ahead of him.

The Magic Square completed, he steps back and it is here, as far as the audience is concerned, that the effect begins. One by one he shows that all the vertical and horizontal columns total the mystical 98. With each row the calculation is done more quickly, and the pace picks up. Prior to the show he often hands out electronic calculators so that members of the audience can participate and check his arithmetic. By the end of this sequence the audience are applauding madly, convinced that this is the conclusion of the feat. This makes what happens next all the stronger and funnier because David takes out lots of different masks, each more peculiar than the previous one, and shows that the number 98 can be found just about anywhere you care to look on the board.

The masks have two purposes. They isolate four numbers at a time so that they can be seen more clearly and added more easily. And they hide the numbers in the rest of the Magic Square so that, once again, the audience cannot leap ahead of the performer. It's this last feature of the masks that becomes all-important in the final revelation. All the masks can be seen in the illustrations and thirty-two different arrangements of numbers can be highlighted using them. However, David doesn't always insist on demonstrating every one. It very much depends on the response of the audience and he uses this as a gauge to pace the routine.

The four-corner mask is kept in reserve and brought out only when the time is right to bring the routine to a close. David places it on the Magic Square, slightly askew, so that the four cutouts only expose half of the corner numbers beneath them.

He turns to Robert and asks him to remind him of his birth date, 16th May 1958, which he writes on the centre of the mask. Then he slides the mask across the Magic Square until the cutouts are, for the first time, aligned with the corner numbers. 16, 5, 19 and 58 slowly appear and the audience are stunned to realise that this is Robert's birth date.

It is a remarkable finish guaranteed to draw tremendous applause. But after waiting just a beat David turns back to the Magic Square, points at the corner numbers and says, "And if you add 16 plus 5 plus 19 plus 58, once again you get 98." It's inevitable really but the audience don't realise this and accept it as yet another punch line to an amazing routine that gets a big laugh and even more applause.

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