Casting out nines

kelper · Feb 9, 2024

Here is an amusing trick to play on your friends.

Ask them to think of a number between 100 and and 1,000. Suggest it has three different digits and tell them not to reveal it to you.
Ask them to multiply it by nine.
Ask them to add the digits together. If there is more than one digit, repeat until only one digit remains.

The answer will always be 9!

Example: 123 x 9 = 1107
digits add up to 9

Example 999 x 9 = 8991
digits add up to 27
add these 2 + 7 = 9

This works for any positive number, even with decimals, so you could use a much longer number and impress them with your mental arithmetic while they use pen and paper.

Example Pi x 9 is 3.1415926535 x 9 = 28.2743338815
digits add up to 54, which add up to 9!

cereberus · Feb 9, 2024

Another game.

Write down this number 142857 and ask somebody to multiply it by 2 and 3 using calculator or long mutiplication as they wish.

Then tell them to multiply 142857 by 4, 5, and 6 without a calculator without a calculator in max 15 seconds.
The trick is to see a pattern after multiplying by 2 and 3.

kelper · Feb 10, 2024

cereberus said:
Another game.

Write down this number 142857 and ask somebody to multiply it by 2 and 3 using calculator or long mutiplication as they wish.

Then tell them to multiply 142857 by 4, 5, and 6 without a calculator without a calculator in max 15 seconds.
The trick is to see a pattern after multiplying by 2 and 3.

I see that the digits just rotate in each multiplication but it seems by a differing number of positions. After 2 and 3 I could not predict 4, 5 or 6.

Where did you find this quirky arithmetic?

cereberus · Feb 10, 2024

kelper said:
I see that the digits just rotate in each multiplication but it seems by a differing number of positions. After 2 and 3 I could not predict 4, 5 or 6.

Where did you find this quirky arithmetic?

142857

Tick is to start with first digit of n x 14 rotating digits as you noticed.

2x14 = 28 thus 285714

3x14 = 42 thus 428571

4x14 = 56 thus 571428

5x14 = 70 thus 714285

6x14 = 84 thus 857142

re. quirky - something I was taught over 50 years ago at school when proper maths was taught.

kelper · Feb 10, 2024

Thanks. I figured that eventually, but thought there might be an easier way. It is a peculiar quirk of arithmetic unless there are other, similar numbers. I remember a way to extract square roots by a form of long division.

Wisewiz · Feb 10, 2024

kelper said:
I see that the digits just rotate in each multiplication but it seems by a differing number of positions. After 2 and 3 I could not predict 4, 5 or 6.

I agree with kelper on this one. I did the math and saw that the same numbers were repeated, but I was looking for a pattern like this:
At x2, the number repeats starting with the second-last position 285714.
At x3, it repeats starting with the last position 428571.
so, if the rotation continues, at x4, it should repeat starting with the first position (142857), but that wasn't possible, and at x5, it should repeat starting with the second position (714285) and that worked, so at x6, it should start at the third position (571428), but that didn't work at all, (857142).
So I abandoned my first hypothesis, and then I never got back to it until you posted earlier today. Thanks.

Dan

cereberus · Feb 10, 2024

Interesting one is when you multiply 142857 by 7 (a connection in a way to 1st post).

Wisewiz · Feb 10, 2024

@cereberus and @kelper
Then when you try x8, you get 1142856, which is one short of being the original, if you drop the 1 at the beginning. But it's exactly that 1 that you need to make it _142857.

cereberus · Feb 11, 2024

Divide 1/ 7 and you get
.142857142857142857142857.... ad inifinitum.

cereberus · Feb 11, 2024

kelper said:
Thanks. I figured that eventually, but thought there might be an easier way. It is a peculiar quirk of arithmetic unless there are other, similar numbers. I remember a way to extract square roots by a form of long division.

You use iteration.

1st guess root 2 equals 1.4 1st guess.
2/1.4 = 1.42857 2nd guess

guess 3 = (guess 1 + guess 2)

2/guess 3 = guess 4

guess 5 = (guess 3 + guess 4)

2/guess 5 = guess 6

keep repeating until guess n = guess n+1

Of course, it is an infinite series if the root of a number is not an integer.

kelper · Feb 11, 2024

Long division is neater

Long Division Method to find Square root with Examples - GeeksforGeeks

Your All-in-One Learning Portal: GeeksforGeeks is a comprehensive educational platform that empowers learners across domains-spanning computer science and programming, school education, upskilling, commerce, software tools, competitive exams, and more.

www.geeksforgeeks.org

OR

Square Root by Long Division Method

Square Root by Long Division Method - Concept - Step by Step Explanation

www.onlinemath4all.com

Casting out nines

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