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### Number Strings and Reversing Numbers

```Date: 02/20/2004 at 00:33:56
From: Ajit or Joe
Subject: Number strings

I need to find an 18-digit number in which no two consecutive digits
are alike and the number is such that it reverses itself upon being
multiplied by 4.

I have found out a 4-digit string (2178) which reverses itself upon
being multiplied by 4 (8712).  By adding together strings of 2178, one
can make 8-digit, 12-digit or any (multiple of 4)-digit numbers such
as 21782178 or 217821782178 which reverse themselves upon being
multiplied by 4. But how does one go about finding an 18 digit number
with the same property?

```

```
Date: 02/20/2004 at 12:46:24
From: Doctor Vogler
Subject: Re: Number strings

Hi Ajit or Joe,

There is one other 4-digit string which reverses itself upon being
multiplied by 4.  You didn't notice it because you thought it was too
obvious, but it is important: 0000.

Now, you can prove using a little modular arithmetic and some simple
bounds that every n-digit number which reverses upon being multiplied
by 4 either has a leading and a terminating zero with an (n-2)-digit
reversing number in the middle, or it starts with 21, ends with 78,
and has a number in the middle which is all nines minus an (n-4)-digit
reversing number.

For example, all 1-digit reverses are:

0

All 2-digit reverses are:

00

All 3-digit reverses are:

000 (zeros on ends, go up two sections to fill in the middle)

All 4-digit reverses are:

0000 (zeros on ends, go up two sections to fill in the middle)
2178

All 5-digit reverses are:

00000 (zeros on ends, go up two sections to fill in the middle)
21978 (21-middle-78, go up four sections to fill in the middle)

All 6-digit reverses are:

000000 (zeros on ends, two ways to fill in the middle)
021780
219978 (21-middle-78, one way to fill in the middle)

All 7-digit reverses are:

0000000 (zeros on ends, two ways to fill in the middle)
0219780
2199978 (21-middle-78, one way to fill in the middle)

All 8-digit reverses are:

00000000 (zeros on ends, three ways to fill in the middle)
00217800
02199780
21999978 (21-middle-78, two ways to fill in the middle)
21782178

All 9-digit reverses are:

000000000 (zeros on ends, three ways to fill in the middle)
002197800
021999780
219999978 (21-middle-78, two ways to fill in the middle)
217802178

and so on....

If you have any questions or need more help, please write back and
show me what you have been able to do, and I will try to offer further
suggestions.

- Doctor Vogler, The Math Forum
http://mathforum.org/dr.math/
```

```
Date: 02/20/2004 at 21:40:22
From: Ajit or Joe
Subject: Thank you (Number strings)

Thanks Dr. Vogler.  As I see it, one could have for an 18-digit number
any symmetrical combination such as 217802178217802178 or
217821978219782178 which has the necessary property.  Thanks once again.

-Aji
```
Associated Topics:
High School Puzzles
Middle School Puzzles

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