The Prisoners and the Chessboard Solution

 Two weeks ago, I posted the following puzzle:

So, the evil guy who kidnaps people and sets them bizarre mathematical tasks which they have to complete or be executed has kidnapped you and your friend Bob. He sets you the following bizarre mathematical task:

I will take you into a room in which there is a chessboard. There is a coin on every one of the squares of this chessboard, showing either heads or tails. I will then tell you which one of the squares has the key to the door of the prison under it. After this, you will be allowed to turn over exactly one of the 64 coins and then be escorted from the room.

Immediately after this, Bob will be brought in, and will be asked to find the key. If he can locate it you're both free to use it to unlock the prison and leave. If not, you'll both be eaten by rabid wildebeest.

You and Bob are free to discuss a strategy before you are taken into the room with the chessboard - what should you do?

Adrianna emailed me a (sort of) solution less than a few hours later. I've posted my solution (and hers) below the fold).

Click here to read the rest

The next number in the sequence

We'll start today's post with what I think is a relatively easy set of puzzles. What is the next number in each of the following sequences?

a. 1 2 3 4 5 6 7 8 9 10 11 12

b. 1 1 2 3 5 8 13 21

c. 8 8,000,000,000 8,000,000,008

d. 31 28 31 30 31 30 31 31 30 31

e. 3 1 4 1 5

f. 1 11 21 1112 3112 211312 312213 212223

g. 1 1 4 1 9 2 16 3 25 5 36 8 49 13

Click here to read the rest

Iterated Prisoner's Dilemma

Just a little note to say that we're running an Iterated Prisoner's Dilemma competition in our department. If anyone is reading this, and would like to enter a strategy, they're more than welcome. I've already written a description of the tournament, which you can find here, so I'm not going to write it again.

Incidentally, while writing that website, I decided I should probably put something on my academic page, you can read what I decided to write here - I'll be updating it with a few bits and bobs that I've done over the past few years (mostly the outreach stuff I've done) as soon as I get back from Jamaica. If it looks familiar, that's because it looks exactly like Andy's academic homepage - thanks for the template Andy!

Toy Boat, Big Whip, Piano Panier

I learned a long time ago that one of the hardest Tongue Twisters in the English language is "toy boat". If you're a native speaker and you don't trust me, try it - say "toy boat" quickly and repeatedly. If you're not a native speaker, go find one and get them to say it - you'll be pleasantly amused by their difficulty.

Other examples of this phenomenon in English are "big whip" and "unique New York", although this latter is far too contrived for my liking - the beauty of "toy boat" is that it's a phrase that you've undoubtedly said at some point in the past without realising it's potential for short-circuiting your voicebox.

Since I discovered this weird phenomenon - not just that "toy boat" makes for a ridiculously good tongue twister, but also that non-native speakers never seem to have any difficulty with it, I've been on the hunt for something similar in other languages. So far, I've found "piano panier". If you're a native French speaker, try it. If you're not (but speak good enough French to give it a go), just see how easy it is. I've yet to get any examples in other languages, but that's probably at least in part because I haven't asked enough people.

I'm also very intrigued as to why there is such a difference. Do our brains actually process speech in foreign languages fundamentally differently? Is it just that I pronounce "piano panier" so badly that I don't come across the same difficulty as a native speaker would? Are there any serious linguists out there who would be interested in studying this phenomenon?

Optical illusions

The BBC has this article about optical illusions which we were discussing in the pub after the seminar today. I thought this would be a good chance to share one of my favourites. Have a look at the picture below:

The blue squares on the top in the left-hand picture are the same colour as the yellow squares on the top in the right-hand picture.
Don't believe me?

That grey line is the same colour all the way along. If you still don't believe me, use some sort of graphics programme and determine what the colours are for yourself.

I can't remember where I downloaded these pictures from, it was ages ago - if anyone knows who this illusion was made by, I'll put in  link.

If you want some more illusions, have a look here I particularly recommend the break of the curveball and the color dove, but the whole site is full of pretty awesome stuff.

The Prisoners and the Chessboard

There's a long tradition in mathematics of puzzles in which someone asks some prisoners to perform some bizarre task. Usually, if they can't complete the task, they are to be executed. Here is one of my favourite examples of the genre (which I only heard for the first time relatively recently).

So, the evil guy who kidnaps people and sets them bizarre mathematical tasks which they have to complete or be executed has kidnapped you and your friend Bob. He sets you the following bizarre mathematical task:

I will take you into a room in which there is a chessboard. There is a coin on every one of the squares of this chessboard, showing either heads or tails. I will then tell you which one of the squares has the key to the door of the prison under it. After this, you will be allowed to turn over exactly one of the 64 coins and then be escorted from the room.

Immediately after this, Bob will be brought in, and will be asked to find the key. If he can locate it you're both free to use it to unlock the prison and leave. If not, you'll both be eaten by rabid wildebeest.

You and Bob are free to discuss a strategy before you are taken into the room with the chessboard - what should you do?