My son came to me a few days ago and showed a trick that a friend from his swim team shared with him.
He told me to:
- pick a 3-digit number that is larger than 600 and where tens digit is smaller than hundreds and singles digit is smaller than tens. I picked 941.
- then he told me to reverse this number. I reversed 941 into 149.
- next I was supposed to subtract second number from the first. I did: 941-149=792
- then reverse the digits in the result: 792 reversed into 297
- and add two numbers obtained in the last two steps: 792+297= 1089
He then gave me a book and asked to look on the page corresponding to the number created by the first 3 digits of my number - 108 and line corresponding to the last digit in my number - 9. The book happened to be a "Book of Manners for Men" and got a smirk from me. I found it interesting that my 12-year old who sometimes can still be caught eating spaghetti with his hands grabbed this book for a trick. On the page 108, line 9 from the bottom there was a word "must". My son then showed to me his palm where the word "must" had been written in ink.
Wow! This was spooky.
We tried again with a different number. The result was again 1089.
How does it work?
I spent a few days trying to figure it out. Even recruited my dad to help me with it. We couldn't find a nice elegant explanation of how the math behind the trick works. There is one but not clean and beautiful enough to be worth these pages. But no worry - my dad came up with a great explanation for a simplified version of this trick when using two-digit numbers.
Here is the trick with two-digit numbers:
- pick a 2-digit number that is larger than 60 and where singles digit is smaller than tens. Say 71.
- reverse this number. Reversing 71 you get 17.
- subtract second number from the first. In our case: 71-17=54
- reverse the digits in the result: 54 is reversed into 45
- add two numbers obtained in the last two steps: 54+45= 99
- no matter what numbers you chose you get 99; to impress somebody you can pick a book, find a word on page 9, line 9 from the top or bottom and copy this word on your palm.
Here is how it works:
We start with a two digit number: AB. Reversing it we get BA.
Subtracting AB-BA = 10A + B - 10B - A = 9A - 9B = 9(A-B)
Now, we know that we get a result that is divisible by 9.
Let's add this result to its reverse. Assume that this result has digits CD.
CD+DC=10C + D + 10D +C = 11(C+D)
Interesting this result is divisible by 11.
But remember that CD was divisible by 9. Remember the rule of divisibility by 9? CD is divisible by 9 if and only if C+D is divisible by 9. Therefore our result 11(C+D) is divisible by 11 and 9, divisible by 99.
Numbers divisible by 99 are: 99, 198, 279, etc.
Can our result be larger than 99? No! Why?
Because for 11(C+D) to be 198, C+D would need to be 18. This is possible only if C=D=9. In other words, after the step 3 we get number 99. But we can't subtract two two-digit numbers (AB-BC) and get 99.
Same happens when we try larger numbers.
Therefore we will always get 99.
Happy Mathematical spooking!
By the way, if you happen to come up with a nice explanation for the original trick, fill free to post it below or email it to me.
Top image by The Real Estreya, distributed under CCL.