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Lecture 3: The secret of the winning streak

Puzzles and problems - mathematicians love studying them and devising equations to solve them! In this lecture Marcus shows you how to come up with strategies to win at games from rock, paper, stone to Monopoly, noughts and crosses to Greco-Latin squares like those in Soduko. Then we move on to the daily problem of finding the quickest way to get somewhere. Lots of people need to find the shortest route that uses the least fuel - doctors visiting patients, ships picking up and dropping off cargo - but nobody knows a quick way to find it! Mathematicians, like Marcus, can help, but sometimes problems are so hard that even they have to guess.

Further and Deeper

Try this out

It's all about strategy. Download the rules and then challenge your friends.

Strategy shoot out (PDF Document 90KB - new window)

 

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More Maths

In this section, as in the others, there are questions which mathematicians cannot solve. Download this PDF to see how a clever idea could win you US $1 million.

The million dollar question (PDF Document 90KB - new window)

 

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Lecture 3 transcript

The secret of the winning streak (PDF Document 158KB - new window)

 

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Web links

Probability can sometimes be surprising. In this game there is a prize behind one of three doors - choose one. You will be shown another door that does not contain the prize: should you change your choice? (Requires Java.)

The Monty Hall problem (external link - new window)

 

This Plus magazine article explains some of the mathematics behind the National Lottery.

The mathematics of the Lottery (external link - new window)

 

A game that tests your thinking skills. Drag the blue spots around until none of the lines cross. The puzzles become more challenging - how far can you get?

Crossing lines puzzle (external link - new window)

 

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Video clip - Play 15

A member of the audience takes up the challenge and Marcus reveals a simple winning strategy.

(Press play to start the video.)

This exciting, interactive site has been designed using Flash 8. In order to get the full experience, you will need to download Adobe Flash version 8.

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So, who'd like to come and play 15 with me? I want somebody, yeah, why don't you come? Right, come on down here, and I want you stand behind your cake stand here. Now what's your name?

Lottie

OK, Lottie. The game here is where I've got loads of chunks of cake behind me. Some of them are just, one slice down here and up here I've got a large chunk which consists of 9 slices of cake, but your cake stand, it takes 15 slices. The game is you've got to take some slices, chunks of cake and try and make a complete cake, but you have to do it with exactly three chunks of cake. So you're going to try and take three chunks where the numbers add up to 15, OK. And we're going to take it in turns to play the game. I'm going to try and do my cake as well. So off you go, you can start in this game, alright? You understand the rules?

Yes

OK. Choose any piece you want. She's going for a big one, first of all, why not, that's good. Try and fill it quickly, so I'm going to go for this one. Right and I have to watch what she's doing, so she's gone for 4. OK, not sure, probably 2, I think I'm going to stop her, so go for 2. You can take your piece off and try a different one because I've just scuppered you with your strategy there. So those are your pieces, you can keep those pieces and you can chose some more pieces, OK. So you've got another piece here. Number 1, 3, what do you need, oh there's lots of people shouting. Don't help her, God it's one against 400, that's not fair. I'm going to take 3 I think, right, well that doesn't work. Has she got, did I manage to get the piece that she needs? OK, your turn, you've got 3 left, what are you going to do. No helping her, shhh, it's not fair. OK, she's looking at mine, that's fair enough, you know. Now I'm going to win now because I've got 7, yes, and I can take that, and 5 plus 3 plus 7 wins me the game. OK, so even against 400 I managed to beat her. So not bad. It's a pretty difficult game because, to start with it's quite easy, you're trying to add up the numbers to 15, but after a while you've got to keep track of so many different things: what I'm doing, what you're doing, different combinations.

Now the reason I won that so quickly, was that I was playing a different game with you. I was actually playing noughts and crosses with you. How come I was playing noughts and crosses? It doesn't look like noughts and crosses at all. Well here's the noughts and crosses board that I was using. And there it is on the screen, it's actually a magic square, some of you may have seen it: magic square. All the ways that the lines that you can draw in this thing - rows, columns, diagonals - they're all the ways that you can add up to 15. So if I played noughts and crosses with you, and I got 3 in a row, which is what I got here, 7, 3 and 5. So 7, 3 and 5s are the middle column, I won the game. I was actually playing noughts and crosses with you, which made this really easy to play. So, what we're going to do now is I'm going to give you this strategy and you're going to play me again, and you will see how much easier it is when it becomes a game of noughts and crosses. So if you could sit back down in your seat for a while, and I'm going to pull you up in a little bit.

Now what I'm going to do is to rearrange this board, so it looks like a magic square up here. And as if by magic, I've got a magic square. So now I'd like my volunteer to come back on again, and we're going to play noughts and crosses, but actually our noughts and crosses game is going to be the same as the game we played, of 15. Here are our noughts and crosses. Now, what do you like? Do you like noughts or do you like the crosses? OK, crosses, and you get to go first, like I gave you the chance to go first last time. Now do you know a good strategy for noughts and crosses, where's the best place to go on noughts and crosses, the guarantee? The corner, I actually think the middle is pretty good. OK, you can play corner, you can play corner. Now I think the really bad move, for me, is to go here. I think, can you force a win from there? Let's see if you can. Right now, if we were playing 15 I would have seen she'd got, so this one is 6, by 5 so I know that she's going for 4, so I'll block that one there. But like in noughts and crosses I can see very quickly that that's what you're going for. So your go now. No that's a good move, isn't it? Because look now there are two different ways for her to get 15, this line here, and this line here, and I can't stop her. So I'm going to go here, try and stop that one. And she wins the game like that! Now isn't that so much simpler than playing 15 and trying to add things up? Let's give her a big round of applause for beating me. Oh and, and we even have a very small cake, I'm afraid for, for the winner, so that's great.

