The Knights Tour

The Knight's Tour is an ancient chess puzzle, the aim of which is for the Knight to travel across the chessboard visiting every single square just once. Mathematically speaking there are literally thousands of solutions. But in practice it is almost impossible for anyone not in the know to uncover even one of them. As a result, magicians have sometimes used the puzzle as a demonstration of prodigious memory and calculation. David has always felt that it was an impressive feat for the right audience but all too often it has been presented with all the excitement of a mathematics lesson in a schoolroom. He has performed the Knight's Tour on stage many times and offers several tips on this venerable routine that transform an intellectual puzzle into a dramatic and engaging presentation.

It begins with an impressive looking board on which to stage the drama. David uses a large chessboard, mounted vertically so that the audience can see every square. The squares

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are numbered from 1 to 64 any one of which can be illuminated on David's command.

He begins by displaying several pieces from a regular chess set (Castle, Knight, Bishop and Queen) and quickly explains how each one is used in the game. Squares on the board light up appropriately as each move is described. Diagonal lines for the Bishop, straight lines for the Castle etc. He drops them into a bag and has one chosen. The Knight is selected and for the benefit of those who don't play chess he once again describes its peculiar L-shaped move. He introduces the audience to the age-old problem of moving a chess piece across the board so that it touches every square just once. It is a puzzle that scholars and mathematicians have debated for centuries and it doesn't take long before the audience appreciate the enormity of the task. Then he adds one more surprise; they get to choose the starting point on the board. "Call out any number between 1 and 64," says David. Let's assume that 27 is called. David repeats the number and it immediately lights up on the board. "That is our starting point."

The routine begins slowly and picks up speed as the audience become familiar with the manner in which the Knight moves. David points out that from square 27 there are a number of possibilities as the Knight can move in several different directions. He chooses one, square 44, and that square lights up. From there he chooses square 61 and that too illuminates. So far the pace has been almost instructional in nature but now it begins to quicken with David calling out numbers, 55, 40, 23, 8 and so on. Square by square the board lights up, the audience following the twists and turns of the Knight's zigzag journey.

It's clear that this is no random walk and after about 20 moves David pauses and draws

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the audience's attention to some peculiarities of the tour. The Knight started on the left of the board and is now way across to the right. "I'm sure you're wondering how we will ever get back," says David. "You'll notice that we have a mirror image. Look at 6 and 62, 16 and 56 and the four squares in the centre 30, 31, 38 and 39." The audience can see he is right. The board is starting to resemble an illuminated Rorschach blot.

He continues to call out numbers, never looking at the board, never pausing to check. He never calls the same number twice, never makes a mistake and yet the Knight continues unimpeded moving from one square to the next. After about another twenty moves David pauses once again. "You're now beginning to wonder how we can ever reach the isolated squares. 10 is completely surrounded, as is number 50. Don't worry, we'll get back to them." However, it seems impossible that he could.

He calls out more numbers and the Knight quickly makes its way around the rest of the board, dodging already lit squares and landing on those that remain. "I think by now you are beginning to see the pattern," says David as the Knight enters the home stretch. "You can see that we can jump from 30 to 13, back to 28. To 38 and to 53. Then to 43 and finally 33." Remarkably, the entire board is now lit. Every square has been visited. "And if we did the Knight's move one more time we would be back at number 27, which is exactly where we started. So we've covered the whole 64 squares using only the Knight's move, touching each number just once and finishing where we started from. Thank you." Which is an obvious cue for applause.

Revelations: The solution to the Knight's Tour was originally contained on a small piece of card that David held concealed in his hand. On that card was a list of 64 numbers which mapped a predetermined loop that took the Knight around the board. When a number was called, David secretly glanced at the list, spotted the chosen number and called out the rest of the numbers from that point. When he reached the last number on the card he started again at the first number. It was a cyclical set up, similar to a stacked deck. He later memorised the numbers, which gave him more freedom. He tried working the effect blindfolded but found it was unnecessary. Without the blindfold he had more control over the situation and contact with the audience. Consequently the effect had even greater impact.

The Knight has become synonymous with the chess problem because of the mathematical possibilities inherent in its unusual L-shaped move. You won't be surprised to know that the Knight was forced using a Change Bag. Having the audience believe that they have selected the chess piece adds another layer of mystery and lessens the feeling that the demonstration is somehow contrived for the performer's benefit rather than their entertainment.

The cyclical tour detailed here is just one of many that have been discovered by mathematicians. Each produces a pattern when mapped across the board. David recognised this and decided to make a feature of it. At some stage during the routine these patterns would manifest themselves. In particular the way one half of the board reflected the other. It was also important to point out how far the Knight had travelled from its point of origin and how difficult its return journey would be. There were also numbers that seemed to be completely isolated. David highlighted these as obstacles to the successful completion of his mission. It is essential

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that in routines of this type the audience fully appreciates what the performer is trying to do and how difficult that is. Pausing at intervals not only prevented the calling out of numbers from becoming less monotonous but, more importantly, it kept the audience informed. It was like the commentary on a horse race and it helped build the drama and gave the audience a genuine interest in how David would complete the challenge.

The board was immensely important and David went to great effort and expense to prepare it. The only other performer David knows of who has used a similar board is his good friend Peter Reveen. It resembled a giant light box in construction. Beneath the translucent plastic cover lay sixty-four light bulbs, each in a separate compartment. The bulbs could be switched on and the square above them illuminated. It was operated by an assistant backstage with a cable running from the chess board to a console holding sixty-four switches, one for each bulb.

To keep the operation simple the switches on the board were laid out in the same order as the numbers on the cue card. The most difficult thing the assistant had to do was find the first number called. After that he need only flick the rest of the switches in the order they appeared on the board for the routine to work. This was a far better arrangement than having the switches in numerical order. Not only would that have been more time consuming, with the assistant hunting all over the board for each number, but it was also more prone to error.

David doesn't reveal secrets unnecessarily and as far as he was concerned there was no need for anyone beyond him and his assistant to know that the Knighfs Tour was always the same. If anyone backstage got a glimpse of the console and its strangely arranged numbers, they might have concluded that the feat was not all it purported to be. So David devised an effective piece of camouflage. It was a kind of keyboard overlay, with holes cut out for the switches. Now anyone looking at the console would see that the numbers above the switches were in the correct numerical order, from 1 to 64. But just before the routine, the assistant would remove the overlay, flip it over and replace it to reveal the numbers in the same order as they would be called. Reputations are hard won and easily lost and David spared no effort in protecting his.

The routine can be worked so that the board starts with all squares lit and are then switched off as the numbers are called or the other way round, it's a matter of choice. David recommends that illuminating the board provides a more positive experience. Today all manner of electronic lighting effects could be built into the board. One modification that David would certainly make is that of having the entire board flash on and off at the finale. That, together with some appropriate music, would make for a better applause cue.

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