Archive for September, 2011

Prisoners with hats

Monday, September 19th, 2011

30 prisoners will line up facing the same way and with a black or white hat on their head. They are not told how many hats there are of each colour. Each prisoner can only see the hats of those in front of him, but not his own or that hats behind him. A guard will ask each in turn, starting with the prisoner at the back of the line, the one who can see all the hats except his own, which colour their hat has. The prisoners can only answer ‘black’ or ‘white’. If they are right they are liberated, if not they are executed. Every prisoner can hear the answers to the other prisoners. The guard does not say who answered correctly until they all have answered.

Before the test starts the prisoners, who know the procedure of the test outlined above, are given time to discuss the best strategy they should follow to make sure that the maximum number of prisoners survice.

What is their best strategy? How many will survive?

The Royal Spanish Mathematics Society is 100 years old this year. The newspaper El País celebrates this with a math problem every week. The problem above is, in essence, one of them. Here is a video presenting the problem (in Spanish).

Quote

Monday, September 19th, 2011

A mathematician is a person who can find analogies between theorems; a better mathematician is one who can see analogies between proofs and the best mathematician can notice analogies between theories. One can imagine that the ultimate mathematician is one who can see analogies between analogies.  – Stefan Banach

Reflections

Sunday, September 18th, 2011

Playful thinking

Saturday, September 17th, 2011

Find the winning strategies

Friday, September 16th, 2011

Here are two games for two players.

Game 1: A move consists in removing 1, 2, or 3 matches from the table above. The one who removes the last match wins.

Game 2: A move consists in removing as many matches as you like, but only from one letter (P, A, I, S) at a time. The one who removes the last match wins.

What are the two strategies and is it the first or second player who can win always?

The Royal Spanish Mathematics Society is 100 years old this year. The newspaper El País celebrates this with a math problem every week. The problem above is, in essence, one of them. Here is a video presenting the problem (in Spanish).

Quote

Friday, September 16th, 2011

A good stack of examples, as large as possible, is indispensable for a thorough understanding of any concept, and when I want to learn something new, I make it my first job to build one.  – Paul Halmos

A watch in two colours

Thursday, September 15th, 2011

– Here is a game you may like.
– I am all ears.
– Colour six of the numbers red and six blue.
– You mean the numbers 1-12 on the watch you have drawn?
– Exactly!
– I can do that!
– Can you do it in such a way that after you have coloured the numbers I am not able to draw a straight line that will have three red and three green blue on each side of it?
– I am willing to try!
– That’s the spirit!

The Royal Spanish Mathematics Society is 100 years old this year. The newspaper El País celebrates this with a math problem every week. The problem above is, in essence, one of them. Here is a video presenting the problem (in Spanish).

Quote

Thursday, September 15th, 2011

If you don’t work on important problems, it’s not likely that you’ll do important work. – Richard Hamming

Solvable with elementary mathematics

Wednesday, September 14th, 2011

Fill in the square with distinct natural numbers in such a way that if you multiply the numbers in a row, a column, or any of the two diagonals, you get the same result.

‘It is not easy, but it can be solved with elementary mathematics.’

The Royal Spanish Mathematics Society is 100 years old this year. The newspaper El País celebrates this with a math problem every week. The problem above is, in essence, one of them. Here is a video presenting the problem (in Spanish).

Quote

Wednesday, September 14th, 2011

Anyone who considers arithmetical methods of producing random digits is, of course, in a state of sin.  – John von Neumann