The Dictionary Trick

This is my favorite trick.   A student of mine from Herndon High School in Fairfax County, Virginia showed this to me in the late 1960s, and I used it nearly every year after that until I retired in 2006.

  1. Ask a student to pick a three digit number (where the first and last digits are different).

  2. Then have the student reverse the digits and subtract the two numbers.

  3. Give the student a dictionary, and have her look up the fifth word on the page corresponding to her last calculation.
    There are only nine words in the dictionary that your student could have chosen.

  4. It is at this point that you will need a clue.
    Ask the student to give you the first letter of the word.

  5. Now, go to the page number in the following list which has words beginning with the letter the student told you: page 99, page 198, page 297, page 396, page 495, page 594, page 693, page 792, or page 891.
    Count down to the fifth word on that page and you have found the word.
    It is more impressive if you memorize the nine words on the above pages.

When I showed this trick to Ray Frantz, an English teacher at Handley High School, he added another feature to the trick.   One of us would bring his class down to the other teacher's room.   I would go outside the room and a student would volunteer to go with me and make sure that I wasn't listening to what was going on in the room.   Then the students would choose the number and select the word.   Then I would come back in the room and guess the word.

Since Mr. Frantz was in the room when the word was chosen, and the only clue I needed was the initial letter of the word, he would "send me the clue" by making a statement when I reentered the room.   The first letter of the third word would be the letter that I needed to know.   For example, he would say, "Mr. Pleacher, come in and show us how this is done" when the word started with a c; or he would say, "The students don't believe that you can do this" when the word started with a d.

Remember that you can only use this trick a few times before it starts to repeat, since there are only nine pages that the word will fall on.


Why does this work?

The difference between the original three-digit number and the number with its digits reversed is always a multiple of nine -- in fact, the middle digit must be a nine (you must always borrow in doing the subtraction) and the sum of the digits is always eighteen.
So, the only possible answers after the student has done the subtraction are: 99, 198, 297, 396, 495, 594, 693, 792, and 891.

As long as you don't choose an unabridged dictionary, each of the page numbers above will be found under a different letter.

Links:

Math Tricks

Mr. P's Puzzles and Games Page