The Amazing 1089 Trick Demonstrated and Explained

Many number patterns are entirely predictable.

If you rearrange any number to form a new number, and find the difference between the two numbers (i.e. subtract), the answer will always be divisible by nine. This formed the basis of the coin trick in my last video (   • Amazing Coin Trick (Mathematical)  ).

If you REVERSE a three digit number and find the difference between the new number and the original one, the answer will always be divisible by eleven AS WELL (and, therefore, by ninety-nine).

This REVERSE AND SUBTRACT pattern ensures that the answer conforms to a particular structure, a9b where the numbers a and b add up to nine, i.e., 099, 198, 297, 396, 495, 594, 693, 792, 891, or 990. Reversing this number (b9a) and adding it to the a9b will ALWAYS produce 1089 as I explain in the video. If you are trying this challenge out with someone, you can have a bit more fun by secreting your answer somewhere and producing it as 6801 ... which will produce a little bit of confusion, before you rotate your answer to reveal the true answer of 1089!

Remember that this trick ALWAYS WORKS provided two rules are adhered to:

1. The starting number must not be a palindrome (i.e. it must not read the same in reverse. If such a number is chosen, when it is reversed and subtracted, the difference will be found to be zero! Technically, this is still divisible by 99 (so the number theory pattern/rule still holds), but it is a redundant sort of result, so we exclude it by starting with a number that looks different when reversed.

2. If, after subtracting, your friend ends up with 99, they must consider it as 099, i.e. as a three digit number, and reverse that to form 990 before adding.

Make these rules clear to your friend, and the trick should work every time.

Because I did not want the video to be unduly lengthy, I did not give a FULL explanation of the number theory principles involved. If you have any questions, please ask them in the comments below the video, and I will do my best to answer them for you.


Number Theory is a branch/aspect of mathematics in which we analyse how numbers work and what makes them behave the way they do. Number theorists are fascinated by prime and composite numbers, divisibility, number types and patterns. A whole world opens up ... and it is fascinating ... much moreso than you might think! I will be sharing more material relating to number theory as I am able.

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