This post is (very late) the 140th Carnival of Mathematics. This blog Carnival is organised by The Aperiodical and is hosted by various maths bloggers from around the world. The last Carnival was hosted by Manji Saikia at Gonit Sora.

It is the tradition that this carnival is always started with some interesting number facts, so here are some facts about the number 140:

- 140 is a Harshad Number. A harsh ad number is simply a number (in a given base) that is divisible by the sum of its digits in that base. As 140 is divisible by 10, 140 is a Harshad number in base 10.
- 140 is the sum of the squares of the first seven integers.
- It is a square pyramidal number – these numbers are the number of spheres in a pyramid with a square base of a given size (think cannonballs).

On to the posts… It’s been a bit of a slow month but there have still been plenty of great things happening in the Maths Blogosphere.

Manan Shah (@shahlock) sent in this post about the concept of “no solution” problems which was prompted by a post he had read concerning the classic “Shepherds Age” problem. If you aren’t familiar with this problem it is a classic problem where useful information is given to enable it to be solved, yet a staggeringly large amount of people still try to solve it by making use of prior knowledge, adding spurious deductions etc. As a teacher, I see this kind of behaviour in the classroom often and I have often wondered if this would be as bad if we exposed children to problems with no valid solution sooner than an undergraduate linear algebra course, as is typically the case in the UK.

Matthew Scroggs (@mscroggs) submitted this post written by Steven Muirhead entitled “Problem Solving 101“. I first read this article in the print version of Chalkdust magazine, and thought it was excellent. It is a very well structured introduction to some typical strategies for mathematical problem solving, namely:

- Get some intuition.
- Find and exploit the structure of the problem.
- Look for quantities that do not change.
- Consider the extremes.

Chalkdust is – if you haven’t already come across it – an excellent, relatively new magazine for the “mathematically curious”. It is available online, but they also produce a printed version which you can order if you pay for postage. In the run up to Christmas they are producing an Advent Calendar – of the days so far I particularly like the mathematical Christmas carol.

I was excited to see a blog I wasn’t aware of in the submissions list this time. It is called Tony’s Maths Blog. Sadly I’m not sure who is behind this blog (all the bio says is that they teach at a University in London) but it has some interesting short posts, such as this one about a book where the murder victim is a mathematician.

The always entertaining mathwithbaddrawings by Ben Orlin (@benorlin) has a great, thought provoking, post about possible ways to arrange the school mathematics curriculum. I’m really not sure how I think the curriculum should be arranged, but I do think something should / needs to be done to stop the large amounts of young children who don’t see the beauty of mathematics and become very disaffected.

Not strictly a blog post, but I wanted to include it as I found it fascinating, is this article on probabilistic programming from Cornell University. It’s fairly maths and programming heavy but a nice, accessible introduction to this research area.

James Hunt has shared an article looking at the frequency distribution of colours of smarties on the site MentalFloss. Many teachers will have done lessons with classes based around a tube of smarties!

Recently Ed Southall (@solvemymaths) has been posting some excellent area puzzles, such as this one.

At the end of November an excellent article was posted on to the AMS Blogs page about the teaching of inverse functions. I’ve hated the whole “swap the variables and solve” approach as students often have no real understanding of inverse functions if they have been taught this way and instead just see it as a black-box algorithm to apply. This article discusses this problem in great detail with a few alternative proposals.

Jemma Sherwood (@jemmaths) posted about Francis Galton’s Wisdom of the Crowds observation here – I’m certainly going to steal this idea for an activity in school. A reproduction of Galton’s original article is available from here.

Stephen Cavadino (@srcav) has shared a question from brilliant.org in his post where he discusses his solution method . Stephen has also fairly recently written a blog post on the use of mnemonics in maths teaching – I’ve never been a fan of these, but they do seem popular. I’d love to know your views.

That brings us to the end of the carnival. The next one is hosted at Ganit Charcha (who incidentally have an interesting article on population growth)- make sure you check it out!