I went walking with a friend of mine in Las Vegas last week. We turned a corner and saw a street magician doing some tricks for a crowd of 12-15 people. In most of his tricks, the magician would make a little money by getting the crowd to bet on something he had set up, such as which cup a ball was hidden under.
On one of the tricks, someone in the crowd won a quick $50. Anyway, after 10-15 minutes of watching him, we volunteered for the next trick. He handed each of us a pen, a small piece of paper, and a calculator. As he turned his back to us so that he couldn't see, we were each told to secretly, legibly write down any 5-digit number, so we did.
He told me to
- Add together all the digits in my number
- Subtract that sum from the original number to get the final number
and he told my friend to
- Rearrange the digits in her number to make another number
- Subtract the smaller number from the larger number to get the final number
Then, he told us to add our two final numbers together, write the sum large and legibly on a piece of paper, and circle it. Lastly, we were told to choose exactly one non-zero digit in that final answer and underline it.
We did that, and he then said he would ask us to read off to him, in any order, all of the digits except the one that we underlined, and then he would tell us which digit we had underlined.
A guy behind us bet him $20 that he would not be able to guess the number in a single guess. He took the bet, and asked if there were any more bettors.
Should we have made a bet? Is there a way for the magician to know the digit for sure?