6. What are the chances?

We’ve recently returned from a few days holidaying in deepest Pembrokeshire – a beautiful unspoilt part of west Wales with rugged coastlines and wind blown trees.

Rugged (and wind blown)

Wind blown

One of the highlights was visiting the cathedral in St Davids, which has roots that go back to the 6th century. 

St Davids cathedral

On our last evening, we tried to find a table at a small fish restaurant in the tiny coastal town of Dale. The owner explained that he had a party of eight people about to arrive but thought he could squeeze us into a corner. 

Dale in Pembrokeshire

Just as we were about to tuck in to our fresh lobster, we heard a shout from behind us, ‘I don’t believe it!’. Who should troop in but eight of our near neighbours and good friends from back home in Warwickshire.

After a fair bit of hilarity, our conversation turned to unlikely events in general and then the national lottery in particular. We were drawn into that old chestnut about what on earth we would do with those awkward Lottery millions and how much of our fortune we would want to give away. I was asked if there was a best strategy for choosing the numbers, and it caused a bit of a surprise when I said that there was! 

The UK National Lottery

But before I reveal this hot lottery tip, there are a couple of issues that need to be sorted out.

Issue 1: Is any one combination of six lottery numbers more likely to crop up than any other?

My Answer: No! Provided that the lottery numbers are chosen randomly from the lottery ball machine, all numbers (and all combinations of numbers) are equally likely to occur.

Issue 2. People say that lightning doesn’t strike in the same place twice. So I’d expect that the six numbers that came up last week are less likely to come up this week?

My Answer: This is a fallacy. As a matter of fact, lightning does sometimes strike in the same place twice, especially if that place is a tall building or tower. Indeed, several people have been hit by lightning more than once. For example, Roy (‘Human Lightning Conductor’) Sullivan who was a United States park ranger in Shenandoah National Park in Virginia was hit by lightning on seven different occasions and survived all of them. 

But anyway, even if it were true that lightning didn’t strike in the same place twice, how would this work? Are we to believe that there is some lightning god from above saying, ‘No, hang on, we hit that particular spot last year – better move this bolt’s trajectory a smidgen so that it hits the next field.  The same principle is true of the selection of lottery numbers from the lottery ball machine – last week’s numbers are as likely to appear this week as any others. Just as was true of the lightning bolt, it’s hard to believe that there’s an invisible force subtly pushing last week’s six balls towards the bottom of the machine in order to give the other 43 balls a better chance, in order to even up the outcomes. 

Now you might be thinking, ‘But surely by the Law of Averages, all the numbers should come up roughly the same number of times, so that should mean that previously favoured numbers are less favoured next time around. Well, sorry but this is not how the Law of Averages works. If you want to know why, have a look at the follow-up to this post in the Blog page of my website, MathinPictures.co.uk.

Issue 3. So if each number is equally likely to crop up, how can I have a ‘best strategy’ for choosing my numbers?

My Answer: Unless you can find a clever way of cheating the system (which would be very hard to pull off), it is true that you cannot find a way of choosing a combination of six numbers that is more likely to win than any other combination. But there is another factor here. You need to remember that if you choose the same six winning numbers as someone else, you have to share the jackpot with them. And if nine other people choose your winning numbers, your share is reduced to one tenth of the jackpot, So if you want to maximise your winnings, try to be aware of which numbers other punters are likely to choose … and avoid them! 

Here then are my hot tips for maximising your lottery winnings.

1. DON’T CHOOSE numbers between 1 and 31: people are more likely to choose numbers linked to birthday dates, so numbers in this range tend to be more widely used.

2. DO CHOOSE consecutive numbers: many people mistakenly think that a consecutive string of six numbers is less likely than any other combination and so they avoid such combinations. However, don’t choose the particular consecutive combination 1, 2, 3, 4, 5, 6 as it seems that several hundred punters regularly choose these numbers each week.

The joke

Finally, no lightbulbs this time – just this:

Teacher: What does “coincidence" mean?

Pupil: That’s funny, I was going to ask you the same thing.

Postscript
This post looked at the idea of chance in the context of the UK National Lottery… If you are interested in finding out a bit more about the mathematics of lotteries, have a look here in my website, MathinPictures.co.uk.

5. The Golden Rectangle

‘Quick, Alan. We’ve got to go NOW or we’ll lose them. They’re only a tenner each.’

I didn’t question Sheila further. We jumped into the car and sped off in the direction of our local second-hand furniture shop. What she had spotted was a collection of ammunition boxes – leftovers from World War II that probably originated from the US army. But of course, what Sheila was after was wooden boards for her paintings. 

