Jumping in Puddles

Answer: Tuesday Teaser #12

Tue, 09 Mar 2010 17:04:47 +0000

It’s easy enough to solve this puzzle by working through all the combinations, but there’s a more elegant and adaptable solution using conditional probability. Let’s define the events:

A: I throw three numbers in strictly increasing order;

B: I throw three different numbers;

Then we can easily see the probability of B and the probability of A given B.

P(B) = 1 x 5/6 x 4/6 = 5/9

P(A|B) = 1/3 x 1/2 x 1 = 1/6

Applying conditional probability [ namely that P(A|B) = P(A∩B) / P(B) ], we see that P(A∩B) = 5/54. But then the probability of A and B is just the probability of A since A can’t happen without B happening.

Therefore the probability of throwing three numbers in strictly increasing order is 5/54.

As a further example, imagine I split a pack of cards into four piles, one for each of the suits. If I draw one card from each in turn, what is the probability that I pick cards in a strictly increasing order (let’s say that Ace is high)? Using the above method, it’s easy. Define the events:

A: I draw four cards in strictly increasing order;

B: I draw four different numbered cards;

Then:

P(B) = 1 x 12/13 x 11/13 x 10/13 = 1320/2197

P(A|B) = 1/4 x 1/3 x 1/2 x 1 = 1/24

So the probability is 1/24 x 1320/2197 = 55/2197 (or just better than 1 in 40).

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Tue, 09 Mar 2010 01:37:34 +0000

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Tuesday Teaser #12

Tue, 02 Mar 2010 20:09:30 +0000

Back to dice probability for this week’s teaser:

If I throw three 6-sided dice one by one, what is the probability that I roll three numbers in strictly increasing order?

As an added bonus, enjoy OK Go’s incredible new video here.

OK Go This Too Shall Pass

Tue, 02 Mar 2010 20:02:47 +0000

OK Go This Too Shall Pass

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Mon, 01 Mar 2010 17:55:34 +0000

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Tue, 23 Feb 2010 20:48:50 +0000

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Quick Sunday Trivia

Sun, 21 Feb 2010 21:15:18 +0000

A few London themed trivia questions for a Sunday evening:

What is the oldest London Underground line?
Can you name the thirteen bridges of Central London?
Fill in the missing blank in this famous quote about a street in London:

“You must not miss _________. At one end you will find a statue of one of our kings who was beheaded; at the other, a monument to the man who did it.That is just one example of our attempts to be fair to everybody.”

Edward Victor Appleton

See the comments for the answers.

“It is difficult to speak adequately or justly of London… It is only magnificent… It is the biggest aggregation of human life – the most complete compendium of the world.”

Henry James, 1881

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Wed, 17 Feb 2010 02:17:27 +0000

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Answer: Tuesday Teaser #11

Wed, 10 Feb 2010 14:34:00 +0000

There are many ways to answer the first three, but here are my favourite answers:

1. “I am not a knight.”

2. “I am a knave.”

3. “If someone asks me whether I’m a normal, I’ll answer ‘Yes’.”

Clearly if a knight or a knave were to say any of 1, 2 or 3, they would have to be lying or telling the truth respectively.

The second of the puzzle is much harder and relies on what method of semantics are used.

4. “If I am not a knight, then this statement is false.”

5. “If I am not a knave, then this statement is false.”

6. “If I am not a knight, then I am not a knave, and this statement is false.”

Let’s work through 4 to see why it’s a satisfying answer. If said by a knight, then the statement is true (under the ordinary rules of logic) since the implication is vacuous. If said by a knave, we arrive at a curious paradox because it has to be a lie (by definition) and hence must also be true (because “this statement is false” must be a lie). Note that it therefore cannot be said by a knave, because the statement would not be a lie (it’s a paradox so is neither true nor false).

Likewise, if said by a normal it could be true (in which case ”this statement is false” must be true) or it could be a lie (in which case ”this statement is false” must be a lie).

At this point, it depends on what your interpretation of the rules are as to whether this is an answer or not. If we assume that everything a normal says is either true or false, then it couldn’t be said by a normal (as it would be neither true nor false) and hence can only have been said by a knight. If, on the other hand, we allow normals to say paradoxical statements, we have no way of discerning whether the statement was said by a knight or a normal. If this is our interpretation, there is no way of answering the final three parts of the puzzle.

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Tue, 09 Feb 2010 22:30:15 +0000

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Quick Monday Trivia

Mon, 08 Feb 2010 23:06:58 +0000

Can you name the ten countries of the world whose names are only four letters long?

