The Two Children Problem

Sorry I haven't posted in so long. For a while I became a bit demotivated and sort of fell out of love with teaching. I also moved house, leaving me without internets for over a month. I'm not gonna lie, that was tough.

So here goes my first post of 2013 (and I'll spare you the mathematical factoids about the number 2013: my students seemed to know them all from Facebook before I told them, anyway).

A few days ago a friend with whom I did my PGCE posted a really rather excellent maths problem on Facebook, claiming it to be better than Monty Hall. It went like this:

I have two children. One of them is a boy born on Tuesday. What is the probability I have two sons?

I managed to work out the answer within twenty minutes (not bragging, just sayin') but I kept thinking about it for several hours afterwards, and then spent all week talking about it to anyone who would listen, and even those who wouldn't (sorry colleagues, sorry boyfriend, sorry dad...).

But that's not actually the problem I'm going to discuss. As interesting as I found it, I found that the people I discussed it with got themselves hung up on a less interesting part of the puzzle. I couldn't believe that something I thought was so clear cut was causing so many people such cognitive conflict.

So I am going to discuss the Two Children Problem (the problem above but without the Tuesday bit). 

I thought it would be fun to lay this out in a Sophie's World kind of format. I am going to discuss the problem via the emails sent between my dad and myself for the past few days. These are almost entirely unedited. Enjoy!

Me: I have a good probability puzzle for you. I have two children. One of them is a boy. What is the probability I have two boys?

Dad: Off the top of my head, I would guess 50 – 50.

Me: It's not.

Dad: I don’t understand.  The other child is either a boy or a girl.  That means there’s a 50% chance of it being a boy. What’s the snow like in Coventry is it bad?

Me: There are four options: BB, BG, GB, GG.

I have eliminated GG by telling you one is a boy.

So the probability would be 1/3.

 

It is a bit like the Monty Hall problem (two goats, one car).

 

The snow is mental!! Got sent home from school at 12:30. Walked home. In un-walked areas it was about 15cm deep.

Dad: No there are only two options because:

 

GG is not possible and GB and BG are the same option (nothing is mentioned here about whether one is older or younger).  Hence the only options are BG or BB.

Me: But they are not the same. I didn't point at a child and say this one is a boy. I said one of them is a boy.

Dad: Yes and so if one of them is a boy, the other is either a boy or a girl. 

 

Me: No!

 

Flip two coins. If both are tails, re-flip them both. If not, then one of them is heads. What is the chance the other one is also heads?

Dad: I have to go home now.  Send a more explicit explanation to my home e-mail.  I still don’t believe you.

Me: f you flip two coins, and reflip them if they're both tails, then at least one of your coins is heads. What is more likely, they're the same or they're different? 

Dad: I don't understand this analogy with coins.  Surely the analogy is that:

 

You flip one coin and it turms out tails (a boy).  That is now a known.  If you then flip another coin it may be heads ( a girl) or tails (a boy).  So the question is if you flip one coin and it turns out to be tails, and then you flip another coin what are the chances of it being the same or different as the first coin?  With the second coin the probability is 50 - 50 of it being either heads or tails.

 

Is this not the case?

Me: This is not the same. You are right that if you flip one coin and it is heads, then the other coin is 50-50. But what we have done is flipped both coins, looked at both of them, and noticed that at least one of them is a head. So there are three possibilities: HH, HT, TH.

Dad: All right, I think I get it now. 

 

If you flip 2 coins at the same time, what are the chances of them both being heads, if one of them is a head.  Then the answer is 1 in 3.  

 

H T

T H

H H

 

Therefore

B G

G B

B B

 

I think that what troubled me here was a notion of sequence.  I am still thinking about this.

What about this? There are 2 goals in a football match. Palace have scored 1 of the goals. What is the probability that Palace have scored both goals?

What is the probability of Palace scoring both goals? Surely the score is either

Palace1 : 1 Bolton or
Palace 2 : 0 Bolton

Therefore there is a 50 - 50 probability that Palace have scored both goals, (and in the real world Glenn Murray probably scored both of them).

What is the fallacy if,

Goals equals children
Palace goals equals boys

Therefore the score is either

1 boy : 1 girl
2 boys : 0 girls

Me: Hmmm this is a really interesting question. I am thinking about it.

