Subject: Strange number sequence
Date: Sat, 07 Mar 1998 19:11:44 +0000
From: Austin Plunkett

Hi. I know you don't runa a questions-and-answers service, but I can't really find anywhere to ask this. I was wondering if you've come across or can explain the following sequence of numbers:

1 4 9 7 7 9 4 1 9

To arrive at this sequence, take the numbers one to ten. For each, square it, and reduce it to a single digit by adding the resulting digits together. So 1 squared is 1. 2 squared is 4. 3 is 9, 4 is 16, which gives 7 because 1 plus 6 is seven... and so on. For results with more than two digits, add them, and keep adding the resulting digits until you arrive at a single digit. It will always be one of the above, in the order given.

Following this, I noticed that if you take any digit from the group which gives the first 4 digits (1, 10, 19, 28, 37 etc) and multiply it by any other then reduce it to a single digit, you get the single figure at the head of that group (1, 4, 9 or 7). To explain a little better, look at the table:

``` 1   4   9   7  |  7   9   4   1  |  9
--------------------------------------
1   2   3   4  |  5   6   7   8  |  9
10  11  12  13  | 14  15  16  17  | 18
19  20  21  22  | 23  24  25  26  | 27
28  29  30  31  | 32  33  34  35  | 36
```

etc

So, in the first '1' column, multiply any of the digits by any other, reduce it to a single digit by addition, and you'll get '1'. (19 x 10 = 190, which gives 10, which gives 1). In the '4' column the same holds true, and in the 9 column, and the 7 column. After that, in the '7 9 4 1' group, this doesn't work, but it does work in the final '9' column.

I realise that this is probably a trivial artifact of a base 10 numerical system (there are 9 digits in the sequence, one less than 10, our standard base). I wondered if similar patterns have been spotted in other base systems, or if this pattern has been studied before, and if it is of any use other than simply being a curiosity.

I am not a mathematician, so go easy on me! ;)