## Mathematical Oddities

### Powers of Three

By adding or subtracting *Powers of Three* (1, 3, 9, 27, 81 etc.) it's possible to make any whole number (integer) that you want.
With a steel bar cut into pieces such that each piece weighs the equivalent of a *Power of Three*, you can accurately weigh any whole number quantity using a Balance. How?

*(Hints: See below. Some weights can be on the same side as the things you're weighing.)*

#### Powers of Three

3^{0} = 1 (*Any Number* greater than Zero, when raised to the power Zero, equals 1 - even π^{0}. That's the way the world works!)

3^{1} = 3 (Three to the power 1)

3^{2} = 3 x 3 = 9 (Three to the power 2)

3^{3} = 3 x 3 x 3 = 27 (Three to the power 3)

3^{4} = 3 x 3 x 3 x 3 = 81 (Three to the power 4)

3^{5} = 3 x 3 x 3 x 3 x 3 = 243 (Three to the power 5) etc. etc.

#### Examples

2 = 3^{1} - 3^{0}

4 = 3^{1} + 3^{0}

5 = 3^{2} - 3^{1} - 3^{0}

43 = 3^{4} - 3^{3} - 3^{2} - 3^{1} + 3^{0}

1,001 = 3^{6} + 3^{5} + 3^{3} + 3^{1} - 3^{0} That was a tricky one. Now think of a fairly low number and give it a try.

#### Googol

Everybody's heard of *Google* but what's a *Googol*? Answer: It's an incredibly large number - equal to 10^{100}. Try *Googling Googol*.

### The Möbius Strip

Bizarre shapes and strange discoveries are two things that make Mathematics really interesting. Take the Möbius strip,
for example. The nineteenth-century German mathematician August Ferdinand Möbius discovered that it was possible to make a surface that has
only one side and one edge.

Although such an object seems impossible to imagine, making a Möbius strip
is very simple: take a strip of ordinary paper and give one end a twist, then glue the two ends together. It's as simple as that.
If you begin drawing a line lengthwise down the centre of the strip, after one full revolution you won't be at the point where you
started – you'll be at the opposite side of the strip! Drawing the line through another full revolution will find you back where you
began.

If you cut a möbius strip lengthwise down the centre until you return to the beginning, what will happen to the strip?
If you can't visualise it, it's definitely worth trying - especially if your kids are watching.

### Only The Lonely

No matter how hard e^{x} tried to integrate, it didn't seem to make any difference!

### The Missing Pound

Some years ago Spelk and his two intrepid cousins, Ronnie & Davey, staggered into a B&B and asked for a room
with three single beds. The gorgeous lady owner informed us that the cost would be £30. We gave her £10 each
and were shown to our room. A few minutes later she knocked on the door; she'd charged us too much. She then handed over
five £1 coins. We appreciated her honesty and kept one coin each, giving her the other £2 as a tip. *(Well,
you never know how things might pan out!)* The lady smiled, thanked us, and left.

Thinking about it with the benefit of hindsight, we each gave her £10 before we took our £1 refunds.
So we gave her £27 plus the £2 tip. That's a total of £29. What happened to the other pound?

### Crossword Clue

Colin Dexter, author of the Inspector Morse novels, published his favourite cryptic crossword clues in the Radio Times.
I like this one:

#### 1. Nothing squared is cubed. (3)

It's only three letters so I can't possibly give you another clue.

to be continued ...