Normally Sheila paints on commercial canvases. She was delighted to have the opportunity to paint on wood but also to be free to make this wooden ‘canvas’ to any size and shape she fancied. Now, as I am a big fan of a particular rectangular shape – the so-called ‘golden’ rectangle’ – and also because, I was the one wielding the saw, I suggested that we should cut up the box lid to create a board of this ‘golden’ shape.

A week later, this little ‘Apple Blossom’ beauty was hanging on our kitchen wall.

Apple Blossom

This isn’t the only art object that Sheila has produced that was inspired by this particular 'golden' proportion. Here is her Golden Window, painted from a wrought-iron window frame in a crumbling fromagerie in Normandy.

Golden Window

So what exactly is a golden rectangle and why is it considered to be special? Have a look at this 3-minute video to get a quick overview.

Golden Rectangle

Maori drawings

A few years ago, we spent a wonderful two months travelling around New Zealand. I had arranged a deal with a number of NZ mathematics advisors that I would provide a series of ‘inspiring’ talks/workshops in return for a night’s board and lodging in each location. That way we had a great time meeting and chatting to many teachers and their families in their own homes. But there was also plenty of time for sight-seeing. We particularly enjoyed having the opportunity to explore Maori culture and art. 

On our return to England, we decided to mark the trip by commissioning, from my friend Roger the glass artist, a pair of small stained glass windows that celebrated the fusion of mathematics (upper) and Maori art (lower). 

The Golden Spiral

A Symbol from Maori Art

The mathematical one shows a logarithmic spiral (sometimes referred to as the ‘golden’ spiral) inscribed inside a golden rectangle. Here's a detail to clarify the mathematical part of the picture.

A Detail of the Golden Spiral

Starting out from the bottom left corner, the spiral takes a path between opposite corners of each square in turn. In order to construct it, Roger used compasses to form an arc inside each square which he drew as a quarter circle, centred at one of the other corners of its square. 

Golden rectangles have been known about and celebrated since the time of Euclid. He didn't use the term 'golden' but referred to this special ratio as the "extreme and mean ratio". It's sometimes said that shapes formed from 'golden' proportions have a particular aesthetic appeal. I'm not totally convinced by this claim but it would be lovely if it were true!. 

Anyway, this 'special' ratio is all very well, but it's not getting these lightbulbs sorted out ... .

Postscript

This post looked at the idea of a golden rectangle, which has proportions of approximately 1.618 to 1. The number, 1.618… is often referred to with the Greek letter Φ (phi). If you're interested in finding out where this number comes from and indeed other matters 'golden', have a look in my website, MathinPictures.co.uk

4. The Simpsons and Mathematics

I recently celebrated my n+1th birthday (yes, a year older…) and received from Sheila, ‘The Simpsons and their Mathematical Secrets’ by Simon Singh. Having already read several of Simon Singh’s books – Fermat’s Last Theorem, The Code Book: The Secret History of Codes and the fabulous Trick or Treatment?: Alternative Medicine on Trial – I was confident that this one would prove to be an entertaining page-turner. And I’m pleased to say that it is indeed a little cracker so I thought I’d share an extract from it in this post.

The Simpsons

Unless you’ve been hiding out in another planet for the past 20 years, you’ll know already that The Simpsons has been one of the longest running and most successful TV series in history. Singh's book explains how it came about in the first place.

Back in 1985, the British entertainer Tracey Ullman was having a successful run of shows in the US. Her sketches were performed before a live audience and many of them required extraordinary makeup and prosthetics. Scriptwriter James L. Brooks thought that it would give Ullman more time between sketches if they could include some cartoon animations. He met with cartoonist Matt Groening and minutes prior to this meeting Groening thought up the characters of the Simpson family. Later, some of the short animations featuring the Simpsons were pieced together into a longer show. To everyone’s surprise, they went down very well. Their first full-length show was ‘Simpsons Roasting on an Open Fire’ which was aired as a Christmas special on 17 December 1989. 

'Simpsons Roasting on an Open Fire'

Where’s the mathematics?

If you are a regular follower of The Simpsons, you’ll be aware of the many amusing mathematical references it has included over the years. This is down to the strongly mathematical backgrounds of many of the writers, some of whom gave up teaching and research posts in mathematics to join the writing team. Singh includes many examples of mathematical references but don’t be put off if your own mathematical background is a bit thin – the emphasis is just as much on the humour as on the mathematics.