Dad: What about this.  If you put two blue counters (boys) and two red counters (girls) in a bag.  If you take out a blue counter, it immediately negates the existence of one of the red counters, because it is no longer a possibility.  Therefore for next pick there is one red and one blue counter in the bag and therefore a 50% chance of it being blue.

 

With the coins, if for example, the first coin is a H then it negates the possibility of

 

T H

 

therefore there are only two possibilities;

H  T or H  H.

 

If the second coin is a H, it similarly negates the possibility of  H T and so there are only two possibilities T H or H H.  Therefore in both cases if you know that at least one of them is a H, there is a 50% chance that the other is also H.

Me: You are correct in what you have said above, but that is a completely different situation.

The original problem can be thought of like this:

Pick a million families that have two children. Discard all those that have two girls.

Are you really saying that half of these families will have two boys, and the other half will have one of each?

Surely you can see that that is not true in real life. 

As for the football question, let's think of it in the same way:

Pick a hundred Palace matches where two goals have been scored, eliminating all those where Palace did not score.

Of these matches, do you think Palace will have won half and drawn half, or do you think they are more likely to draw? 

Dad: Let's say 990,000 families instead of 1,000,000.  Discard families with two girls. Is that 330,000?  That leaves 660,000.  Half of those are families with two boys.  That is 330,000.  Wouldn't that fit with real life?  You end end up with 50% boys and 50% girls.  (Actually in the real world I think there are slightly more female births than male births).

Me: You are wrong. I think it's more like this:

1 000 000 families.

250 000 two girls

500 000 mixture

250 000 two boys.

If you don't agree, consider a family with 4 children.

Combos are:

0 boys 4 girls

1 boy 3 girls

2 boys 2 girls

3 boys 1 girl

4 boys 0 girls.

You might think these 5 are equally likely. But actually they're not.

The 2 of each option is actually 3 times more likely than having 4 of one. 

This is based on the Binomial theorem. Are you familiar with that? Do you know what Pascal's triangle is?

Dad: If you flip a coin and it turns out H does it make it more likely that when you flip another coin, it turns T? If so, why?

Yes I can see that the combinations are

B B
G B
B G
G G
And therefore a 25 percent distribution. But again if each birth invoves a 50 - 50 chance, how does one birth apparently influence another? 

Is pascal's triangle a kind of musical instrument?

Me: One birth does not influence the other. But knowing that one of them is boy only tells you that GG is not an option. There are still three equally likely options left. Note that by "one of them" what we really mean is "at least one of them". 

Dad: Interestingly in 40 football matches where there were at least two goals, 23 of which were 1- 1 after 2 goals and 17 were 2-0 or 0-2. Therefore approx 58 perc to 42.

Me: Well that supports my argument! Convinced yet?

The conversation ends there (for now). Do you think the answer is a third or a half? I find it weird that my dad still doesn't quite believe me. I suppose that is the difference between a mathematician (me) and a linguist (him). I look for meaning by extracting the bare minimum of information to simplify the problem, whereas he looks deeper in between the lines to find meaning. 

It goes without saying that my way of thinking is better. :-)

Emma x x x 

PS Look out for part two, where I'll discuss the Tuesday boy problem. 

How to Deal with Being Inadequate

It's been a while since I've written a diary-style blog post. Recently I've been sticking to maths and the odd bit of teaching and learning. I decided after finishing my NQT year that I would no longer discuss my CPD publicly, for the sake of my readers more than anything: reading somebody's moans can't be interesting. But then again I suppose most people wouldn't find absement interesting either.

But this week I felt the need to write something that is somewhat more personal. It's been a bad week for this QTpi. I've experienced something that I knew would probably come eventually, although I'd hoped it never would: I have had a lesson observation graded "inadequate".

In case you didn't know, I'm somebody who likes to do well. I liked getting A*s at school. I didn't like getting anything else. So it bugs me that I'm no longer a straight A student. I have just received my first ever U. I don't know how to respond. This does not compute. Syn error. 404 page not found.

I don't want to talk about what went wrong in the lesson, or anything like that. It's personal, and besides, it wouldn't be interesting. But I do want to talk about what happened after I received my "Inadequate".

I felt sad. Annoyed. Frustrated. Victimised. Like a failure. And after 10 minutes of all that (I'm a fast thinker), I started to just feel glum. What an amazing word glum is! It sounds exactly like the feeling. Glum. It's almost impossible to say glum without slumping your shoulders and sticking out your bottom lip.

