Number Strings and Reversing NumbersDate: 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 |
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