Jumping in Puddles

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Mon, 26 Jul 2010 21:10:39 +0100

Tuesday Teaser #20

Tue, 20 Jul 2010 15:51:51 +0100

It’s the Salem witch trials and two villagers are accused of being witches.

The witch-finder-in-chief says “There is a very simple test to tell whether you are witches. Each pick a card from a deck, and you can look at it if you want. But what I want you to do is predict the color of the other person’s card.

I’m going to put you in separate rooms so there is no communication between you. And if you’re both wrong, or one of you is wrong, then you are free to go. But if you both correctly predict the color of the other person’s card, then you are in league with the devil and will be burned at the stake.”

You have one minute to discuss strategy before the game begins. What is the strategy to survive?

I’ve stolen this week’s puzzle from the excellent game theory blog Mind Your Decisions. For anyone with an interest in maths, economics or game theory, I’d definitely recommend this as being worth following.

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Mon, 12 Jul 2010 21:02:53 +0100

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Answer: Tuesday Teaser #19

Thu, 08 Jul 2010 18:00:00 +0100

The textbook answer to this is to name the North Pole as the only place where it can be done. Indeed, it is definitely true that the North Pole is the only place where you can walk a mile south (to a point A, say), then a mile east (to a point B, say) and then a mile north, arriving back where you started, such that point A is not equal to point B. However, what if point A is equal to point B? Imagine drawing a line around the Earth (close to the South Pole) that is a mile in circumference. If you were to start anywhere on the line and walk a mile east you would arrive back where you started. Such a line definitely exists, so we can conclude that if we start at any point a mile north of the line, we obtain another place with the property that we want. This gives us an (uncountably) infinite number places on Earth that have the property.

We can even go one further than this: instead of walking once around the line and arriving back where we started, we can draw lines 1/2 mile, 1/3 mile, 1/4 mile, etc. long and walk around them 2, 3, 4 times to arrive back to where we started.

In conclusion, there are infinitely many points on Earth from which we can walk a mile south, a mile east and a mile north and arrive back at the place where we started.

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Thu, 01 Jul 2010 13:22:55 +0100

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Tuesday Teaser #19

Tue, 22 Jun 2010 22:58:00 +0100

This week’s puzzle is an absolute classic with a great (and perhaps surprising) solution:

How many points are there on Earth where you can walk one mile south, then one mile east, then one mile north and end in the place that you started? For the sake of argument, assume the Earth is a perfect sphere (of course).

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Tue, 22 Jun 2010 20:08:51 +0100

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Answer: Tuesday Teaser #18

Tue, 22 Jun 2010 11:00:00 +0100

The trick to this puzzle is to look at the glass side-on and imagine it in two dimensions as a simple rectangle. Now, if you tip the glass so that the water touches the rim of the glass, the water will form a line that approximately meets the opposite diagonal. If the line meets the glass above the opposite diagonal, then the glass must be more than half full and vice-versa.

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Tue, 15 Jun 2010 06:36:44 +0100

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World Cup 2010 Preview

Thu, 10 Jun 2010 21:56:00 +0100

With less than 24 hours to go, I’m incredibly excited about this year’s World Cup. It looks to be one of the most open tournaments that I can remember, with plenty of genuine contenders for the title. There are nine countries who I think could realistically win the trophy (Argentina, Brazil, England, France, Germany, Italy, Netherlands, Portugal, Spain), but there are plenty of good teams who could also spring surprises along the way. I wouldn’t be surprised if the likes of Paraguay, Chile, Mexico, Denmark and Slovakia were to gain a big scalp at the expense of one of the “Big Nine”.

Come on England!

There has been much hype about England ahead of the tournament, with many pundits predicting that they could win it. But are they being realistic? They certainly have a similar squad to 2006 in terms of personnel, but under Fabio Capello, they look a far more balanced team than they ever did under Sven.

One huge positive for England is the draw they’ve been given. Having being drawn alongside Algeria, Slovenia and USA, England should comfortably qualify as group winners. It’s from there that it gets interesting. In the second round, England would then play the runner-up of Group D. Germany should win that group, leaving England with a game against Ghana, Serbia or Australia, all of whom they would expect to beat. The seed who they’d be due to play in the quarter finals is France. Not only do England stand a great chance of beating France, but France are also the most vulnerable of the group seeds, having to contend with Mexico and Uruguay in Group A. As an illustration, they’re only 6/4 to win their group, whereas England are 2/5 to win theirs. If France were to fail to win their group, England’s QF would be against Nigeria, Greece, South Korea, Uruguay or Mexico (assuming Argentina win their weak group) - again, a game they should win. This seems an extremely comfortable ride to the semi finals when compared against tournament favourites Spain’s. If they win the weak Group H and all goes to seed, Spain would face Portugal, before going on to play the holders Italy. Winning both of those games would only get them to the semi finals!