This glumness, unfortunately, has lasted all week. It has hovered around me like a bad smell. I walked into lessons thinking "what's the point? They're obviously not going to learn anything anyway". And this made me so sad. I love my students. I want them to learn. I'm letting them down.


Things people tell you when you get an inadequate:
-It's not you who's inadequate, it was the lesson.
-It was only one lesson out of thousands of good ones.
-It only went badly because you were nervous about being observed.
-It happens to everyone.

At any other point in my life I would have said that those four things are all true. However, in my current state of glum, they all seem like the kind of cruel lies that parents tell their children (I ate my crusts religiously for years and my hair remained as linear as ever).

Taking these cruel lies one by one:
-You are what you teach, hence I am inadequate.
-All of my lessons are exactly like that. Hence I am always inadequate.
-I never get nervous. Literally never. Unless fire is involved.
-It does not happen to everyone. I can name someone it has never happened to. She is an outstanding teacher and her hair is nice and curly too. I bet she never had to eat crusts. I hate her.

As you can see I have reverted to my fourteen-year-old self.

So if these nice things my colleagues have said (I do appreciate it guys!) have failed to make me feel better, what will? The answer is this: time. That's it. I just have to get back on the horse, and stay on it for as long as it takes to forget I ever fell off.

Maybe one day I will be able to look back at this observation and learn something from it. At the moment I don't even want to read the feedback sheet. I'm not in the mood for learning from my mistakes. I'm in the mood for focusing on what I'm good at. Like integration by parts, and matching my shoes to my dress. My colour co-ordinations and calculus will always be outstanding.

Have you ever been told you're an inadequate teacher? How did you cope?

Emma x x x

A Moral Dilemma: Teach Them Less So They Learn More?

At a *theoretical* school last *theoretical* Monday in a *theoretical* sixth form teaching meeting, somebody *could have* mentioned an idea for raising achievement in A level students. *Theoretically*, this could have caused some disagreement between staff.

This is all theoretical, of course.

The idea was (I mean, might have been) this: if you have a group of students who are unlikely to achieve a high grade in their AS or A level, and are aiming to scrape a pass, why not teach them less of the subject matter, and instead focus on just a few topics? So in A level maths, for example, teach them the calculus chapters, but nothing else. If the students got most of the marks allocated to those sections, they should get an E.

This idea caused a lot of debate. Once we'd got past the cries of "that wouldn't work in this subject!" we got to  some more interesting discussions: if it were possible, would it be morally right to do it?

Let me present some arguments for and against (I'm going to stay neutral, in case any theoretical colleagues are reading this):

For:

We do it at GCSE

In GCSE maths, you will probably have a few groups who are all sitting the higher tier paper. Let's say sets 1, 2, and 3. Well set 1 are obviously going to have to learn everything. With set two, you might choose to leave some of the circle theorem proofs out, and focus mainly on the grade A topics instead. With set 3, whose target grades might be a C or B, you would probably leave out all the A* topics completely, and only look at a few of the A topics. You would want to use your teaching time to make sure they fully understand the grade C and B stuff.

This makes a lot of sense, and few people would disagree with this. On a larger scale, you wouldn't teach your bottom set kids the A* stuff. That's why we have tiers in maths!

It benefits the students

Getting the grade E in their AS or A level would make a big difference to the student. OK, maybe their understanding of maths will be insufficient to actually help them in any way in the future, but at least they have something else to put on their CV. And these students obviously wouldn't be planning on studying maths at Uni anyway, so who cares? Who uses infinite series in real life anyway?

Besides, if pupils fully understand two topics, isn't that better than not really understanding six topics?

It decreases the school's number of fail grades

Enough said.

Against:

It's not fair on the students

Telling a student: "you're not going to learn that, there's no point because you won't get it", is a pretty demoralising thing to tell a student. Especially if the rest of the class are being taught it.

Also, limiting their knowledge in this way automatically stops them from progressing in the subject. If you teach them a limited number of topics at AS level, there is absolutely no way that student could progress to A2, even if they end up with a grade D in the end. Putting a cap on student progress will surely be a self-fulfilling prophecy? And who are we to assume that this student who appears to be on track for a U won't turn it around in the last month by working their socks off?

