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 Number prediction (Posted on 2016-09-26)
A summer series on TV has been a program called "Penn and Teller: Fool Us". Five or Six magicians perform their tricks on stage, hoping to get a trophy and a guest appearance on Penn and Teller's Las Vegas show, with famous magicians Penn and Teller front and center in the audience to see if they can figure out how the tricks were done; they usually can; I usually can't.

But sometimes I can get a particular aspect of the main trick. Here's a case.

First the magician asks if anyone (gesturing to Penn, Teller and the Emcee) has a cell phone he can borrow. Penn offers his and gives the contender, who has gone to where Penn and Teller are sitting, permission to look at the contents of his phone.

The magician holds the phone up, saying he's taking a picture, while doing something on the phone's screen (it's an iPhone, actually), and continuing some patter.

When he shows the screen to Penn, it already shows a calculator app on the screen. The magician asks Penn to enter any 4-digit number and press the "times" button; then he asks Teller to press any 3-digit number and the times key; finally he goes back to the stage and asks the emcee to key in any 2-digit number and press the "=" button. She does so and hands the phone back to the magician. He shows the result to the audience (the camera zooms in so we can see it: 577,345,663). He hands the emcee a lecture-sized pad and reads off the digits to her as she writes them so the theater audience can see. He pulls out some blank paper supposedly related to the prediction he made about the number that would be produced and puts it into a wine goblet and swirls it around. As a separate trick from the one now being considered, the paper is replaced by an egg, which he cracks open and discards the raw contents, leaving only the shell.

The emcee is then asked to rotate the number 180°. As you can verify by the number shown above, it's one of those numbers that spells, in this case, EggShELLS when turned upside down.

2. Could the participants (notably Penn and Teller) see the fishiness? What's the probability they'd choose numbers that would make it extremely obvious to them?
3. Do you notice anything very out-of-whack with the "eggshells" number?

 See The Solution Submitted by Charlie No Rating

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Well, for starters, 577,345,663 has only two prime factors, and so cannot have been produced by multiplying three numbers.

Perhaps the magician enters this number during the initial patter and stores it in memory, and then recalls and clears the memory after the emcee's entry but before showing it to the camera? I don't know how clear the iPhone's calculator app makes it that there's something stored in memory, but that could certainly have been a clue for P&T.

Moreover, if Penn and/or Teller had chosen numbers that were even, or multiples of 5 (or both) then the last digit of the "product" would be a dead giveaway that it was not as it was claimed. The probability of one doing that is 1/2 + 1/5 - 1/10 = 3/5 and so the probability that  neither did is (2/5)^2 = 4/25 and so the probability that at least one did is 21/25.

 Posted by Paul on 2016-09-26 16:16:09

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