The Octal Pencil

Effect: A spectator is handed a special pencil that has a series of six numbers printed on each of its eight sides (Figure 57, next page). These numbers are all different and some are printed in red, others in black. The spectator is told to choose any one of the sides of the pencil he wishes and to call out only the colors of the numbers there. However, as he does so, he is to lie about the color of one of the six numbers. This number is his choice as well.

The performer turns away from him, making visual cues impossible, and listens to the spectator's recitation of colors. Then, though no numbers have been mentioned, the performer correctly names the number at the miscalled color. The feat can be repeated with unvarying success.

Method: This astonishing effect is based on an ingenious mathematical arrangement that Mr. Elmsley calls a "liar's matrix". Consider the following eight sequences:

Examination will show that each of these sequences is unique. More interesting still, if you transform any single element in any one of the sequences from an I to an O or vice versa, that sequence remains unique. Other such matrices can be constructed, but we will work with the one above.

Next, octal notation must be explained. It is a simple idea. You will probably have noticed that the sequences shown are built in the

binary digits of Is and Os. With the age of personal computers fully arrived, binary mathematics is recognized by a much larger population than it was a few years ago. Hence, the following binary notation for the numbers 0 through 7 may already be familiar,

If binary notation is new to you, the logical progression of these eight binary numbers can be quickly understood, and they are easily memorized. Now, let's analyze the first of the sequences in our liar's matrix: O-O-I-I-I-O. These six digits, or "hits", can be split into two binary numbers: the first three bits, O-O-I: and the last three bits, I-I-O. O-O-I = 1 and I-I-O = 6. 1-6 = 16. 16 is the octal notation for this sequence. Octal is nothing more than a simple shorthand for bit patterns. Here are several more examples:

It is important that you understand the octal system, as this trick and the two subsequent ones rely on it.

Now to the trick itself. The eight series of numbers the spectator finds on the pencil handed him are shown in Figure 57.

The outline numbers in italic are printed in red and the solid numbers in black.

An eight-sided pencil or pen is required to hold all these numbers. In the States, at least, six-sided pencils are more common, and the trick can be done with six sequences of numbers, rather than eight. The reason for placing the numbers on a pencil, aside from novelty, is that there is an observable pattern in the formation of the numbers in these sequences. Within each row, the second digit of the first three numbers is the same; as is the first digit of the last three numbers. It is unlikely that these regularities would be perceived unless the spectator were given enough time to examine the sequences closely. However, by wrapping the sequences around a pencil, this element of the matrix is made more difficult to discover. If the preparation of such a pencil is not appealing to you, the eight number sequences can instead be written one on each side of four blank cards.

Hand the pencil or cards to someone and ask him to decide on one of the eight series of numbers, letting no one else know which it is. He is then to choose one of the six numbers in that series. Turn your back as he does this and have him recite just the colors of his chosen numbers, reading from left to right. But as he does this, he is to lie about the color of his chosen number. "If it is red, say it is black; if black, say it is red. Don't, however, get carried away and tell me the number is green or purple. Since everyone knows the numbers are all either red or black, it would take no special powers to recognize your lie. Try to make your lie as difficult to detect as possible." This precaution against a predictable jest is necessary, as such a fledgling attempt at humor by the spectator can only diminish the desired effect.

Astonishingly, it is the spectator's lie that identifies the chosen number for you. As an example, assume he calls out, "Black, red, red, black, black, red."

Your reply: "Ah, the subtle overtones in your voice tell my trained ear that you were lying about the number forty-six." How do you know? Think of black as I and red as O in binary. The sequence BLACK-RED-RED-BLACK-BLACK-RED just called out translates into I-O-O-I-I-O. If this is converted to octal, it becomes 46, the very number he lied about. I-O-O - 4 and I-I-O = 6. Here are several more examples:


By recording the sequence on your fingers as it is called out, your mental work will be made easier. Simply use the first three fingers of the left hand for the first three colors, and the corresponding fingers of the right hand for the second three colors. Bend the finger into the palm to signify red, and leave it extended for black. Since your back is turned, this ticking off of the colors won't be observed.

Concerning the liar's matrix employed in this trick, Mr. Elmsley comments:

"I consider myself to be a veiy good programer of the second class. I keep inventing wonderful techniques, and then discover that someone else has already invented them (but I'm catching up with him).

"Not long after inventing the liar's matrix I discovered that a man named Hamming had been there first, a long time before. Hamming codes are widely used in the main stores of computer mainframes as a sophisticated security mechanism. If a store failure causes a single-bit error, it cannot only be detected, but corrected in flight— because the erroneous bit can be identified.

"I worked out the liar's matrix by trial and error. Later I read an article on Hamming codes that gave the mathematical analysis and a general method for their construction.

"All the same, I invented the liar's matrix all by myself!"

With the method of the "Octal Pencil" understood, a step can be taken to divorce the presentation from numbers, completely obscuring the mathematical basis of the method. This is done in the next trick.

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