Christmas Baubles

Santa is hanging baubles on his Christmas tree that are spheres, cubes and tetrahedra. He notices that for spheres the formula for the surface area:

*A*=4 π*r*^{2}

is the derivative of the formula for the volume:

*V*=4/3 π*r*^{3}

This doesn't work for cubes with side length *a*; however, if Santa defines the "radius" of a cube, *r*, to be half the side length then the formula for the surface area:

*A*=24*r*^{2}

is the derivative of the formula for the volume:

*V*=8*r*^{3}

How would Santa need to define the "radius" of a tetrahedron so the formula for the surface area is the derivative of the formula for the volume?

When dates align

10th November 2012, i.e. 10/11/12, is an 'AP date': a date whose numbers are in arithmetic progression.

Some questions:

When was the first AP date (ddmmyy) in your lifetime?

When did/will your birthday fall on an AP date?

What is the shortest gap between two AP dates this century?

What is the longest gap between two successive AP dates this century?

What is the probability that somebody who is currently under 18 was born on an AP date?

Factor in all the solutions

Factoring *x ^{n}*–1 (over the integers) for the first few positive integer values of

*x*^{1}–1 =* x*–1

*x*^{2}–1 = (*x*–1)(*x*+1)

*x*^{3}–1 = (*x*–1)(*x*^{2}+*x*+1)

*x*^{4}–1 = (*x*–1)(*x*+1)(*x*^{2}+1)

*x*^{5}–1 = (*x*–1)(*x*^{4}+*x*^{3}+*x*^{2}+*x*+1)

Will the factors of *x ^{n}*-1 always have coefficients of –1, 0 or 1 when

Siders

Kites are taken from the following tile:

These can be arranged in the following ways:

or

Would either of these tilings meet-up to form an exact polygon?

This activity is taken from the Carom Maths site. For more activities like this please see the Carom Maths website.

Linear matrices

for all natural numbers *n*.

Can you find other matrices, **M**, such that all the elements of **M**^{n} are linear functions of *n* (where *n* is a natural number)?

Which is the best circle constant: pi or tau?

The circle constant π is defined as the ratio of the circumference to the diameter of a circle. An alternative is to define the circle constant as the ratio of the circumference to the radius.

This number is τ ≈ 6.283185307179586…

If τ is used as the circle constant then angles measured in radians are the same multiple of τ as they are fractions of a circle.

Please use the links below to navigate to pages on external websites containing more information about the following:

An argument in favour of using τ

A counter-argument in favour of sticking with π

A conference session taster

If each vertex of a triangle is connected to a point one third of the way along the opposite side you create a new triangle. What is the ratio of the area of the smaller triangle to the area of the larger one? Can you prove it?

Taken from the the MEI 2012 conference session Mathematical Proof in Core Maths.

Identity problem

How many trigonometric identities can you find in the following diagram:

Taken from the TAM resources. To find out more about TAM courses please see our TAM page.

Think these are true? Don't be fooled.

Find counterexamples to the following conjectures:

- Every odd integer,
*N*>1, is expressible in the form*p*+2*n*^{2}where*p*is prime. - If
*n*is an integer, then 991*n*^{2}+1 is not a square number. - Polya's conjecture: For
*n*>1 , the number of integers up to and including*n*with an odd number of prime factors is never less than the number of integers with an even number of prime factors.

The Circle Line

In the diagram below AB is the diameter of a circle, C is the centre of the circle, CBD is an equilateral triangle and a second circle is constructed with diameter AC. The largest possible circle that can fit within the gap has centre E. Show that E lies on the line that is perpendicular to AB through C.

Does the point E lie on the line that is perpendicular to AB through C if CBD is an isosceles triangle? In this case C is a point on AB, CD=BD and the second circle still has diameter AC.

How odd is Pythagoras?

Pythagorean triples are sets of positive integers (*a*,*b*,*c*) where *a*^{2}+*b*^{2}=*c*^{2}. A Pythagorean triple is primitive if there isn't a common factor that divides *a*, *b* and *c*.

(3,4,5) and (5,12,13) are primitive Pythagorean triples but (6,8,10) is not as 2 divides 6,8 and 10.

- Is the smallest number in a primitive Pythagorean triple always odd?
- Is the largest number in a primitive Pythagorean triple always odd?

Happy New Leap Year

2012 is a leap year, i.e. it has 366 days. 366 can be written as the sum of consecutive square numbers:

366 = 8^{2} + 9^{2} + 10^{2} + 11^{2}

Non-leap years have 365 days. 365 can be written as the sum of consecutive squares in two different ways:

365 = 10^{2} + 11^{2} + 12^{2}

365 = 13^{2} + 14^{2}

Neither 366 nor 365 are square numbers themselves. Are there any square numbers that can be written as the sum of consecutive squares?