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Mobile Numbers: Multiples of nine

In this series of posts, Katie investigates simple mathematical concepts using the Google Sheets spreadsheet app on her phone. If you have a simple maths trick, pattern or concept you’d like to see illustrated in this series, please get in touch.

We’re all (hopefully) aware that a pleasing property of numbers that are divisible by nine is that the sum of their digits is also divisible by nine.

It’s actually more well known that this works with multiples of three, and an even more pleasing fact is that the reason three and nine work is because nine is one less than the number base (10), and anything that’s a factor of this will also work – so, in base 13, this should work for multiples of 12, 6, 4, 3 and 2. Proving this is a bit of fun.

Once when I was thinking about this fact, an interesting secondary question occurred.

Puzzlebomb – December 2016

Puzzlebomb LogoPuzzlebomb is a monthly puzzle compendium. Issue 60 of Puzzlebomb, for December 2016, can be found here:

Puzzlebomb – Issue 60 – December 2016 (printer-friendly version)

The solutions to Issue 59 will be posted around one month from now.

This will be the last regular monthly Puzzlebomb – in future, there will be occasional one-offs but regular editions are taking a break. If you have any ideas for puzzles, please send them in! Previous issues of Puzzlebomb, and their solutions, can be found at Puzzlebomb.co.uk.

CP’s solid used as the basis of a puzzle

Back in 2013, our own Christian Lawson-Perfect came up with a way of making a solid from the smallest non-Hamiltonian graph, the Herschel Graph. Called the Herschel Enneahedron, it’s got nine faces (three squares and six kites) and the same symmetries as the graph itself.


The most recent news is that Spektrum magazine – sort of a German version of New Scientist – has included in its regular puzzle column a Herschel Enneahedron-related challenge. Here’s Google’s best effort at translating it:

Please make a polyhedron of 3 squares and 6 cover-like kite rectangles with suitable dimensions (in your thoughts, drawings or with carton). What symmetry properties does it have, how many corners and edges? Is it possible to make a (Hamilton-) circular path on its edges, which takes each corner exactly once and does not use an edge more than once?

Before you get out your cartons and start working on this, given that we started from a graph which isn’t Hamiltonian, you may have a slight spoiler on the answer here… but the solution given includes some nice videos and explanation as to how the solid is formed.

Treitz Puzzles 313, at Spektrum.de

The maths of the Grime Cube

Not content with already having five cubes named after him, internet maths phenomenon James Grime has now developed a new Rubik’s cube-style puzzle for internet maths joy merchants Maths Gear. I’ve been slightly involved in the development process, so I thought I’d share some of the interesting maths behind it.

Another name for a Rubik’s cube is ‘the Magic Cube’ – and Dr James Grime wondered if you could make a Magic Cube which incorporates its 2D friend, the Magic Square.