Happy and sad numbers

In Pickering I was browsing a second-hand bookshop / toy shop and came across the most unlikely of books to buy – Oxford Study Mathematics Dictionary.  At £3.99 it was a bit expensive for an obscure read, but what magic is contained within!

It’s a maths dictionary.  ie All the maths words and phrases, with a simple definition.  As I work my way through the entries I will post any interesting entries.  For example happy numbers and sad numbers.

19, for example, is a happy number because you can square all its digits and add them together to make another number and by repeating this process eventually you get a result which is the number 1.

For example 19 -> 1 + 81 -> 82 -> 64 + 4 -> 68 -> 36 + 64 -> 100 -> 1 + 0 + 0 -> 1

 If this process doesn’t eventually result in the number 1, then the number is said to be ‘sad’!  The number 18 is an example of a sad number!  Do the maths yourself!

See Wikipedia to blow your mind further.

Should I have been a mathematician?

Many, many years ago I noticed that the square of a number consisting of all sixes had a particular pattern.

With pretensions to be a mathematician I set out to prove why this was so! Every ten years or so I would re-visit the previous solution and attempt to improve on it. Now I have decided to publish it on my blog, for posterity.

If you understand it, comments are welcome. See image files below or a PDF file here