## Archive for the ‘Problem’ Category

### Playing with license plates

Wednesday, March 7th, 2012

My 6-year old son and I have been playing with license plates lately. Here are some of the games:

a) If all the four digits have the same parity you get x point.

b) If all the digits are squares (0, 1, 4, 9) you get y point.

c) If the three first digits and the last three digits add to 1000 you get z point.

Find x, y, and z.

Problem 371 on Project Euler asks how many license plates one has to see for the first three digits of one plate plus the first three digits on another plate to equal 1000.

### Happy New Year!

Tuesday, March 6th, 2012

- 10*9*8+7+6-5+4*321 Happy New Year!
- It is a bit late for that!
- May be, but isn’t it beautiful?!
- That 10*9*8+7+6-5+4*321 = 2012?
- Yeah, I bet you it does not happen every year!
-  How long will it be until another year arrives that can be similarly expressed?
- You mean with interspersing +, -, *, /, or nothing between the numbers in order from 10 to 1?
- Yes.
- May be in one thousand years.
- That is a long time. Amazing!

Problem source: The Museum of Mathematics and the Wolfram Blog.

### Monkeys, Shakespeare, and mathematics

Monday, March 5th, 2012

I came over the phrase  ’the Shakespeare of mathematics’ a few days ago and it has made me think.

If you search the Internet for the phrase this page turns up. It is an article written by Marcus du Satoy, 2009, and starts with a question.

How do you spark off an interest in maths when the curriculum seems dreary?

Here are a few excerpts.

The teachers are required to teach a utilitarian and unadventurous curriculum that leaves them no room to explore the creative side of the subject. Indeed, most people are utterly surprised to discover that there is any creativity in mathematics.

I am not an educationalist. I am a mathematician. But I know what turned me on to the subject. It was being shown what mathematics is really about. It was being exposed to the big stories, the Shakespeare of mathematics that inspired me.

Why are more children not given the key to this secret garden? Why can’t we include the Shakespeare of maths in the curriculum?

A few months ago, he repeated the same thing at a Wolfram conference.

I disagree a bit. It is OK to bring up new exciting topics in maths class, but only if it is done in a way that allows the students to act like mathematicians.

I like better what Erich Witman said:

Modern mathematics teaching should start with problems and attempts at solving them, which should lead to a mathematics as a strategy of such attempts. – More

How does one excite the students into learning useful and/or beautiful mathematics while using their brains creatively?

That is my question given birth by the phrase ‘the Shakespeare of mathematics.’

Of course, it takes a village to educate a child, and if the rest of the village oppose the question little can be done.

Talking about Shakespeare and maths, a piece I read a long time in The World of Mathematics (1956) came to mind. It was about monkeys writing Shakespeare. Since my copies of the four volume opus by James R Newman is in another country I downloaded the work and a dvju reader, and found that it was called Inflexible Logic and was written by Russel Maloney (1940). You find it here on the Internet, thanks to the University of Houston.

The best way I can think of to show maths teachers (many of whom has not read or does not believe in George Polya’s ten commandments) how the problem may be used in class is through a dialogue. The students should not read the dialogue, because they should be set free to investigate any way the like. Then they will communicate to each other what they find of answers and new questions and the teacher will be the master of ceremony and not the source of all knowledge.

Here are Polya’s ten commandments:

1. Be interested in your subject
2. Know your subject
3. Know about the ways of learning: The best way to learn anything is to discover it by yourself
4. Try to read the faces of your students, try to see their expectations and difficulties, put yourself in their place.
5. Give them not only information, but “know-how,” attitudes of mind, the habit of methodical work.
6. Let them learn guessing.
7. Let them learn proving.
8. Look for such features of the problem at hand as may be useful in solving the problems to come – try to disclose the general pattern that lies behind the present concrete situation.
9. Do not give away your whole secret at once – let the students guess before you tell it. Let them find out by themselves as much as is feasible.
10. Suggest it, do not force it down their throats.

You should also read what Paul Lockhart has to say about the state of maths in schools. More quotes on my philosophy of maths education here.

My interest at the moment is to write dialogues to show how one may excite students into learning useful and/or beautiful mathematics while using their brains creatively.

The dialogue on monkeys and Shakespeare will appear in this blog soon, but of course, there is room for your version of the dialogue in the comments section.

A Google search for ‘monkeys and Shakespeare’ gave some interesting hits worth reading:

### Is 1 a prime number?

Friday, March 2nd, 2012

- Is 1 a prime number?
- Is 0 a natural number?
- Is it polite to answer a question with a question?
- Who am I to know?

What are your answers? What do the questions mean? Are they important? Operators are standing by for your comments.

### What does GeoGebra prove?

Thursday, March 1st, 2012

A tangent touches a circle at B and BD is a diameter of the circle. E and F are points on the tangent. ED intersects the circle at H and FD intersects the circle at J. Prove that EHJF lie on the circumference of a circle.

This was a problem in the textbook for my Year 11 class. Solve it, and do as we did, create a GeoGebra demonstration for the theorem.

If you are a maths teacher you may like to read this.

### Excuse me for asking!

Wednesday, February 29th, 2012

If a golf ball was a sphere with radius 4.3cm and it had 350 half-spheres indented in it with radius 0.2cm what would be its surface area?

### A Google search a day

Tuesday, February 28th, 2012

Our good friends at Google run a daily puzzle challenge and asked us to help get them out to the geeky masses. Each day’s puzzle will task your googling skills a little more, leading you to Google mastery. Each morning at 12:01 a.m. Eastern time you’ll see a new puzzle, and the previous day’s answer (in invisitext) posted here.

SPOILER WARNING:
We leave the comments on so people can work together to find the answer. As such, if you want to figure it out all by yourself, DON’T READ THE COMMENTS!

Also, with the knowledge that because others may publish their answers before you do, if you want to be able to search for information without accidentally seeing the answer somewhere, you can use theGoogle-a-Day site’s search tool, which will automatically filter out published answers, to give you a spoiler-free experience. – WIRED

What is Romanian for “good morning”?

### Maths competition for high school students

Monday, February 27th, 2012

### Nobel Prize in Mathematics

Friday, February 17th, 2012

- I had a bad dream last night.
- You had? Me too!
- I was sitting on the floor inside a big square painted on the floor.
- I was dividing by zero.
- I was 3m from one corner, 4m from another, and 5m from a third.
- And I was awarded the Nobel Prize in Mathematics.
- But that prize does not exist!
- Does your square exist?
- I am trying to find out its size.
- One million Swedish kronor.

Problem inspired by dan’s problems.

### A timely question

Thursday, February 16th, 2012

Some months have 31 days. How many have 28?