A number is said to be a Harshad number if it is divisible by the sum of its digit. For example, consider 156. The sum of its digits will be 1 + 5 + 6 = 12. 156 is divisible by 12. So, 156 is a Harshad number.

Needs more than a diagram

While some written math problems are easier solved if you draw a diagram, some written math problems are beyond solution. Not because they are intractably difficult, but because the author is an idiot, the publisher is an idiot, the ‘education’ body that chose the idiot question by the idiot author, and the idiot teacher (using that word in it’s broadest sense), is innumerate and unable to see that the question cannot be solved with the information given. Here’s an example:

Go ahead. There is no solution to the problem as presented. If you see crap like this showing up in assignments then COMPLAIN long and hard. Ask your child’s math ‘teacher’ how to solve it with only the information given. Students aren’t born stupid but crap like that make them give up math ASAP.


2021 was not a good technology year. Lost my original iPhone 8. Worst part of that was nobody had turned on the ‘save to Cloud’ option so photos were lost. Flip side to that was that the post-Covid period didn’t include visits to photogenic foreign parts. Back in business now so feel free to call me – if you have my number 🙂

Sadly my not-so-old ASUS began suffering intermittent shut downs and didn’t always respond well to kind words, so I purchased a larger HP laptop which came with Windows 11. I wish there was a downgrade available because Windows 11 is not-ready-for-prime-time in the same way as Windows 8. Alternatively, a switch to get rid of stupid stuff – why does the ghost outline of a second(!!!!) desktop render that corner of my screen unclickable? While I could use more real desktop (being a horizontal filer) I cannot see any benefit to have multiple virtual desktops.

I continue looking for perfect add-ons and equivalent software to overpriced commercial stuff at

Forward to the past or whatever

As a Windows user, I suffer frequent irritations when the Mother Ship sends down ‘updates’ often at inopportune moments. The whirling dots of updates are provided for your entertainment. Last month I got the opposite of an update – presumably a downdate – when the Mother Ship wanted to undo the damage. Thanks, I suppose.

Draw a diagram!

More than sixty years ago, I was taught mathematics by a relatively young teacher named Dan Drummond who was responsible for Pure and Applied Mathematics in later grades. One of the things I recall is his advice on written problem solving – “Draw a diagram, boy!”. The assumed logic behind it was that by visualising a problem you were disregarding all the extraneous clutter that came with the problem as stated and could concentrate on the essential elements of the actual problem … which would make it simpler to solve the problem.

The use of diagrams for explaining or defining issues was recognized by Euclid. If it was good enough for one of the math gods, it was good enough for me.

You can read more about the Oxyrhynchus Papyri here

On a more modern note, while staying at the Lan Tian hotel in Shanghai 10 years back there was a problem with the digital alarm clock in my room. My Mandarin wasn’t up to writing a note to the room service lady but a diagram crossed the language barriers and resolved the issue as well as getting a ‘written’ response from the room service lady.