See the comments for the answer.

A Winter’s Day in London

Mon, 08 Feb 2010 22:58:30 +0000

A Winter’s Day in London

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Thu, 04 Feb 2010 00:08:34 +0000

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Tuesday Teaser #11

Tue, 02 Feb 2010 11:00:00 +0000

Consider three types of people in a medieval kingdom - Knights, Knaves, and Normals. Knights always tell the truth, Knaves always lie, and Normals can do either. The King wants his daughter to marry a nice normal Normal, not one of those goody-goody Knights or devious scoundrel Knaves.

How can you make a true statement that will convince the King you are a Normal?
How can you make a false statement that will convince the King you are a Normal?
How can you make a statement that will convince the King you are a Normal, but he won’t know whether it’s true or false?

For a (much) harder twist, suppose the King doesn’t trust those unpredictable Normals and wants his daughter to marry a Knight or a Knave instead.

How can you, in one statement, convince the King you are a Knight?
How can you, in one statement, convince the King you are a Knave?
How can you, in one statement, convince the King you are not a Normal, but leave him unable to deduce whether you are a Knight or a Knave?

This puzzle can be found in it’s original form in Raymond Smullyan’s What is the Name of This Book? although this slightly modified version is taken from the excellent [wu:riddles].

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Tue, 26 Jan 2010 09:31:37 +0000

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Catch-22

Tue, 19 Jan 2010 23:18:14 +0000

Over the last couple of weeks I’ve been reading Joseph Heller’s classic Catch-22, which I’m currently about halfway through. An FT quote on the back of the book reads: “Blessedly, monstrously, bloatedly, cynically funny, and fantastically unique.” I wasn’t sure what that meant until I started reading the book, but it’s now clear to me that the description is indeed very accurate. Take, for example, the following passage which is typical of Heller’s cynically funny style throughout the whole book so far:

“From now on,” he said, “I don’t want anyone to come in to see me while I’m here. Is that clear?”
“Yes, sir,” said Sergeant Towser. “Does that include me?”
“Yes.”
“I see. Will that be all.”
“Yes.”
“What shall I say to the people who do come to see you while you’re here?”
“Tell them I’m in and ask them to wait.”
“Yes, sir. For how long?”
“Until I’ve left.”
“And then what shall I do with them?”
“I don’t care.”
“May I send them in to see you after you’ve left?”
“Yes.”
“But you won’t be here then, will you?”
“No.”
“Yes, sir. Will that be all?”
“Yes.”
“Yes, sir.”
“From now on,” Major Major said to the middle-aged enlisted man who took care of his trailer, “I don’t want you to come here while I’m here to ask me if there’s anything you can do for me. It that clear?”
“Yes, sir,” said the orderly. “When should I come here to find out if there’s anything you want me to do for you?”
“When I’m not here.”
“Yes, sir. And what should I do?”
“Whatever I tell you to.”
“But you won’t be here to tell me. Will you?”
“No.”
“Then what should I do?”
“Whatever has to be done.”
“Yes, sir”
“That will be all,” said Major Major.
“Yes, sir,” said the orderly. “Will that be all?”
“No,” said Major Major.

I’d certainly recommend it so far.

Answer: Tuesday Teaser #10

Tue, 19 Jan 2010 08:00:00 +0000

The answer is surprisingly simple. If at any stage somebody sits in your seat (number 50), then you will inevitably be forced to sit elsewhere. Similarly, if at any stage somebody sits in seat number 1, then each subsequent person will have their seat available and so you will end up sitting in your own seat. Now at each stage, while seats 1 and 50 are free, there is an equal chance of a person choosing to sit in seat 1 as seat 50. Hence, your chance of sitting in your own seat is a half.

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Mon, 18 Jan 2010 22:03:11 +0000

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Tuesday Teaser #10

Tue, 12 Jan 2010 10:52:09 +0000

You’re taking a coach trip up to Scotland and are standing in line waiting to board the coach as it prepares to depart from London. There are 50 seats on the coach and each passenger standing in line is assigned one of these seats (for simplicity, suppose they are lined up in order with the first person in line being assigned to seat number 1, etc). Now the first passenger in line is horribly drunk and randomly chooses a seat. The subsequent passengers are all sober and so they’ll sit in their allocated seat if it’s free. If not, they’ll choose one at random to sit in. If you’re at the very back of the line, what are the chances of you sitting in your allotted seat (number 50) when you eventually get to board?

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Tue, 12 Jan 2010 06:42:01 +0000

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