It's not fair on universities

How do you think secondary school teachers would feel if their year 7 pupils came to them with level 4s in their KS2 maths SATs, but had never learnt about, say, decimals. These students would be placed in a middle ability set with "true" level 4 students, and will have absolutely no idea when it comes to anything involving decimals. This is the problem that universities are having. They call it grade inflation. Students are coming in with the same grades every year, but the students' knowledge is getting weaker and weaker year after year. We do not want to be the cause of this! This applies even if the subject they are studying at university is completely unrelated to maths.

It's "teaching to the test" which everyone knows is a BAD thing

Learning A level maths should not be about the grade you receive. It should be about the journey of discovery, of honing your skills, and developing your way of thinking mathematically. It is not a means to an end.

And I might as well say it: OfSTED do not like this sort of thing. They explicitly criticise teaching in a way that allows students to pass exams without sufficient understanding. You do not want OfSTED to catch you doing this.



What do you think about this idea? Do you think it's morally wrong to restrict the amount of content you teach certain students, in order to teach them a select few topics really really well?

Do you have any more arguments for me to add to my lists above? Please leave a comment!

Emma x x x

The Area Under A Distance-Time Graph

Teaching Mechanics

I love teaching A level maths, but I must admit to feeling a bit disappointed when I found out I would be teaching the Mechanics module (M1 from MEI, fact fans) because... Should I really be saying this? OK, I'll just say it: I find it boring. No, that's not the whole truth, I'm making excuses now. I have to admit: I don't always get it.

I did Mechanics 1 at A-level (also using the MEI exam board) and that's the full extent of my knowledge. No Mechanics 2, none at Uni, I didn't do Physics A level or even separate Physics GCSE. I've never really been a "scientist", preferring to see myself as a creative type. I was always better at English and French than at science.

Anyway, this year (2012) I've been team-teaching M1, sort of. I've basically been watching someone else teach it. And suddenly I'm learning things I never really understood the first time round. I'm starting to get it!

The Area Under a Distance Time Graph

But then I went and spoiled it all by doing something stupid like Googling "What's the area under a distance-time graph?"

I would hope that those reading this are aware of the concepts illustrated below:

When you draw a velocity-time graph, for example, the area underneath the curve gives you the displacement, and the gradient of the curve gives you the acceleration.

The other teacher of the class posed the question to the class "What's the area underneath a displacement-time graph?". My mind was immediately blown. With a bit of jotting down (the diagram above), I could quite easily work out that the units had to be ms (metre seconds, a bit like kilowatt hours), so I knew it was something to do with distance multiplied by time.

I knew that what I was looking for would be the blank in this sentence: "displacement is the rate at which BLANK changes". What could fit? Nothing seemed intuitive.

Absement

That was where that much-relied on search engine came in. After a bit of research (Googling), I found that the word I was looking for was "absement" (oh of course, you all exclaim) and that I was by no means the only geek in the world wondering what it was.

Absement is a port-manteau of the words absent and displacement (knowing which makes it no easier to understand) and there are not many results on Google for it (the eighth result is an English to Urdu translation page. Erm, cheers for that).

I will attempt to explain absement using an example. For another example, visit Wearcam.

You live 2km from school. You walk to school in 30 minutes, stay there for six hours, then return home, also taking 30 minutes.

A displacement-time graph would look like this:

Now let's consider the area under this graph. It would be given by integrating the curve above. You can sketch this curve easily: in the first section, the gradient is constant and positive, so the corresponding bit of our new graph would be increasingly increasing (curving upwards). The middle bit has zero gradient so our new graph will have a constant positive gradient. The last bit is the opposite of the first bit, so our new graph's last section will have a decreasingly increasing bit.

My sketch:

The next thing to do is calculate the numbers on the vertical axis.

Your absement at any point is given by the (average) distance you are from home multiplied by the time that you're there for.

So when you've just arrived at school, your absement for that point is 1000m (your average distance from home) multiplied by 1800. So that's 1 800 000 ms. Up to that point, your absement has been increasingly increasing. When you're halfway to school, for example, your average displacement was 500m, your time is 900, so your absement is 450 000ms (note that this is not half of 1 800 00ms). At the end of your second part (when you've just finished school), your displacement has been constant at 2000m, your time has been 6 hours which is 21600 seconds, so your absement at that point is 43 200 000 ms. You can calculate your absement at many different points so that you can plot a nice, smooth curve. It looks a bit like a cumulative frequency curve. I'll leave this to you as an exercise.

What's the point of absement?

Well if you were on a spaceship and you had some kind of mobile communication device, you might imagine that the further from Earth you are, the more power the device uses. Therefore you might measure its battery usage in metre seconds.



