Two tricks based on the circularity of 9

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3 years ago (Last updated: 2 years ago)

This post is inspired by the response to my previous article, The Knights Templar & The Circular Number Nine , which generated comments about number 9 far beyond the original context. And numbers are fun!

Two tricks

- Write down a 3 or 4 digit number. Make it random.

- Change the order of the digits to create another number.

- Subtract the smaller number from the larger one.

- Draw a circle around one digit of the result (although not a zero), and keep it secret.

- Change the order of the digits of your answer and write them down, except for the one you circled (removed).

-> And I can tell you which number you circled (removed).

But how is it done?

This is a trick that can be quite fun, but it is not difficult to do. It is based on a couple of straightforward principles.

If you...

- form a number consisting of two digits or more (no upper limit).

- create a new number by rearranging the digits of the first one, randomly or in any way you want.

- subtract the lesser number from the larger one.

Then the result will always be a number that is divisible by 9!

Examples:

(i) 81-18=63 = 9x7 / divisible by 9

(ii) 84-48=36 = 9x4 / divisible by 9

(iii) 213-132=81 = 9x9 / divisible by 9

(iv) 6297-2967=3330 = 9x370 / divisible by 9

(v) 59643125-43619255=16023870 = 9x1780430 / divisible by 9

Further, as already stated above:

A number is divisible by 9 if, and only if, the sum of its digits is divisible by 9.

Let us now look at an example.

Assume you choose: 59643125 (without showing it to me).

Then you create a second number by rearranging the digits of the first: 43619255.

You subtract the lesser one from the larger one:

59643125-43619255=16023870.

Now you are "drawing a circle" around , say, the "3" – that is, you remove that number and keep it secret.

Then you tell me the other digits, in any order you wish: 7106802.

I quickly calculate: 7+1+0+6+8+0+2=24. I know you have removed something, with which this should be a number divisible by 9. The closest number higher than 24 that is divisible by 9 is 27. So I set 27-24=3. This was the digit you withheld from me!

Since, due to certain properties of the number 9, we can continue to add digits until we have only one left, without changing the result, I could have been continuing with the sum I got, 24, one step further. 2+4=6. And 9-6=3. The same result!


There is another similar trick, which can be done like this:

- Ask someone to choose a number, consisting of at least 2 digits, without telling you which.

- Tell him to rearrange the digits, so he gets two other numbers.

- Now he should add the three numbers (the original one, and the two he created by rearranging the digits of the first one), and multiply the result by either 3, 6, 9, 12, or any other multiple of 3.

- From the result, ask him to remove one digit, except a 9.

- Now you ask him to tell you the number he still has, and you can tell him which digit he removed.

An example:

Assume he chooses 1732, and he forms two other numbers by the same digits, 3172 and 2137.

These are added: 1732+3172+2137=7041

He multiplies this by 3: 3x7041=21123

He removes a 2, and still has 2113, which he tells you.

You add 2+1+1+3=7, 9-7=2, which was the digit he removed.

One of the two principles behind this is that, if you add three numbers, each one formed by the same digits, the sum of these three numbers is always divisible by 3. And if you multiply the sum by 3, or any multiple of 3, the result is always divisible by 9.

The second one is the same as above: A number is divisible by 9 if, and only if, the sum of its digits is divisible by 9.

(This material has previously been published in TMA/Meriondho Leo and in my e-book “Numericon”.)

Copyright © 2019, 2020 Meleonymica/Mictorrani. All Rights Reserved.

If you are interested in numbers, you find all my articles related to numbers here, and please also join my community Numbers (7b6a).

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3 years ago (Last updated: 2 years ago)

Comments

Yeah love these cool little math tricks. It is amazing how many there are but then again its about numbers and well there are a lot of them :) lol

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3 years ago

This is fantastic! This must have taken you some time.....I guess? Keep up the good job!

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3 years ago

Hmm! That is critical thinking. We are learning from each other, is good as you solved it that way.

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3 years ago

This is brain teasing in deed. If you can develop this into a software program for students, you'll make cool money.

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3 years ago

you think so? I never really thought of that. It's very simple really. The trick is not as advanced as it looks.

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3 years ago

This is pure mathematics and not magic, but such things can be performed well at children's birthday parties (the children must of course be a bit taller to be able to calculate). Then you are admired as a great magician. Thanks for the article.

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3 years ago

Of course it is not magic and I had not planned in advance to write that article. It was the comments to the article about the Templars that inspired me and reminded me of these old tricks. A little playfulness now and then cannot harm, not even for the "tallest" children.

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3 years ago

New ideas usually emerge after discussions and in exchange with other people. I like such number games very much and have read a few books about them in the past. Maybe I will dust them off at some point (if I find them at all 😉).

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3 years ago

Yes, why not dust them off? I am also very amused by such games. So are a lot of people.

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3 years ago

Lovely

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3 years ago