Many of the posts so far on this blog have been problems to solve.  This is more of an article to read, but it still involves some opportunities for thought and action. Here are some of the divisibility rules I find useful.  Others may exist too.  Some of these will be familiar to you but I hope there will be something here that will be new to you and that you can make some new connections as well.  Finally, it would be good to be able to prove these rules. (Part 2 will appear at some point and will give some further thoughts on divisibility rules.) These are primary school rules: Divisible by 2: final digit is 0,2,4,6 or 8 Divisible by 10: final digit is 0 Divisible by 5: final digit is 5 or 0 Then there are the rules for 4 and 8: Divisible by 4: final pair of digits forms a 2-digit number that is divisible by 4 Divisible by 8: last three digits form a 3-digit number that is divisible by 8 Hmm.  There is a pattern here.  ...

This is a lovely question that I was emailed by brilliant.org If you haven't seen that site then do go there.  I suggest signing up for the free access and having fun exploring some of their frankly brilliant problems. This one involves some nice techniques and will allow you to delve into different A-level maths ideas.

How did you do it?

Here is a question I came across recently.  Looks easy. What is the problem with it? The question was presented without the prompt "What is the problem".  Is this a fair question?

A problem to solve: Let a be the answer to this problem, where a is a positive integer, and let b be the sum of its digits.  Calculate 2 a – 2 b .

Lots of ways have been proposed, with plates affixed to space probes and messages being broadcast to the universe.  The movie ' Contact ' has one method.  (I enjoyed the book too.) The 'Arecibo Message' is interesting.  Here it is: Before you look it up ... 1) How could it be sent? 2) How will aliens know how to reconstitute the image? 3) What do the different parts of the message mean? 4) Does it include sensible things? Find a full explanation here (but have a think about it before looking): https://en.wikipedia.org/wiki/Arecibo_message

It's back!  Very exciting! The Cipher Challenge is run by the Maths Dept at University of Southampton and is sponsored by Trinity College in Cambridge, IBM and others. Each week during the challenge teams of under-19s work to crack codes.  There is always a story and it is great to get immersed in the scenario that has been created. Some use of ICT has been helpful in the past.  Using 'search and replace' in Word, or using a few Excel formulae, or using some fairly easy Python. Do go to the website and have a look: https://www.cipherchallenge.org/ I strongly suggest you work with others and sign up.  The website has some teacher resources too - they are worth a look. Enjoy!

The square ABCD is a unit square.  Next to it is square BEFG.  I am interested in the triangle AFC. What is its area? Why is this a surprising question? What happens if square BEFG is smaller than ABCD? There is an interactive version of this image at https://ggbm.at/BphKr2mG