10 days of Christmas

We are used to writing numbers in base 10 (we have digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 and use powers of 10 for the columns when writing out numbers) but is this the most convenient number base? It corresponds well to our number of fingers but it is less convenient when dividing because 10 only has 2 factors (other than 1 and itself): 2 and 5.

An alternative is to use base 12. This uses the *digits* 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, X, E and "twelve" is written as 10 but often said as "do" (short for dozen) to avoid confusion. 12 has 4 factors (other than 1 and itself): 2, 3, 4 and 6 and so using base 12 can make much arithmetic easier.

Base 12, or dozenal, is not as bizarre as it may first sound - we already use base 12 when counting hours and for feet and inches.

... and finally some dozenal maths:

If you take any prime number > 3 and square it does the result always end in a 1?

e.g.

5^{2} = 21

E^{2} = X1

15^{2} = 201

2X1^{2} = 80981

For more information about base 12 see the Dozenal Society of Great Britain and Alex Bellos's blog.

- On the same page, use Autograph to draw the graphs of and .

Can you explain what is going on?

- Now use Autograph to draw the graphs of and .

Can you explain what is going on now?

Maths Jam

Place the numbers from 1 to 8 in the grid so that numerically adjacent numbers are not in adjacent cells. In this case, we say two cells are adjacent horizontally, vertically and diagonally.

- If you can't do it, why not?
- If you can do it, in how many ways can it be done?

Tom's Buttons

Survival

Two people have been accused of being 'in league with the devil'.

To test this they know they are going to be locked in separate sealed rooms where they cannot communicate with, or hear, each other.

Every minute for an hour they will each flip a coin and then make a prediction of the other person's coin flip.

If on any one of the 60 predictions they are both right then this is sufficient evidence that they are 'in league with the devil' and they will both be killed.

There is a strategy they can use which will guarantee their survival.

What is it?

Dissections

A square can be cut into six pieces that can then be arranged to form two different Greek crosses as shown below.

Can you cut the square into only five pieces that can then be arranged to form two different Greek crosses

Solution

Different Distances

Arrange *n* counters in the cells of an *n*x*n* grid in such a way that distances between pairs of counters are all different. (Distances are measured in a straight line connecting the centres of the occupied cells.)

Solutions for *n*=3 and *n*=4 are shown below.

*n*=5,6,7 are also possible. Can your students find any of these arrangements?

Solution

Geogebra Demonstration

Broken Reciprocal Key

The 1/x key on my calculator is broken. How can I use the trigonometry buttons to calculate reciprocals?

Solution

MEI Mathematics Conference 2010

Please see our MEI Conference 2010 page.

Difference of two square roots

(√2−1)^{2} = √9−√8

(√2−1)^{3} = √50−√49

Is every positive integer power of (√2−1) the difference between the square roots of consecutive integers?

Almost regular

Here is a technical drawing method for constructing a 'regular' polygon with *n* sides.

Draw an equilateral triangle ABC and a circle with diameter AB.
Use the standard construction to find point D on AB such that BD: BA is 2:*n*. (The case *n*=5 is shown below)

Join C to D and extend this line to cut the circumference of the circle at E.
BE is one side of the *n*-sided 'regular' polygon inscribed in the circle. The other sides can be swept out using compasses set at radius BE.

- Show that the method is not exact.
- Explain why it is a good approximation for small values of
*n*. - Investigate the method for large values of
*n*.

The medieval Italian painter Giotto is said to have sent the Pope a perfect circle that he had drawn freehand in evidence of his ability to do some decorative work. This became known as Giotto's 'O'.

How would you judge a freehand circle drawing competition?

Discuss! You can assume that, in judging the drawing competition, you have access to any measuring instruments. You can also assume that you are allowed to set any rules of the drawing competition.

This task was set for the poster round of 2008 national final of the Senior Team Mathematics Challenge organised by the Further Mathematics Network and the United Kingdom Mathematics Trust. Teams of four sixth form students were given one hour to produce a response to this question in the form of a poster. The content of the winning poster was used to produce this poster which was distributed to schools in England.

You can download the winning poster.