Emma x x x

Praise and Rewards: Overrated?

How often do you praise your students? I bet most of you would answer "not enough", because we're constantly being told that we should be praising and critiquing in the ratio of 4:1 (which I'm sure is completely arbitrary. Where are the calculations?) and that we should be constantly reinforcing good behaviours and boosting confidence.

But really? Really? The hype over praise has reached such a ridiculous level that we're now told we have to seek out things that students are doing and praise them for it. It's called "catching them being good" but it should really be called "searching frantically for something they do which isn't completely moronic". 

I sound quite bitter don't I? I'll dial it back a bit. I'm no educational expert, and I'm by no means an experienced teacher, but I still feel quite strongly that the advice we have been given about praise, and more so about rewards, is bad.

I have a confession to make. My name is Emma and I'm an over-praiser. I will meet every child's contribution to the lesson with a "well done!" or an "excellent answer!" or, more often than not, "that's not correct, but I'm impressed by your creativity!" I actually found myself praising a student for remembering to draw his margin with a pencil today. And this was a middle ability set. I hate the fact that I do this. It's a habit I got into during my training year, before I was confident enough to challenge any of the advice I'd been given. I'd spend hours trying my hardest to think of three "stars" for little Johnny whilst trying to cram all of my hundreds of "wishes" into one line. Result: Johnny walks away thinking he's done a good piece of work. It doesn't matter to him that one of the stars was "You wrote the date! (Smiley face)"

My revelation came during my NQT year. I was talking to a particularly inspiring teacher, and he admitted to me that he never gave out "points" for good work, good behaviour etc. He said he had never even logged into the online points system. I felt really smug then, because I had painstakingly given out every single point in my account every week since the start of the year. My smugness didn't last long, however. He told me he didn't really agree with the points system. He said that students shouldn't be behaving well and doing good work so that they can earn points, they should be doing it for their own self-satisfaction. This really struck a chord with me.

Giving out points encourages extrinsic motivation, where people act to gain external rewards and avoid external punishments. Intrinsic motivation is where people act for their own satisfaction. Studies have shown that using extrinsic motivation to get people to do a certain activity can lead people to see that activity as "work", a job for which they are paid. I have heard that it is common advice to tell parents not to reward their children for reading, because if they do, they will stop reading just for the fun of it. 

We all know that Pavlov et al have shown that using praise and rewards allows us to control behaviour, but I personally would rather teach students who are in control of their own behaviour. There's something quite sad really about a dog who salivates when a bell sounds. Would you like to teach a bunch of robots who immediately start working the second you say "VIVO miles"? Are they the kind of people we want in society? There's no one around to reward adults for not dropping litter on the floor, not sticking chewing gum under cinema seats and wearing appropriate clothing in public (I'm talking to you, large blonde lady from the number 5 bus).

And when you throw praise and rewards around willy-nilly, you devalue it. I don't think any of my kids ever really experience the feeling of pride puffing up in your chest. I'm sure my comments wash over most of them, a lot of the time. I watched the teacher mentioned above teach an A level lesson the other day. He asked the class how they thought speed cameras worked. After some answers and some discussion, a boy contributed an answer that I thought was pretty good (embarrassingly, I didn't know the correct answer). The teacher, without saying anything to the boy, explained the boys answer to the rest of the class, rewording it a bit so they would all understand. He then finished by pausing for a second, and then simply saying "that's exactly how they work". He didn't add "well done", he didn't even smile. He just said that, in a slow, low voice. But even from where I was standing, the pride that boy was feeling was palpable.

I must admit, I stopped giving out my points a while ago, and since then I haven't noticed a single bit of difference in my classes' behaviour, effort or achievement. My next step is to stop being so generous with my praise. I'm thinking I might aim to give out one really good bit of praise per lesson. 

Think about yourself as a student. What bit of praise really affected you? What moment of pride do you still remember? For me, it's the time my year nine history teacher (also the head teacher) wrote that my end of year project is the best he had seen for as long as he could remember. And he gave me a pen (my schoolmates will remember how scarcely these pens were given out). Embarrassing admission of the week: I still have the pen. It's in my "special box". 

Am I on my own here or do you agree? Comment below!

