As Christmas approaches, I always try and get my cards out in good time. This year there is significant industrial action by the postal workers in the UK: the Communication Workers Union (CWU) members are taking national strike action on 9, 11, 14, 15, 23 and 24 December. As a result, the last dates for posting cards in the UK have been moved forward to the 12^{th} December for second class and 16^{th} for first mail. This is quite a bit earlier than usual.

In this house we have worked hard to get our cards written and into the post so that they go by second class. A second class stamp costs 68p and a first class 75p. Second class should reach its destination between two and three days and first class should arrive at its destination the next day.

Rather thean being annoyed I do try and take the trouble to understand why people are striking. The CWU has said that its members want a pay rise that matches inflation (well over 10% here at the moment), and that members feel their management wants to turn Royal Mail into an organisation similar to Uber: the latter applies mainly to parcels.

For this blog post I decided I would write about either the number 68 or the number 75 as prices of stamps are in the forefront of my mind at the moment. Both are interesting – all numbers are! But 75 has a property that is described by a word that is known to all of us.

# Self Numbers

The number 75 is a self number. This means that it cannot be written as the sum of any other positive number plus its individual digits. (Note: Whenever I write about numbers, please assume that I am writing about them in base 10 unless I specify otherwise.) I hadn’t heard of self numbers before writing this post and have really enjoyed getting to know them.

As an example, the number 21 is not a self number as it can be expressed as: 15 + 1 + 5 = 21.

However, the number 20 is a self number. Consider the numbers 1 to 19. I’ll begin with the numbers 10 to 19 as the working is easier to see:

10: 10 + 1 + 0 = 11

11: 11 + 1 + 1 = 13

12: 12 + 1 + 2 = 15

13: 13 + 1 + 3 = 17

14: 14 + 1 + 4 = 19

15: 15 + 1 + 5 = 21

16: 16 + 1 + 6 = 23

17: 17 + 1 + 7 = 25

18: 18 + 1 + 8 = 27

19: 19 + 1 + 9 = 29

20: 20 + 2 + 0 = 22

Single digits are, I think, a little less obvious. But you are, effectively, just adding each number to itself – i.e., to its one digit:

1: 1 + 1 = 2

2: 2 + 2 = 4

3: 3 + 3 = 6

4: 4 + 4 = 8

5: 5 + 5 = 10

6: 6 + 6 = 12

7: 7 + 7 = 14

8: 8 + 8 = 16

9: 9 + 9 = 18

You can see that no number is going to match 20. It has fallen through the gap between 14 and 15.

And anything larger than 20 is too large before we even get to the stage of adding its digits. So, 20 is a self number.

The first few self numbers are:

1, 3, 5, 7, 9, 20, 31, 42, 53, 64, 75, 86, 97, 108, 110, 121, 132, 143, 154, 165, 176, 187, 198, 209, 211, 222, 233, 244, 255, 266, 277, 288, 299, 310, 312, 323, 334, 345, 356, 367, 378, 389, 400, 411, 413, 424, 435, 446, 457, 468, 479, 490 and so on.

You can see some patterns. Consider the numbers 20 to 97 in the list above. There is one self number in the twenties, one in the thirties, one in the forties and so on all the way up to the nineties. This is interesting. Why is this? And the same is true from 110 onwards.

There is no doubt a neat mathematical explanation somewhere but I decided to write out all the potential self numbers between 1 and about 120 and take a look at them. I hope you find my approach useful. The LONG table at the end of the post helped me to understand how the self numbers emerge. As you can see, I wrote out the numbers in groups of 10 with the sum of the number itself and its individual digits to the right of each one, e.g. 20: 22 means that 20 + 2 + 0 = 22. I’ve separated the columns into the twenties, thirties, forties etc. so that you can see how the self numbers emerge in each. I’ve also placed gaps between each number in each column so that you can see how numbers from both earlier and later columns fill up the gaps of the current column.

For example, we looked at the teens above and got:

10: 10 + 1 + 0 = 11

11: 11 + 1 + 1 = 13

12: 12 + 1 + 2 = 15

13: 13 + 1 + 3 = 17

14: 14 + 1 + 4 = 19

15: 15 + 1 + 5 = 21

16: 16 + 1 + 6 = 23

17: 17 + 1 + 7 = 25

18: 18 + 1 + 8 = 27

19: 19 + 1 + 9 = 29

We know that 20 is a self number, but where will we find 12, 14, 16, 18, 22, 24, 26 and 28?