Various mathematical topics are touched upon in the book but I’ll mention just one – the number known as pi. An episode, first aired in 2001 and called “Bye, Bye, Nerdie” is about the bullying of nerds, a question that Lisa decides to investigate. She discovers and isolates a pheromone emitted by ”every geek, nerd and four-eyes” that could be responsible for attracting bullies. She decides to deliver a paper on this theme to the ‘12th Annual Big Science Thing’. The conference is hosted by John Nerdlebaum Frink Jr, Springfield’s favourite absent-minded professor. Frink attempts to introduce Lisa but the audience is so excitable that he is unable to bring them to order. In desperation, he eventually calls out:

“Scientists… Scientists, please!  

I’m looking for some order.  

Some order, please, with the eyes forward…  

And the hands neatly folded…  

And the paying of attention…  

Pi is exactly 3!”

Suddenly the noise stops. Professor Frink has correctly judged his audience. Stating that there was an exact value for pi has stunned this audience of geeks into silence!

If you have only a hand-waving grasp of this number, pi, about where it comes from, or indeed, why this story is funny, you might like to look at the 3-minute video below where I have provided a cheap and cheerful explanation of pi and how it is linked to a fundamental question about what makes a circle a circle.

The Circle and Pi

Of course, this story just scratches the surface of the many gems contained in Singh’s book – definitely 5 stars from me.

Now on to weightier matters. How are those lightbulb operatives doing, I wonder.

Postscript

If you are interested in reading more stories about pi and this delightful book about the Simpsons, have a look here in my website, MathinPictures.co.uk.

3. A Troublesome Tweet

I come from a very sporty family and anything that involves chasing after a ball generally catches my interest. I couldn’t avoid reading about the following tweet from cricketer Stuart Broad (the England fast bowler) that made a stir in the news recently.

Stuart Broad, England Fast Bowler

"I've heard if you earn minimum wage in England you're in the top 10% earners in the World. #Keep Humble”

The backlash caused by this tweet resulted in Broad having to issue a rather grovelling apology. 

Now setting aside for the moment the tactfulness of how he worded his tweet, was his claim actually true? There are two parts to this – the earnings of someone on the minimum wage and the pattern of all earners, globally. 

Minimum Wage Earnings: A quick scan in Google revealed that the national minimum hourly rate for UK adults, at the time of writing, is £6.50. Working, say, a 40 hour week, this gives a weekly wage £260 or roughly £13,500 annually. (Assume a 37 hour week and this figure drops to £12,500 annually.)

Global Earnings: A harder task was to get a picture of world earnings. Again, with the help of Google, I was able to find a Daily Mail article which provided some useful tidbits of information – for example:

* You need to be earning just £22,000 per annum to be in the world’s top 1% of earners.

* Half of the world’s richest 1% live in the USA.

Now this is all interesting stuff, but it didn’t quite resolve whether Stuart Broad’s claim was true. Some further googling turned up what I was looking for – an on-line calculator called the Global Rich list, posted by the charity, Care International. You simply enter a figure for your annual earnings and it tells you where you fall within the global rich list. 

First, I tried entering the earnings of a UK pensioner receiving the full state pension, which is currently £113.10 per week – i.e. roughly £5880 per year. Using the online calculator, this puts a British pensioner into the top 18% of earners, globally.

Next, I entered the earnings of someone on a minimum wage – £13 500 annually. 

The on-line calculator put this earner into the top 5.41% of earners, globally – a figure even smaller than the 10% claimed by Stuart Broad. 

So these calculations do seem to bear out Stuart Broad’s claim with something to spare. I wondered afterwards why making what appears to be a factually correct statement could cause such a furore. A likely explanation is how Broad presented it – you may have noticed that he had tacked onto the end of his tweet the words: #Keep Humble. He claimed afterwards that this hashtag was aimed at himself. But for someone who reputedly earns upwards of £1million per year, he is open to the accusation of lecturing at those earning a tiny fraction of what he earns by telling them how lucky they are!

Finally, here's what the lightbulb team are getting up to.

Postscript

I wondered whether these figures from the Global Earnings calculator were to be believed? When it comes to statistical claims, it’s always sensible to look for loopholes and alternative explanations. I consider some of these here in my website, MathinPictures.co.uk, along with the sources I used.

2. The rat, the pallet and the logs

The rat: The recent cold snap has encouraged Sheila and myself to be more enthusiastic about putting seeds and nuts on the bird table. But in recent weeks, an uninvited guest has joined the party. Yes, Mr. Rat has been busy helping himself as well. Last week, I had my camera at the ready and here’s the moment when he realised he was rumbled. 

Rumbled

In a flash he was off, making a rapid leap for freedom, disappearing around the side of the house.

The leap for freedom

‘I bet your life he’s living under that tarpaulin where we store the logs,’ Sheila said straightaway and I rather felt that she was right. We removed the tarpaulin and carefully started reorganising the logs. But all we uncovered was a very sleepy, plump frog. 