England’s progress will inevitably depend on Wayne Rooney. In fact, they are probably the only team of all the potential winners who rely so heavily on a single player. If Rooney can recover some of his Manchester United form from earlier in the season, England could be a match for anybody.

Tactically speaking however, I’d argue that England will be more reliant on Gareth Barry. Capello is likely to deploy Heskey/Crouch and Rooney in a 4-4-2 formation for the group games, the same system that worked so well throughout qualification. However, against world class opposition he’ll surely play a 4-5-1/4-3-3 with Rooney as a lone striker and Lampard, Gerrard and Barry in the centre of midfield. For this reason, it is imperative to England’s chances that Barry can regain fitness and form before the crucial knockout stages.

The only significant worry I have about England is at right-back. Glen Johnson is simply not good enough to be an international defender. As comfortable on the ball and threatening as he is going forward, it can never make up for being defensively solid - I hope he doesn’t get caught out in South Africa the way he has done for both Liverpool and England this season.

All in all, I’m hugely optimistic about England’s chances. Given the draw they’ve got it would be hugely disappointing if England were not to progress to the semi finals and their best finish since 1966.

Teams to Watch

Netherlands - Having had a faultless qualifying phase, the Oranje arrive in South Africa with the possibility of fielding a “Fab Four” of Wesley Sneijder, Rafael van der Vaart, Arjen Robben and Robin van Persie, four of the most exciting attacking players in world football. Unfortunately, perhaps, Dirk Kuyt seems likely to start, but if Bert van Marwijk can get the Fab Four on the pitch together, I’d be surprised if they didn’t play some absolutely scintillating football during the tournament.
Chile - Tactically, Chile are by far the most interesting team to have qualified for South Africa. Coach Marcelo Bielsa usually plays a 3-3-1-3 formation, similar to that which spectacularly failed when he was coach of Argentina in the 2002 World Cup. Tactics website Zonal Marking has produced some excellent pre-tournament analysis of Chile’s system. If their qualification campaign is anything to go by we can expect their matches to contain goals too. Their 18 games yielded 54 goals, 10 of which came from Monterrey’s prolific striker Humberto Suazo.
Nigeria - Though they won’t necessary play aesthetically pleasing football, Nigeria will be spearheading Africa’s challenge after the other African sides have all been left with uphill tasks. Africa’s best hope would perhaps have been the Ivory Coast but they have a horribly tough group and the world-class front-man whom their team is built around, Didier Drogba, has since suffered an injury that makes his participation highly doubtful. As Zonal Marking writes: “To get to the knockout stages, the Ivory Coast must get past Brazil and Portugal. To do this, they have appointed Sven-Goran Eriksson – England fans will appreciate the irony there”. Algeria, Cameroon, Ghana and South Africa all have groups from which they are unlikely to progress, so it may well be that Nigeria are the only African representative in the second round. Nigeria will need to take points off Greece and South Korea, but they have every chance of doing so.

Players to Watch

Instead of naming the superstars here, I thought I’d mention some of the lesser known players who I’m looking froward to seeing over the next four weeks:

Mesut Özil (Werder Bremen, Germany) - One of a number of exciting young German players, the 21-year-old is a skillful left-footed creative midfielder who is likely to play off Miroslav Klose as Germany’s main playmaker. The injury to Michael Ballack has opened the door for Özil at this World Cup, where he should get ample opportunity to show off his excellent technique and eye for a pass. Earlier in the season, The Guardian described him as “best player in the [Bundesliga] at the moment, a one-man source of creativity and surprise”.
Lucas Barrios (Borussia Dortmund, Paraguay) - Argentine-born Barrios switched nationality in March in order to have a chance of playing at the World Cup. Qualifying as Paraguayan through his mother, he’s already made an immediate impact having scored 3 goals in his first 3 games. On the back of an excellent debut season in the Bundesliga, he’s a key part of a solid-looking Paraguay side who’ll have hopes of reaching the quarter finals at least. I’ll be closely following Paraguay’s progress having followed their excellent qualifying campaign (in which they gained victories over Argentina and Brazil) and having bought a shirt on my visit to Asuncion in 2008.
Riccardo Montolivo (Fiorentina, Italy) - With no Francesco Totti and injury worries hanging over Andrea Pirlo, Italy look likely to turn to Montolivo, Fiorentina’s 25-year old captain. A highly-rated creative midfielder, he has the potential to shine playing ahead of Daniele De Rossi as Italy’s attacking midfielder in the group stages.

