December 2011
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Albums of the Year 2011
This year has been a truly excellent year for music, with loads of great records, brilliant singles and some fantastic gigs. Here are my top ten albums of 2011; each one is a gem.
My Album of the Year
Bon Iver Bon Iver
The Rest of the Top Ten
(in no particular order)
Burst Apart The Antlers The Rip Tide Beirut Hurry Up, We’re Dreaming M83 Tamer Animals Other...
September 2011
2 posts
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The Horrible Crowes (43)
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August 2011
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Tuesday Teaser #36
Look at the diagram below:
The problem is this: can you draw a continuous curve, without taking your pen off the paper, that crosses each line once and only once? So, for example, one attempt might look like this:
Unfortunately, this attempt misses one line, which I’ve highlighted above.
Hint: If you need a hint, notice the similarities with the very famous Seven Bridges...
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Answer: Tuesday Teaser #35
In order to answer this teaser, we need to calculate the probability of the three man jury coming to the correct verdict.
In order to come to a correct verdict, two of the three must independently reach the correct verdict. This happens when either:
The two “serious” jurors come to the correct verdict; or
The two “serious” jurors disagree, but the “flippant”...
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Tuesday Teaser #35
Suppose there is a three man jury which has two members who each independently have probability p of making the correct decision, and a third member who flips a coin for each decision. In this three man jury, the majority rules.
Also, suppose there is a one man jury with a probability p of making a correct decision.
Which of these juries offers the best probability at arriving at the correct...
2 tags
Answer: Tuesday Teaser #34
This puzzle is pretty tough: the best way to think about it is to generalize what happens if a company hires one more person than it already has.
Let’s let D(n) be the expected number of working days for a company with n employees. If the company has 1 employee, it is clear that there is always 1 unique birthday. So D(1) = 364.
Now, inductively, adding an extra employee will have a...
July 2011
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June 2011
4 posts
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Tuesday Teaser #34
A very tough, but extremely interesting, probability problem for my first teaser in a while.
Let’s forget the standard working week and assume that companies decide when people have to work using a completely different method. Let’s suppose that they have to give a holiday to every single employee at the company whenever any one employee has a birthday. Other than those holidays,...
2 tags
Answer: Tuesday Teaser #33
This can be solved using the classic trial and error method, but we can also do it mathematically by using the information to set up a couple of simultaneous equations.
Let’s call the number of dogs x, the number of cats y, and the number of mice z.
The first thing we’re told is that there must be 100 animals, so x + y + z = 100.
The second thing is that the 100 animals must cost...
May 2011
1 post
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April 2011
7 posts
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Tuesday Teaser #33
You’re given $100 and told to spend it all purchasing exactly a hundred animals at the pet store. Dogs cost $15, cats cost $1, and mice are 25 cents each.
The other only other criterion is that you have to purchase at least one of each animal.
How many of each animal do you have to purchase to equal a hundred animals purchased at exactly a hundred dollars?
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Answer: Tuesday Teaser #32
It is clear from the format of the game that by playing entirely by random, the chance of winning is 1/3. For example, an entirely random strategy could mean simply choosing a random piece of paper and sticking with it, whatever it may be. But we can do a lot better than this by improving our strategy.
A helpful concept when trying to come up with optimal strategies is the concept of information....
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Tuesday Teaser #32
A simple game for this week’s teaser. Three different numbers are chosen at random, and each is written on a slip of paper. The slips are then placed face down on the table. Your objective is to choose the slip upon which the largest number is written.
In order to achieve this goal, you can turn over any slip of paper and look at the amount written on it. If you think this might be the...
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Answer: Tuesday Teaser #31
For the first problem, let’s consider the properties of the tournament rather than trying to deal with a messy tournament bracket. The tournament is straight knockout. That means that all players except the winner, end up losing one (and only one) match, so we can immediately conclude that if there are 128 players, the total number of matches must be 127. Nice.
For the second problem,...
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March 2011
6 posts
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Elbow (61)
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Tuesday Teaser #31
A nice mixture of sports and probability this week - ideal if you ask me :)
In a straight knockout tennis tournament involving 128 players, how many matches are played? N.B. This is very easy if you think about it in the right way…
Now imagine each player is ranked 1 to 128. If the tournament bracket is selected randomly, what are the chances that the top-ranked player will play the...
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Answer: Tuesday Teaser #30
This puzzle relies on simple logic, so let’s go through the statements step by step and see what they tell us.
Tom said: “If I don’t know, then Jerry definitely doesn’t know either.”
Jennifer told Tom the month in which she was born. Tom then asserts that Jerry cannot possibly know at this stage. If Jerry were able to tell the exact date, it would be because the day which he was...