Emma x x x 

My Mathematical Jewellery Collection

Today I'm going to a barbecue hosted by one of my colleagues. The theme is maths. We will be doing maths puzzles and games, and there is a prize for the most mathematical outfit. On reading this I exclaimed "What shall I wear?!" Not, as you might think, because I don't own any mathematical clothing. In fact it was because I own so much, that choosing was going to be difficult.

Two of my absolute favourite things in the world are maths and fashion. When you combine the two, you get mathematical fashion. And I LOVE it. I have a few mathsy clothing items, but I'm not going to show you those in this post. This post is dedicated to perhaps my favourite aspect of fashion: jewellery.

At my academy I'm known by both staff and students for having a massive collection of bold and bright rings and necklaces. A subset of this collection I will be showing to you today.

First up, my absolutely amazing, custom-made SOHCAHTOA necklace;

I bought this necklace from one of my favourite jewellery shops, Tatty Devine. As well as stocking quirky and unique perspex jewellery, they make name necklaces (like Carrie's from SATC). Seeing this feature gave me the genius idea of turning my favourite maths word into a necklace. I thought long and hard about what word (containing nine letters or fewer) to choose. I settled on SOHCAHTOA because it's so recognisable, not to mention useful. I've had several strangers see it and proudly exclain "I remember that! It means..." which is so nice to hear!

I think my favourite thing about this necklace is my students' reactions when I tell them I had it custom-made. I think to them it just confirms my status as geekiest teacher ever.

Onto my next necklace: The Infinity necklace:

I bought this necklace on Amazon after a lot of searching. I had decided I wanted a necklace in the shape of the infinity symbol, and then set about trying to find one, which is the complete opposite of how I normally shop, where I see something I never knew I wanted and then buy it immediately. This necklace is sterling silver and contains an actual real diamond, which sadly my poor eyesight inhibits me from seeing. Amazon assures me it is there, though.

Necklace number three: the Rubik's Cube:

Everyone loves my Rubik's cube necklace. I have been nearly strangled countless times as students grab it as  I walk away. Nobody can resist turning it a bit, which is why it is a jumbled mess. I am actually rubbish at solving them, although I keep telling myself I will learn the algorithms soon. My friend Emma appeared to learn overnight when she got one for her birthday, but knowing her she probably spent hours and hours learning it obsessively.

Oh, I almost forgot to tell you where this amazing necklace is from! I got it from Folksy, a brilliant website where random people sell all kinds of handmade and vintage stuff. I have bought so many things from there. The necklace was from a seller called Mary Quite Contrary.


And now, a bracelet:

I bought this bracelet quite recently from Folksy, from a seller called I heart my art. It was the only one of its kind, which makes me feel special. Check out her other items though. I like how the bracelet contains "e" because most of my students don't know about the exponential constant, so they'll ask questions about it. Any excuse to talk about e and logs! Although I bet most of them just assume it stands for Emma.

Next, my Origami necklace:

Boy was this hard to photograph! This little stunner really is origami: it was made from a single square of silver and folded into the iconic crane. I saw a very similar necklace on Anthropologie, one of the most beautiful shops in the world, but it cost way too much so I went looking elsewhere. I found this on Etsy, which is basically the American version of Folksy. I bought it from the hugely talented AllegroArts who handmakes all of these stunning origami pieces. This exact necklace can be found here.

.
So that's my mathematical jewellery collection as it stands at the time of writing. I fully expect it to be twice the size this time next year!

I will leave you with one last photo: a little cutie I just had to buy from Ryman's as I was picking up my back-to-school stationery:

I just can't resist cute geekiness.

Emma x x x

Playing to Lose

(I was contemplating calling this post "Every Loser Wins", but I hate that song).

As all of you should know by now, I'm a winner. I win things. So when my friend Stacy challenged me to a game of Noughts and Crosses today, I was prepared to win. Until she said this:

"The winner is the person who loses. The aim is to NOT get three in a row".

As you can imagine, I was flummoxed. Winning is easy, losing is difficult. She told me to go first, which, now that I think about it, was a sneaky way of increasing my chance of winning - and hence losing. I obviously avoided the middle square. I figured the corner squares were also too good to use. So I opted for a side square. That decision was pretty easy. The rest was not so trivial. Stacy lost the game, and hence won. Apparently I am such a winner that even when I'm trying to lose, I win.

Please please please grab your partner/your child/your flatmate/the guy next to you on the bus and challenge them to lose a game of Noughts and Crosses. It's the only way you can really think about the strategy involved.

Have fun!

Emma x x x