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

Earlier today I set you four puzzles, and a challenge to devise a rhyme about a pint being 568ml. The solutions first and the best pint-sized poems afterwards.

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.

Solution: a rhombus (a quadrilateral with four sides the same length). Simple geometry shows that the angles a + b = 90 degrees, so 2a + 2b = 180, a straight line

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.

Solution: If the ‘envelope’ is still rectangular, the side flaps can never overlap the other flaps. They will perfectly fill the triangular gaps. You can see this by looking again at the solution diagram for (1). If you make the rectangle shorter and wider, keeping it centred, the angles a and b stay the same, so the edges of the flaps must still share the same lines. And if they do, the side flaps must fill the triangular gaps perfectly.

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?

Solution. There are several ways to do it, one example is illustrated left, but you must always start at one of the bottom corners, because these are the only two junctions (nodes) with an odd number of lines. In this case the bottom two corners have three lines, and every other node has four lines.

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

Solution: No it is not possible, because there are more than two nodes with an odd number of lines (there are four nodes with three lines each). You can only trace a path that never crosses or retraces a line (what’s called an Euler path) if there are either two or zero ‘odd’ nodes.

In today’s other challenge I wrote that, in the 1970s, cornflakes packets had the following mnemonic:

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

I invited you to come up with a version that converts the units the other way round (1 pint = 568ml).

About 100 entries came in and in the opinion of the judges (me and Rob Eastaway, whose book Maths on the Back of an Envelope inspired the challenge) the two winners are the following. (We couldn’t decide on a singe winner so we decided on two.)

A pint of Pils is 568 mls

and

“How many litres in a pint?” someone tweeted

Just think of the Steps song with the 7 deleted

The first one (by Andrew Viner) scans, is to the point and memorable. The second one (by Bob Craddock) is less concise but funnier – and I like the fact that it describes the digits. A copy of Maths on the Back of an Envelope on the way to both.

Some of our other favourites are listed here:

(1) A pint in heaven – leave out 7,

At the pearly gate say 568

(2) One pint o’ waterz

Juz taz neat az

5 6 8 milly leetaz

(3) A pint in Britain sounds much neater,

than the Aussie equivalent: four sevenths of a litre

Authors: Rose Murphy, David Thomas, and Dan Jessen (in Queensland)

The next one has kudos for lateral thinking, but sadly doesn’t rhyme.

Some other tweeted responses

And finally….

More great suggestions can be read below the line on the original post. Please continue adding new ones here.

Thanks to Rob Eastaway for today’s 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.