The Magical Number 9

NUMBER 9 was first brought to my attention as possessing unusual properties by a rather simple parlor trick. My host asked me to write down my telephone number, leaving off the exchange. This I did, and at his further advice I added two zeros after the number and subtracted the original number from this new figure. Thus: —

726700

7267

719433

Then I was told to add the digits in the answer, the result being 27. The telephone book was brought forth, and, after turning to page 27, I was instructed to look in the second column and to note the name and address 7 lines from the top of the page. This I did, and to my great surprise my host uttered the correct name and address.

Not daunted, however, I immediately thought that he had prepared himself for this trick, having already known my phone number. Asking him to repeat the performance, I offered a fictitious number, which was a good deal smaller than the first. When made to pass through the same procedure, this gave the following result: —

147900

1479

146421

The sum of the digits equaled 18. The telephone book was again brought forth, and we turned to page 18, first column, 8 lines from the top of the page. My host once more uttered the correct name and address.

By this time I was bewildered as well as suspicious. No stretch of the imagination could make me believe that my host had memorized the telephone book! At a third attempt, again with a fictitious number, the result was: —

896700

8967

887733

The summation of these digits equals 36; and on page 36 of the phone book, third column, 6 lines from the top, I read the correct name and address as my host recited them.

I hope that by this time the reader is more advanced in the solution of the problem than I was. Upon repetition of the trick, however, I discovered that 18, 27, and 30, multiples of 9, were the only numbers ever obtained from the summation of the resulting digits, and to my great relief I realized that my host had to memorize only three names with their corresponding addresses.

Upon investigating this problem, using smaller figures than telephone numbers, it became clear that any number subtracted from another number which is 10 or 100 times the original number will result in a figure whose digits total 9 or a multiple of 9.

When the problem is so put, the reason is obvious, but its practical application to the telephone book makes a very good trick with which to mystify one’s friends.