Predictions

Finally, it wouldn’t be much of a preview without giving some predictions. So here goes…

Winners: Brazil - A winning machine under Dunga. They should be the favourites given that Spain’s draw is extremely tough.
Top Goalscorer: David Villa - His goalscoring record for high-scoring Spain is incredible. Wayne Rooney also has an excellent chance given England’s cushy draw. If they make the semi-finals, he’ll get more than enough action.
England’s Stage of Elimination: Knocked out in the semi finals (on penalties, obviously). This bet is currently available at 4-1 on Betfair.
My World Cup Knockout Stages Bracket - You can view this after the tournament begins.

Tuesday Teaser #18

Tue, 08 Jun 2010 23:53:12 +0100

An easier and less mathematical riddle this week for the benefit of @aliholli and @nixenpixen:

Suppose you’re in an empty room with no method of escape. The only object in the room is a transparent glass of water. The glass is a perfect cylinder which looks half full, though you can’t be sure exactly. You are only allowed to leave the room if you can accurately determine whether the glass is half full, more than half full, or less than half full. How would you do so?

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Tue, 08 Jun 2010 04:10:31 +0100

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Answer: Tuesday Teaser #17

Tue, 01 Jun 2010 22:50:14 +0100

Incredibly, they aren’t the same. Let’s calculate the two using conditional probability:

In the first problem, we want to find the probability that both children are boys given that at least one of them is. Using the formula for conditional probability this is equal to:

P(Both children are boys AND at least one child is a boy)/P(at least one child is a boy)

It’s clear that this is equal to P(Both children are boys)/P(at least one child is a boy)

There are four possibilities: GG, GB, BG, BB, so the probability of at least one boy is 3/4 and the probability of both being boys is 1/4.

Hence we conclude the first problem gives us an answer of 1/3.

In the second part, using the same method we get that the probability that both children are boys given that at least one of them is a boy born on a Tuesday is:

P(Both children are boys AND at least one child is a boy born on a Tuesday)/P(at least one child is a boy born on a Tuesday)

There are 296 possibilities: G(Mon)G(Mon), G(Mon)G(Tue), etc., of which 13 give us the situation where both children are boys and at least one child is a boy born on a Tuesday. Similarly, 27 give us the situation where at least one child is a boy born on a Tuesday.

So, amazingly, we end up with the probability having increased to 13/27, much closer to the intuitive answer of 1/2.

The New Scientist provides a similar explanation of how to arrive at this answer without resorting to conditional probability.

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Tue, 01 Jun 2010 21:26:39 +0100

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Tuesday Teaser #17

Tue, 25 May 2010 09:00:00 +0100

A brilliant, although completely useless, counter-intuitive probability puzzle this week.

If I have exactly two children and at least one of them is a boy, what is the probability that both of them are boys?
If I have exactly two children and at least one of them is a boy born on a Tuesday, what is the probability that both of them are boys?

Are these two probabilities the same or does the seemingly arbitrary “born on a Tuesday” change matters?

A controversial puzzle taken from this week’s New Scientist.

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Mon, 24 May 2010 17:32:46 +0100

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Answer: Tuesday Teaser #16

Sun, 23 May 2010 09:00:00 +0100

Tricky one to explain this one, you may need a pen and paper to cumulate the totals. So we start with two 8-litre jugs, let’s call them A and B, and an empty 3-litre jug, let’s call that C.

3 litres A => C => Pool 1
3 litres A => C
2 litres A => Pool 2
3 litres C => A
3 litres B => C => A
3 litres B => C
2 litres C => A
1 litre C => Pool 3
2 litres B => C
1 litre A => C
3 litres C => B
3 litres A => C => B
3 litres A => C
1 litre A => Pool 4
2 litres C => B
1 litre C => Pool 1 (which now has 4 litres in it)
3 litres B => C => Pool 3 (which now has 4 litres in it)
3 litres B => C => Pool 4 (which now has 4 litres in it)
2 litres B => Pool 2 (which now has 4 litres in it)

Done!

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Mon, 17 May 2010 17:32:13 +0100

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Mon, 10 May 2010 22:24:48 +0100

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Tuesday Teaser #16

Tue, 04 May 2010 22:08:51 +0100

A Die Hard with a Vengeance style puzzle this week, but significantly harder (sorry @aliholli)!

Suppose you have two 8-litre jugs full of water and an empty 3-litre jug. Your aim is to transfer exactly four litres of water into four separate pools. Assume that:

The only source of water is the 16 litres in the two 8-litre jugs;
Once water has been added to any of the pools, it cannot be removed.

Can you find a way?