I want to let you in on one of our favourite mathematical mysteries... To get started, choose a four digit number where the digits are not all the same (that is not 1111, 2222,...). Then rearrange the digits to get the largest and smallest numbers these digits can make. Finally, subtract the smallest number from the largest to get a new number, and carry on repeating the operation for each new number.

We'll show you what we mean with the number 2005. The maximum number we can make with these digits is 5200, and the minimum is 0025 or 25 (if one or more of the digits is zero, embed these in the left hand side of the minimum number). The subtractions are:

5200 - 0025 = 5175

7551 - 1557 = 5994

9954 - 4599 = 5355

5553 - 3555 = 1998

9981 - 1899 = 8082

8820 - 0288 = 8532

8532 - 2358 = 6174

7641 - 1467 = 6174

When we reach 6174 the process repeats itself, returning 6174 every time.

Now try with your four digit number... what do you get? I bet you'll get to 6174, every time, no matter what number you chose! Try a few more and see if you believe me!

Do you think we always reach the mysterious number 6174? If we do, why do you think that happens? If this mystery piques your interest, then you can find out why in this excellent article by Yutaka Nishiyama. This question has been intriguing *Plus* readers for years, and Yutaka's article remains one of our most popular articles, generating reams of comments, emails and discussions. And spoiler alert - 6174 isn't the only number with these special numbers - but you'll have to try the same process with three digit numbers, or read the article, to find out!

## Comments

## Magic number 6174

This is very interesting, thanks for posting it!

I understand that the digits of the four-digit number chosen should not be all equal (AAAA) for this process to work but are there other four-digit number exceptions that won't yield 6174 through this process? For example, what about number 1000?

Thanks again,

Aziz Inan

## Magic number 6174

I now realize that I must always keep the result of each subtraction as a four-digit number. The trap I fell into was 1000 - 0001 = 999 instead of 0999. :-(

Thanks again!

Aziz Inan

## Mathematics

I tried with 2035. I found true. Amazing!

## yes 2035

yes 2035

## Mysterious 6174

Yeah! It's really amazing. I too tried it with 1704.

7410-0147 = 7263

7632-2367 = 5265

6552-2556 = 3996

9963-3699 = 6264

6642-2466 = 4176

7641-1467 = 6174

## Mysterious 6174

Do you think we always reach the mysterious number 6174? If we do, why do you think that happens

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## 2-digit and 3-digit number

I've followed the same approach with two and three digit numbers. Here are some interesting observations after a few iterations:

* three digit numbers

is 495 the magic number for three digit numbers?

example 1 example 2

100 - 001 = 099 920 - 029 = 891

990 - 99 = 891 981 - 189 = 792 (eventually 495)

981 - 189 = 792

972 - 279 = 693

963 - 369 = 594

954 - 459 = 495

954 - 459 = 495

* Two digit numbers

For two digits the process does not "converge" to an exact result or mysterious number instead you may get caught up in a loop:

i(1) 84 - 48 = 36

(2) 63 - 36 = 27

(3) 72 - 27 = 45

(4) 54 - 45 = 09

(5) 90 - 09 = 81

(6) 81 - 18 = 63 this is the largest digit from the result of iteration (1).

Example 2:

91-19 = 72 ( largest from iteration 2 from example 1)

72 - 27 = 45

54 - 45 = 09

90 - 09 = 81

81 - 18 = 63

what about 5,6, ..., n digit numbers? Is there a way to find all the "mysterious numbers"?

## 6174 The easy way

If you want to try it the easy way you can choose the actual magical number 6174, it renders down to only one subtraction:

6174 =>

7641-1467 = 6174