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Tuesday Teaser #30
Back to some good old-fashioned logic for this week’s teaser.
Jennifer told Tom and Jerry that her birthday is one of the following ten dates:
March 4, March 5, March 8, June 4, June 7, September 1, September 5, December 1, December 2, December 8.
She then told Tom in which month she was born and Jerry on which day she was born, and asked them to guess which one of the above ten dates...
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Answer: Tuesday Teaser #29
There are various strategies that could be employed to improve the expectation from simply pressing the + button several times. The best such strategy is to separate the tracks into two distinct groups - “far away” tracks and “close” tracks. For the “far away” tracks, the strategy is to hit shuffle until you end up on a “close” track. For the...
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February 2011
10 posts
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The National (25)
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Beach House (12)
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Tuesday Teaser #29
You have a playlist of 11 tracks on the MP3 player connected to your stereo. Each track in the playlist is clearly numbered, but there are just three buttons. The “+” button takes you to the next track in ascending order, looping back to track #1 after the highest numbered track. The “-” button takes you through the tracks in descending order, looping back to the highest numbered track after track...
2 tags
Answer: Tuesday Teaser #28
I think there are four possibilities for you as the cyborg shooting first. You can either:
Shoot at Cyborg B (who has 50% accuracy)
Shoot at Cyborg C (who has 100% accuracy)
Shoot into the air at nothing/nobody.
Let’s explore these in turn. If you shoot at Cyborg B, you will kill him one-third of the time, in which case you will be immediately killed by Cyborg C. So your probability of...
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Tuesday Teaser #28
Back to good old probability this week.
Suppose you’re a cyborg in a pistol duel with two other cyborgs. Unfortunately, you’ve only been programmed to fire pistols with an accuracy of 33%. The other two cyborgs shoot with accuracies of 50% and 100%, respectively. The rules of the duel are one shot per-cyborg per-round. The shooting order is from the worst shooter to the best shooter....
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Answer: Tuesday Teaser #27
1. You have only one such rope. How will you measure 30 minutes?
For this problem, simply burn the rope from both ends. It will then burn twice as quickly as usual and hence will take 30 minutes to burn through.
2. You have two such ropes. How will you measure 45 minutes?
For this problem, burn the first rope from both ends, and at the same time light one end of the second rope. When the first...
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Tuesday Teaser #27
There is a certain kind of rope which takes exactly one hour to burn from end to end. The rope is not uniform, not symmetric - thicknesses throughout its length vary randomly. No two such ropes will be identical.
You have only one such rope. How will you measure 30 minutes? [Easy]
You have two such ropes. How will you measure 45 minutes? [Medium]
You have only one such rope. How will you...
3 tags
Answer: Tuesday Teaser #26
Whatever distribution Pirate A proposes, Pirates B-E will vote on it according to whether or not they can survive/get more money from the next distribution (and so on), so in order to think about this problem we need to work backwards.
Let’s consider the case that Pirates A, B and C have been thrown overboard and it’s Pirate D’s turn to propose a distribution. Then it is clear...
January 2011
9 posts
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Tuesday Teaser #26
This is perhaps my favourite brainteaser of all time. Whilst the solution is readily available on Wikipedia, it’s worth spending some time thinking about as the solution is both beautifully simple and counter-intuitive. Here goes:
There are five rational pirates, A, B, C, D and E. They find 100 gold coins in a treasure chest and must decide how to distribute them.
The pirates have a strict...
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Answer: Tuesday Teaser #25
Apologies for how tough this teaser was. It relies on an extremely cunning trick.
Bourbaki’s strategy relies on looking at all the possible ways in which the names can be arranged in the boxes (the permutations).
He starts by giving each person a number from 1-6 and labelling each of the boxes 1-6, such that each person has a corresponding box. The exact order does not matter here, so...
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Tuesday Teaser #25
This week’s (very tough) teaser is taken from the New York Times’ excellent Numberplay series:
Many years ago, six people — Alfred Dreyfus, Jean Calas, Emile Outreau, Jeanne d’Arc, François Outreau and Nicolas Bourbaki — were sentenced to solitary confinement in a French prison — unfairly, many thought. Fortunately, they had a sympathetic warden, who gathered them together in the...
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Answer: Tuesday Teaser #24
The strategy that Player A needs to use is to try to find a way in which he can always hold a higher straight flush than Player B. This immediately requires that Player A’s first draw prevents Player B from choosing an A-K-Q-J-10 straight flush (or Royal flush), because such a hand would guarantee Player B at least a tie. This means that Player A needs to choose at least one of the Ace-to-10...
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