Items of the month archive

July 2008

Finding the highest card

On 30 cards I write 30 different numbers, shuffle them thoroughly and place them face down in a pile on a table. You have no idea how big any of my numbers are. You turn over one card at a time, look at the number and decide whether you want to stick with that card or discard it and turn over the next card. Once a card has been discarded you cannot go back to it. What strategy should you use in order to maximise your chances of selecting the card containing the biggest number?

A possible strategy:
Do not keep any of the first 10 cards you turn over but remember the biggest number and call this number N. From the 11th card on, as soon as you see a number bigger than N, stop there. Of course, you might not see another number bigger than N in which case you will end up turning over all the cards without ‘winning’.

With this strategy, what is the probability that you will select the biggest number?

June 2008

Roots of a cubic in the Argand diagram

Find a cubic with a single stationary point at x=a (i.e. dy/dx=0 has a repeated real root).
Plot the roots of the cubic and x=a on an Argand diagram.

What do you notice?

What happens if the restriction of a single stationary point is removed?

May 2008

MEI conference

The 2008 MEI Conference will take place at the University of Hertfordshire, Hatfield from Thursday 3rd - Saturday 5th July. Bookings are now being taken.
Brochure and booking form

April 2008

Phive points

The curve y = x⁵ – 5x³ + 5x passes through these points.

What are the x-coordinates of the stationary points on this curve?

March 2008

When is a triangle a square?

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 6²

Are there an infinitely many triangle numbers that are also square numbers?

February 2008

Mathematical podcasts/webcasts :

There are a number of mathematical podcasts/webcasts on the internet. The following are recommended:

• Plus magazine podcasts
  The Plus podcasts will bring you the latest news from the world of maths, plus interviews and discussions with leading mathematicians and scientists about the maths that is changing our lives.
• More or less
  Radio 4's journey through the often abused but ever ubiquitous world of numbers.
• Royal Institution Christmas lectures
  These video webcasts include Marcus du Sautoy (2006), Ian Stewart (1997) and Christopher Zeeman (1978).
  ( They are free but you need to register and add them to your "shopping basket").

January 2008

Two's company; three's a crowd:

What is the smallest positive integer that cannot be expressed using exactly three 2s and any mathematical operations you wish?

Here are some examples:

A hint
Can you adapt the following for other numbers?

December 2007

A mathematical Christmas message from MEI:

November 2007

Finding the tangent to a parabola geometrically

To find the tangent to a parabola at a point P:

Draw a vertical line through P.
Draw a horizontal line through Q, the vertex of the parabola.
R is the intersection of these two lines.
M is the mid-point of Q and R.
MP is the tangent to the parabola at P.

Can you prove that MP is the tangent to the parabola?

October 2007

An implicit equation of a line?

The graph of x³ + 3xy + y³ = 1 is shown:

Can you prove that the graph is a straight line?

Are there any points that satisfy the equation which aren't on the line?

Item submitted by Dr Jeremy D. King, Tonbridge School

See http://www.meiresources.org/tam/tpc3_dq04.pdf for a detailed investigation of this question.

September 2007

Cats

Bill’s cat has 8 kittens: 3 female and 5 male. Alex is going to have two of them. Alex decides that she must have both kittens of the same gender. One kitten is chosen at random then the other will be chosen at random from those of the same gender.

What is the probability that she gets two female kittens?

Two possible answers:

1. There is a 3/8 probability that the first kitten is female. If the first kitten is female then so is the second kitten.
   
   The probability that she gets two female kittens is 3/8.
   
   
2. Suppose the female kittens are A, B, C and the male ones are D, E, F, G, H. The two kittens she gets (in order) could be:
   AB, AC, BA, BC, CA, CB, DE, DF, DG, DH, ED, EF, EG, EH, FD, FE, FG, FH, GD, GE, GF, GH, HD, HE, HF, HG.
   
   The probability that she gets two female kittens is
   6/26 = 3/13 .

Both these answers cannot be correct. Which one is wrong and why?

August 2007

Equilateral triangle in a rectangle

The diagram shows an equilateral triangle in a rectangle. The two shapes share a corner and the other corners of the triangle lie on the edges of the rectangle.

Prove that the area of the green triangle is equal to the sum of the areas of the blue and red triangles. What is the most elegant proof of this fact?

July 2007

Motivating proof by asking impossible questions

If you ask students to find an example of something which is impossible their natural reaction is likely to be to ask "Why?". To be fully satisfied that there are no such example most students will want to see a proof.

e.g. (1) "Find a Pythagorean triple where two of the numbers are even and one is odd."

Many mathematical ideas requiring proof can rephrased in such a way.

e.g. (2) "Show that any graph (in discrete mathematics) has an even number of odd nodes" can be rephrased as "Find a graph with an odd number of odd nodes".

e.g. (3) "Show that the difference between the squares of two consecutive numbers is always odd" can be rephrased as "Find two consecutive numbers such that the difference between their squares is even".

It is important not to tell the students that the question is impossible for the question "Why?" to arise naturally.

