A Spooky Math Trick


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:

  1.  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.
  2. then he told me to reverse this number. I reversed 941 into 149. 
  3. next I was supposed to subtract second number from the first. I did: 941-149=792
  4. then reverse the digits in the result: 792 reversed into 297
  5. 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:


  1. pick a 2-digit number that is larger than 60 and where singles digit is smaller than tens. Say 71.
  2. reverse this number. Reversing 71 you get 17. 
  3. subtract second number from the first. In our case: 71-17=54
  4. reverse the digits in the result: 54 is reversed into 45
  5. add two numbers obtained in the last two steps: 54+45= 99
  6. 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.

35 comments:

  1. Well, what defines elegant? If the three-digit number is abc, the key is that a>b>c, which forces you to borrow with each subtraction in step 3.

    a b c - c b a =...

    First the ones digit. c-a<0, so you have to borrow, so that digit's answer is 10+(c-a), and the first number (a b c) becomes a b' c, where b' = b-1

    The tens digit was b-b, but since we borrowed, b'-b<0, so you have to borrow from the hundreds. So you get 10+b'-b or 10+b-1-b = 10-1 = 9. The middle digit is 9. Always.

    The hundreds digit is the only one where borrowing doesn't happen, and the answer there is (a-1)-c = -1+(a-c) = -1-(c-a). (That odd regrouping makes sense in a minute).

    So your new number is -1-(c-a) 9 10+(c-a)

    Reverse and add.

    Ones digit is 10+(c-a) + -1-(c-a) , and then the +(c-a) and the -(c-a) cancel, and you're left with 10 + -1 = 9. The ones digit is 9. Always.

    Tens digit is 9 + 9 = 18, so you have an 8 and carry the 1 over to the hundreds. The tens digit is 8. Always.

    Hundreds digit is -1-(c-1) + 10+(c-a) + 1. This is the same as the ones digit, plus the carried 1 from the tens. So since the ones is always 9, the hundreds is always 10 (well, zero, and then the thousands is always 1).

    So you have to get 1089 every time, so long as your start number meets the criteria of a>b>c.

    Cool trick!

    ReplyDelete
  2. I put the spooky 99 in my spreadsheet of these. http://bit.ly/NumberTricks Will think more about Rob's 3 digit. So close to being 10,9,8 or 8910...

    ReplyDelete
  3. First of all, any number will work as long as the first and last are not the same.

    102 is a nasty but workable example.

    201 - 102 = 099

    The reverse of 099 is 990

    990 + 099 = 1089. You must subract the larger from the smaller. I went through a contortion with this until I discovered the method.

    Let's consider a number where a and c are not equal and (without loss of generality) a>c

    100A + 10B + C will always require that 2 borrows take place when subtracting 100C + 10B + A from 100A + 10B + C thus:

    100(A-1) + [100 + 10*(b-1)] + 10 + c - a + 10]- 100C + 10B + A

    But when you remove the brackets you are left with

    99A - 99C

    That's really amazing fact number 1.
    Now what you get is 99(A - C)

    A- C> 0 so you can always represent A-C with a single letter say X and add 1. That's the key to solving this problem.

    99(X + 1) Now remove the brackets again.
    99x + 99 Now we have to make this look more like 100H + 10T + U (hundreds tens Units)
    Add an x to get the hundreds and subract x from the units.

    100 x + 90 + 9 - x Reverse this number
    100(9 - x) + 90 + x And add
    ==========================
    100x + 90 + 9 - x + 900 - 100x + 90 + x

    900 + 90 + 9 + 90 = 900 + 180 + 9 = 1089

    This is more than just spooky.

    ReplyDelete
  4. Jerome - this is exactly why I love math. The trick with (X+1) is very cool and makes everything simpler.

    John - when I click on the spreadsheet link it opens another math trick (your shoe size predicts your age). Not this one:)

    Rob - I like your explanation. Much better than what I would have done but still a bit complex. See Jerome's trick.

    ReplyDelete
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