We could see that the logs at the bottom of the pile had started to rot and gather moss. What we needed was a large pallet to go underneath, but where would we get one of those?

The pallet: Two days later, I was driving into town to pick Sheila up from a little light shopping. The footpaths were jammed with pedestrians and the roads were busy with commercial vehicles trying to unload goods. As I approached our usual pick-up point, I could see that there was a bit of a commotion. It seemed that the footpath was blocked – mothers with prams and pushchairs were having to step into the road to get past. It was then that I spotted the source of the blockage, waving at me frantically. There she was, making her way slowly along, staggering under the weight of the largest pallet I have ever seen – it looked like it weighed a ton. It turned out that she had found it at the back of a local TV store (what was she doing there?).

With difficulty we managed to get it into the car and home. It now forms the base of our new log store. So I just need to send Sheila out to liberate a few more pallets of a similar size and shape and our log store will be complete.

Logs: While we were working on the wood pile, I asked Sheila what she remembered of logarithms at school.  She laughed. “I remember a book with big numbers in. I don’t know what we did with them though.”

The book of mystery from schooldays

So, what does the average person understand by the word logarithm – a term I’ve worked with all my life? I emailed a few of my friends and they all graced me with interesting and entertaining responses, from which here's a selection.

* Umm, log tables, looking things up that didn't make sense, then I had an exotic looking calculator and I still have no idea. (age 44)

* I know that a logarithm is something to do with sums, and that there are log tables, but I can’t help you any further. I have to say that I would have expected you to know what a logarithm is without having to ask. (age 65)

* Absolutely no idea whatsoever. (age 42)

* Something from school days -– slim book with columns of very long numbers in. Some kind of crazy multiplication? (age 67)

* I cannot think of any other bit of schooling where the books on the memory shelves have snapped so firmly shut. (age 62)

* Without looking it up I wouldn’t know the word logarithm, other than thinking it is a mathematical equation of some sort. (age 31)

One reply was a perfect textbook definition and it came from my neighbour Andy: 

* It's the power to which the base number is raised to get the number in question.

Have a gold star Andy and go to the top of the class! 

So, overall, the little book and its columns of numbers was well remembered by most of the over sixties age group, but very little else of any substance!

What IS a logarithm?: Although Andy's definition is correct, it may be too dense for most people to grasp at a single reading. Learning about logarithms needs a little patience. If you're interested in knowing what a logarithm is, I've made a short YouTube video which you can view here.

But a trickier question is, what’s the point of logarithms and why do we need them. Relax – I won’t explain that now but I hope to return to it in a future post. Instead, here's another 'lightbulb' joke.

Postscript

If you are interested in finding out more about logarithms and their properties, have a look here in my website, MathinPictures.co.uk.

1. Twice as warm

Our son, who is an actor has just come back from a 3-month tour and this week he moved into a new flat in south London. So, very early last Sunday, Sheila and I filled the car with possessions that he had been storing in our spare bedroom and set off for London to help him move them in.

The first thing we noticed on stepping out of the car was the temperature in London that morning – decidedly brisk. Leo was waiting to greet us outside the modern, red-bricked, 6-storey apartment block. ‘Top floor, Leo’ I asked, joking. ‘That’s right.’ Leo replied, but unfortunately he wasn’t joking!

Returning to the Midlands that afternoon Sheila remarked, 'It's twice as warm here as it is in London!' But is it really possible for one temperature to be twice as warm as another? We keep a thermometer in the garden. It is marked in Celsius down the left side and Fahrenheit down the right and the reading was 10ºC. 

The garden thermometer

I googled the weather details for that day and the temperature in London had been 5ºC. 

So Sheila was pretty pleased to hear this information. 
'Just as I said – twice as warm here as in London!'

But temperature is one of those funny measures where it doesn't make sense to talk about 'twice as warm', 'four times as warm' and so on. The evidence is revealed when you shift to a different unit of measure – say to the Fahrenheit scale. We looked closely at the two scales side by side. 

Celsius on the left, Fahrenheit on the right

When using the Celsius scale on the left, 10º was twice 5º. But coming across to Fahrenheit, the two corresponding temperatures were 50ºF and, roughly, 40ºF. Clearly these two readings did not share the same pattern: 50 is not twice 40.

Anyway, that's measurement for you. I'll end this post on a 'light' note with this little rib-tickler.

Postscript

So, when comparing two temperatures there's no such thing as 'twice as warm'. The same issue crops up when talking about time of day – you can't sensibly talk about 'twice as late'. But you can talk about 'twice as far', 'twice as heavy', twice as many', etc.
If you are interested in finding out more about measuring scales and their properties, you’ll find an explanation here in my website, MathinPictures.co.uk.