Can you solve it? Maths on the back of an envelope

Today, we’re pushing the envelope.

1) A piece of paper is folded to make a shape that looks a bit like the back of an envelope, illustrated below. If the paper is unfolded again to make a flat sheet, what shape will it be? Prove it.

2) The same sheet is now folded again, this time increasing the amount folded over so that there is some overlap between the top and bottom flaps, illustrated below. This looks more realistic (it resembles the so-called ‘Baronial style’ envelope). By how much can the side flaps be made to overlap the top and bottom flaps, so that the rectangular envelope can be stuck down? Prove it.

3) The ‘envelope’ from question (1) is now opened as below. Can you draw all the lines this image without taking your pencil off the paper, and without re-tracing or crossing over one of the lines you’ve already drawn?

4) Can you draw the image of the folded envelope from question (2) in the same way?

Today’s puzzles were devised by the maths writer Rob Eastaway in a nod to his latest book, Maths on the Back of an Envelope, a joyful primer about the lost art of calculating without a calculator. His book full of insightful tips on how to work things out quickly and sensibly, including a guide to arithmetical rules of thumb and a discussion of the importance of estimation. It’s full of anecdotes too. For example, he writes that back in the 1970s, in the pre-pocket calculator era, cornflakes packets used to display the following mnemonic:

A litre of water’s a pint and three-quarters.

How useful! Can anyone think of a good mnemonic that does the conversion the other way round (1 pint = 568ml)?

A pint of….

A copy of Maths on the Back of an Envelope to the entry Rob and I deem the best. Write your suggestion below the line, email me or tweet me.

In other pre-calculator era maths news, geometers will be interested to know that the largest piece of Islamic geometric design to have been built in the last six centuries was recently unveiled in….Yorkshire!

The exterior of the Zakaria Education Centre in Dewsbury is a 9m x 4m mosaic of three thousand of pieces of ceramic tile, designed by the UK-based Dutch artist Eric Broug based on a design from Meknes, Morocco. The original design, made in 1341, would have been made using only compass, ruler and templates.

And here’s a close-up.

I’ll be back at 5pm UK with the solutions.

PLEASE NO SPOILERS. Instead talk about your favourite mathematical mnemonics, the beauty of Islamic geometric design, or your favourite stationery.

Thanks to Rob Eastaway for the puzzles. His new book is Maths on the Back of an Envelope. He is also runs Maths Inspiration, a national programme of interactive maths lecture shows for teenagers.

I set a puzzle here every two weeks on a Monday. I’m always on the look-out for great puzzles. If you would like to suggest one, email me.

I’m the author of several maths and puzzle books, as well as the book series Football School, for 7 to 12 year olds, that opens up the curriculum through football. The latest in the series, Football School Season 4, is out now. It contains chapters on the links between football and classical music, spitting, magic, timekeeping, the International Space Station, etymology and much much more.