The answer is from two separate places. The numbers 12 to 18 come from the single digits as follows:

6: 6 + 6 = 12

7: 7 + 7 = 14

8: 8 + 8 = 16

9: 9 + 9 = 18

And where do 22, 24, 26 and 28 come from?

The answer is the twenties as follows:

20: 20 + 2 + 0 = 22

21: 21 + 2 + 1 = 24

22: 22 + 2 + 2 = 26

23: 23 + 2 + 3 = 28

Bringing this information together – and keeping your eye on the numbers on the right – gives

10: 10 + 1 + 0 = 11

6: 6 + 6 = 12

11: 11 + 1 + 1 = 13

7: 7 + 7 = 14

12: 12 + 1 + 2 = 15

8: 8 + 8 = 16

13: 13 + 1 + 3 = 17

9: 9 + 9 = 18

14: 14 + 1 + 4 = 19

NO ENTRY HERE AS 20 IS A SELF NUMBER DENOTED self: 20

15: 15 + 1 + 5 = 21

20: 20 + 2 + 0 = 22

16: 16 + 1 + 6 = 23

21: 21 + 2 + 1 = 24

17: 17 + 1 + 7 = 25

22: 22 + 2 + 2 = 26

18: 18 + 1 + 8 = 27

23: 23 + 2 + 3 = 28

19: 19 + 1 + 9 = 29

You may find it easier to appreciate the self number pattern by looking at the first three columns of the table below. I have put a red circle to illustrate where the number 20 is and to show that no numbers are adding up to 20. The self numbers seem to occur mainly because the top number of each column – 20, 30, 40 etc. – has a sum of itself plus its digits which is too large to fill the gap left in the column before. You may be able to express this much better than I’m doing so please let me know if you can by using the comments section below.

In each column the number / digit sums begin to add up to numbers in the next column about half-way down. As I’ve just said, in most cases the self numbers appear at the top of each column. There are more self numbers between 1 and 10 and between 101 and 110. The single digits are effectively doubled and so all the odd digits are self numbers. When you get to the 90s column and the 100s column you can see that there’s some double-counting and this leads to more opportunity for self numbers to emerge in the gaps.

I’ve highlighted the number 75 so that you can see how it fits into the pattern. If you take 70 you get 70 + 7 + 0 = 77 and you can see from the seventies column of the table below that 75 drops off as a self number.

To summarise, we have 13 self numbers between 1 and 99: 1, 3, 5, 7, 9, 20, 31, 42, 53, 64, 75, 86, 97. In the table below I’m looking at this relationship and pointing out where there is no lefthand side – i.e. self: 20 means there is no positive number that, when added to its digits, forms 20. I’m left wondering where these 13 numbers have gone? If you look at the eighties column and the nineties column you’ll see some numbers that, when added to their individual digits, take them to 100 or above. There are, in fact, 13 of these. What I am trying to say is that the self numbers have filled in 13 gaps that have been left by the numbers to the right of the table that are adding up to totals of 100 or more. This makes sense to me. Does it to you?

Why are these numbers called self numbers? Perhaps it is because they are “self-contained”. I like to think that this is the reason rather than because they are regarded as “selfish” for messing up a neat pattern. These numbers were first described in 1949 by the Indian mathematician D. R. Kaprekar and you can read more about them here.

# Final Thoughts

Fortunately we got all our cards out second class this year. We don’t send that many but a difference of 7p per card does add up. And these are not plentiful times. Often, when I write about numbers, they don’t contain a neat pattern in their sequence as is the case with self numbers. I could no doubt have drawn a diagram to illustrate self number – you’ll find diagrams elsewhere – but the table below provides an explanation that works for me.

This is my last post for 2022. I hope you all have a very Merry Christmas and a load of fun over the New Year period.

Please pop any comments you have about my posts below – or send an email to anna@ammauthorship.com. Occasionally I make errors in the posts – I don’t ask anyone to proofread them for me – and I find it helpful if people let me know. I don’t have oodles of cash and prizes to dish out but if you do spot an error, I’ll seek your permission to mention you in my newsletter.