June 2007

Circles with integer co-ordinates

The circle x² + y² = 5² has 12 points with integer co-ordinates, as does the circle x² + y² = 13².

To find a circle with more than 12 points with integer co-ordinates you could multiply 5² and 13² to obtain x² + y² = 65² or you could multiply 5 and 13 to obtain x² + y² = 65 (as 65 can be written as the sum of two distinct squares in two different ways).

Does this result generalise: can the product of the largest values in two Pythagorean triples always be written as the sum of two distinct squares in two different ways?

May 2007

MEI Branches

MEI has a number of "branches" around the country. These are local groupings of MEI teachers who meet two or three times a year.

Branch meetings are intended to give teachers an opportunity to discuss issues and help them in their delivery of MEI courses. They are a valuable form of professional development and often include sessions from MEI professional officers as well as discussion of recent examination papers.

Branches are a vital communication link within MEI. There is an annual Branch Chairmen's conference for Chairmen to receive information and for MEI to hear what is going on in the Branches and the concerns of their members.

For more information about MEI's branches see: About us > Branches

April 2007

Zeckendorf representations

The first few Fibonacci numbers: F₁=1, F₂=1, F₃=2, F₄=3, F₅=5, F₆=8, F₇=13, ...
1, 1, 2, 3, 5, 8, 13, 21, …

Zeckendorf representations: Let the notation (a_na_n-1...a₄a₃a₂)_F where each a_r = 0 or 1, represent the number a_nF_n+a_n-1F_n-1+...+a₄F₄+a₃F₃+a₂F₂

Such a Fibonacci representation is called a Zeckendorf representation.

For example, 17=13+3+1 and also 17=8+5+3+1. Therefore there are two Zeckendorf representations of 17: 17 = 11101_F = 100101_F

Theorem (Edouard Zeckendorf 1972):
Every positive integer n has a unique Zeckendorf representation with no consecutive 1s.

For more information see: http://en.wikipedia.org/wiki/Zeckendorf%27s_theorem

March 2007

Past papers

Did you know that all of the past papers on this site are provided with the markscheme and the examiner's report as a single file? This is to make it easy for teachers to see the standard expected of their students and identify common mistakes made in examinations.

February 2007

Possibly the best counter-example in the world!

Hypothesis: An irrational number to the power of an irrational number cannot be rational.

Disproof

First of all consider the number x = .

Is x rational or irrational?

This is a very difficult question to answer but we do know it’s one or the other!

So we have two cases to consider:

If x is rational then we have found our counter-example because we know  is irrational and so x = is an example of an irrational to the power of an irrational with a rational answer.

If x is irrational then think about the number . By our assumption (that x is irrational) then this is an irrational to the power of an irrational. But we know the value of :

and this is clearly a rational number.

Therefore one of  and must be an example of “irrational to the power of irrational equals rational”, but we don’t know which one and nor do we need to know, our counter-example is in there!

(In fact it was proved as recently as 1930 that is irrational and so it is the latter case which is the counter-example.)

January 2007

Sums of cubes

See www.meidistance.co.uk/tfm/tfmfp1_pt12.pdf for details of how to make a practical demonstration of this.

December 2006

19 not out

Some positive numbers add up to 19. What is the maximum product?

See: www.meidistance.co.uk/tam/tpc3_dq06.pdf for a "solution" to this problem.

November 2006

Coursework in mathematics

Following the recent decision to discontinue coursework in GCSE Mathematics, MEI have produced a discussion paper to promote general discussion that will inform national policy.

Summary

• Coursework was introduced for sound educational reasons; it is essential to understand these in order to be able to make sensible decisions about the most appropriate means of assessment.
• Coursework in GCSE Mathematics will soon be discontinued; it became increasingly unpopular following the introduction of the data handling coursework.
• Existing mathematics coursework at A-Level is fit for purpose; it should, therefore, be allowed to continue.
• Statistics 1 coursework in the MEI A-level had similar aims to the GCSE data handling coursework but was much more successful in achieving them. The reasons for this are explored in the paper.
• It is important that the skills we want our young people to acquire are measured and encouraged within the scheme of assessment.
• All specifications should be required to assess these skills; there may be flexibility in how this is done, but all approaches should be properly evaluated.

To read the paper in full click the link below:
Coursework in Mathematics

October 2006

Power Darts

On a dartboard where the "doubles" count as squares and the "trebles" count as cubes how many 3 dart finishes are there starting from 501, finishing on a "double".
e.g. 20, "double" 9, "double" 20: 20 + 9² + 20² = 501

Self-describing numbers

Using these values, you will find O+N+E=1 and T+W+O=2. How much further can you go?

Here we again have O+N+E=1 and T+W+O=2. Now how far can you go?

It turns out that (T+W+O)+(E+L+E+V+E+N) =13. Why does this guarantee that T+W+E+L+V+E =12?

Finally, if we insist that each letter takes a different value from all other letters, how can you prove that it will be impossible to find any system in which all the integers from one to thirteen describe themselves?

September 2006

Double angle formulae