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

30 = 1     (Any Number greater than Zero, when raised to the power Zero, equals 1 - even π0.   That's the way the world works!)
31 = 3     (Three to the power 1)
32 = 3 x 3 = 9     (Three to the power 2)
33 = 3 x 3 x 3 = 27     (Three to the power 3)
34 = 3 x 3 x 3 x 3 = 81     (Three to the power 4)
35 = 3 x 3 x 3 x 3 x 3 = 243     (Three to the power 5)   etc. etc.

Examples

2 = 31 - 30
4 = 31 + 30
5 = 32 - 31 - 30
43 = 34 - 33 - 32 - 31 + 30
1,001 = 36 + 35 + 33 + 31 - 30     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 10100.   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 ex 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 ...