Place three distinct objects in a line. Any three distinct objects will do the job. The applet below uses three US coins: a penny, a nickel and a dime, so it would be a convenience if you use the same. However, you may just as well write numbers 1, 2, 3 on three pieces of paper.

The order of the objects does not matter, but do remember what it is. It will be called the original order. Think hard of one of the objects. The computer is going to read your mind and announce which one you've been thinking about. You have to help it, though. This is what you have to do:

First, swap the two objects other than the selected one. Swap again any two objects but this time tell the computer which objects have been swapped. You communicate with the computer by clicking on the two objects on the screen. Please continue swapping objects, two at a time, and telling the computer which ones have been swapped on every step. Do this in any way you may want to, but see to it that you finally arrive at the original order. At which point, think again of your selected object and press the "Announce" button.

### This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.

 What if applet does not run?

Explanation

This magic trick has been invented by Bob Hummer long, long ago [Gardner].

Your very first action is to select an object and swap the remaining two. Now, if the computer knew which one has not changed its position, it would be immediately able to announce your selection: you selected the one that has not moved. In mathematical terms, you applied a permutation that left one of the objects fixed. Each of your successive actions can be described by another permutation and the total result is their product, which is yet another permutation. When you arrive at the original order, this product is exactly the inverse of the very first (unannounced) permutation. (As a matter of fact, the permutation and its inverse coincide in this case.) So the computer that easily keeps track of your actions, knows exactly what it is. When the "Announce" button is pressed, the computer checks its stored permutation for the one object that has not been moved from its original position. This is the object you thought of! If more than 1 or none of the objects has been moved, the computer knows that you made a mistake in your bookkeeping and warns you accordingly.

Bob Hummer invented the trick when computers were a novelty the size of a good classroom. At the time, the trick had to be performed by a human magician. So an important part of Hummer's invention was a mechanical procedure that made keeping track of your (i.e. one of the spectators') actions simple.

But let's reverse the roles. Assume now you are a magician and wish to impress your audience. Call a volunteer spectator to lend you a hand.

Select three fingers on one hand and assign them numbers: 1, 2, 3. Press the thumb of the hand to the finger #1. Ask the volunteer to arrange three objects in a line, think of one and swap the other two. Ask him to proceed swapping two objects at a time and calling out either their names or their locations by number. If the names are called, you will have to interpret the calls as referring to two numbers. On the first call, if the numbers were 2 and 3, do nothing. But if the spectator called 1 and 2 or 1 and 3, move the thumb to the finger #2 or #3, as the case might be. On the next call, if that number was called out, move the thumb to another finger, otherwise keep where it was. Apply the same procedure to all successive steps.

When the volunteer claims to finish by returning to the original order, your thumb points to one of the fingers. If it points to finger #1, the object that occupied the first location has not moved. This is exactly what you are looking for. If the thumb points to, say, finger #2, then the very first (unannounced) permutation swapped fingers #1 and #2 and left finger #3 fixed. Similarly, if the thumb points to finger #3, finger #2 is the one to announce.

(Colm Mulcahy of Spelman College, Atlanta, GA wrote an excellent column where the trick has been presented with cards rather than coins.)

### References

1. M. Gardner, Mathematics Magic and Mystery, Dover, 1956, pp. 63-64

• An Arithmetic Magic Trick
• Bachet's Magic Trick
• Gergonne's Magic Trick
• Magic in Square
• Math Telepathy
• Number Guessing Game. All by Computer
• Barcode Magic
• Arithmetic magic matrix
• Calendar Magic: an Interactive Gizmo
• Two Numbers Guessing Game