In fact, this is one of the powerful things in mathematics, to try and change a game, or a problem you've got, into something else. And suddenly, from a new perspective, the thing might become much easier. It was quite difficult when we were trying to add all the numbers up to 15, but playing noughts and crosses - anyone can play noughts and crosses.

So, who'd like to come and play 15 with me? I want somebody, yeah, why don't you come? Right, come on down here, and I want you stand behind your cake stand here. Now what's your name?

Lottie

OK, Lottie. The game here is where I've got loads of chunks of cake behind me. Some of them are just, one slice down here and up here I've got a large chunk which consists of 9 slices of cake, but your cake stand, it takes 15 slices. The game is you've got to take some slices, chunks of cake and try and make a complete cake, but you have to do it with exactly three chunks of cake. So you're going to try and take three chunks where the numbers add up to 15, OK. And we're going to take it in turns to play the game. I'm going to try and do my cake as well. So off you go, you can start in this game, alright? You understand the rules?

Yes

OK. Choose any piece you want. She's going for a big one, first of all, why not, that's good. Try and fill it quickly, so I'm going to go for this one. Right and I have to watch what she's doing, so she's gone for 4. OK, not sure, probably 2, I think I'm going to stop her, so go for 2. You can take your piece off and try a different one because I've just scuppered you with your strategy there. So those are your pieces, you can keep those pieces and you can chose some more pieces, OK. So you've got another piece here. Number 1, 3, what do you need, oh there's lots of people shouting. Don't help her, God it's one against 400, that's not fair. I'm going to take 3 I think, right, well that doesn't work. Has she got, did I manage to get the piece that she needs? OK, your turn, you've got 3 left, what are you going to do. No helping her, shhh, it's not fair. OK, she's looking at mine, that's fair enough, you know. Now I'm going to win now because I've got 7, yes, and I can take that, and 5 plus 3 plus 7 wins me the game. OK, so even against 400 I managed to beat her. So not bad. It's a pretty difficult game because, to start with it's quite easy, you're trying to add up the numbers to 15, but after a while you've got to keep track of so many different things: what I'm doing, what you're doing, different combinations.

Now the reason I won that so quickly, was that I was playing a different game with you. I was actually playing noughts and crosses with you. How come I was playing noughts and crosses? It doesn't look like noughts and crosses at all. Well here's the noughts and crosses board that I was using. And there it is on the screen, it's actually a magic square, some of you may have seen it: magic square. All the ways that the lines that you can draw in this thing - rows, columns, diagonals - they're all the ways that you can add up to 15. So if I played noughts and crosses with you, and I got 3 in a row, which is what I got here, 7, 3 and 5. So 7, 3 and 5s are the middle column, I won the game. I was actually playing noughts and crosses with you, which made this really easy to play. So, what we're going to do now is I'm going to give you this strategy and you're going to play me again, and you will see how much easier it is when it becomes a game of noughts and crosses. So if you could sit back down in your seat for a while, and I'm going to pull you up in a little bit.

Now what I'm going to do is to rearrange this board, so it looks like a magic square up here. And as if by magic, I've got a magic square. So now I'd like my volunteer to come back on again, and we're going to play noughts and crosses, but actually our noughts and crosses game is going to be the same as the game we played, of 15. Here are our noughts and crosses. Now, what do you like? Do you like noughts or do you like the crosses? OK, crosses, and you get to go first, like I gave you the chance to go first last time. Now do you know a good strategy for noughts and crosses, where's the best place to go on noughts and crosses, the guarantee? The corner, I actually think the middle is pretty good. OK, you can play corner, you can play corner. Now I think the really bad move, for me, is to go here. I think, can you force a win from there? Let's see if you can. Right now, if we were playing 15 I would have seen she'd got, so this one is 6, by 5 so I know that she's going for 4, so I'll block that one there. But like in noughts and crosses I can see very quickly that that's what you're going for. So your go now. No that's a good move, isn't it? Because look now there are two different ways for her to get 15, this line here, and this line here, and I can't stop her. So I'm going to go here, try and stop that one. And she wins the game like that! Now isn't that so much simpler than playing 15 and trying to add things up? Let's give her a big round of applause for beating me. Oh and, and we even have a very small cake, I'm afraid for, for the winner, so that's great.

In fact, this is one of the powerful things in mathematics, to try and change a game, or a problem you've got, into something else. And suddenly, from a new perspective, the thing might become much easier. It was quite difficult when we were trying to add all the numbers up to 15, but playing noughts and crosses - anyone can play noughts and crosses.