## Things To Ponder

We’ve all looked at our reflection in fun house mirrors. Our distorted, grotesque image -reminiscent of Edvard Munch’s painting “The Scream”- is an example of rubber sheet geometry.

Now, can a coffee cup be a doughnut? And can you remove your jumper without first removing your overcoat?

In a branch of mathematics known as topology, the answer to both questions is ‘yes’.

Topologically, there is no difference between a coffee cup and a doughnut because one shape can be transformed by stretching and without tearing to obtain the other shape.

(Admittedly, I would still prefer to eat a doughnut rather than ingest a coffee cup!)

If you’re a fan of Mr Bean you may recall the episode at the beach when he removed his trousers from under his swim trunks.

And how is this trivia relevant to mathematicians at Monty?

I often observe students in the same predicament as Mr Bean. They struggle at first but manage to extricate themselves from seemingly impossible situations. Via judicious “kneading”, they transform tasks into manageable processes that remain faithful to the concept of topological invariance.

For example, is it easy to recognise that the sum of the first N odd numbers is the square of N?

Or is it appreciated what would happen if the Mobius strip was cut along the dotted line? Try it.

Why did the chicken cross the Möbius strip?

To get to the same side!

Finally, what would be a maths page without puzzles and activities? Here are four tantalising teasers.

Activity

A famous sequence of numbers consists of 1, 1, 2, 3, 5, 8, 13, .. where each number is the sum of the two preceding numbers. Many writers have composed poems whose number of syllables in each line are the numbers in the sequence described. A Fib is a poem of six lines where the number of syllables in each line is 1, 1, 2, 3, 5, 8.

For example: (Sung to the tune of the Beatles song, Octopus’ Garden)

1          I

1          like

2          to be-

3          at the sea,

5          where the octopus

8          crawls quietly, underneath the waves.

Create a Fib of your own. Be as outlandish as you can!

Puzzler 1

because if you add 1 to infinity it is still infinity. Now subtract infinity from both sides:

which leaves us with. How can this be?

Puzzler 2

The shape has 22 coloured squares around the outside

and 8 clear squares inside. What size square or rectangle

is required so that the number of coloured squares is twice

the number of clear squares?

Puzzler 3

You are on an island and there are three crates of fruit that have washed up in front of you. The first crate contains only apples, the second crate contains only oranges and the third crate contains both apples and oranges.  One crate reads "apples", one crate reads "oranges", and another crate reads "apples and oranges".  However, each crate is incorrectly labelled. If you can only take out and look at just one of the pieces of fruit from just one of the crates, how can you correctly label all of the crates?