# Maths item of the month

## Curriculum mapping

A list of Maths Items of the Month categorised by GCSE/A level topics can be seen at: Maths Items of the Month Curriculum mapping.

## Recent Maths Items of the Month

### September 2018

Spiral of Theodorus

The Spiral of Theodorus is constructed using an isosceles right-angled triangle with shorter sides of one unit. Successive right-angled triangles are then constructed with the previous hypotenuse as the base and a height of one unit.

How many triangles can be constructed before one of the triangles overlaps an existing triangle?

Show that the ring created between each pair of triangles has the same area as the circle at the centre (with radius one unit).

### July 2018

Square cubes?

For which values of *k* is it possible to draw a square using four points that lie on the curve with equation *y* = *x*^{3} − *kx*?

### June 2018

MEI Conference 2018 – A couple of taster problems

The 2018 MEI Conference takes place at the University of Keele on 28-30 June. To see details of the conference and the wide variety of sessions on offer visit the conference website: conference.mei.org.uk

The following problems featured in the 2017 Conference sessions Squaring the circle and other shapes and The history of logarithms.

In the diagram below AB is a diameter of the circle with centre O. The point C lies on AB and the points D and E lie on the perpendicular to AB through C, with D being on the circle and CE=CB. Show that the area of the rectangle CAHE is equal to the area of the square CFGD.

In the graphs below the area under the curve *y*=1/*x* between *x*=1 and *x*=*a* is represented by A(*a*). Use the graphs to show geometrically that the A(*ab*)=A(*a*)+A(*b*)

### May 2018

Rich Tasks for Further Maths

The loci for |*z*–(*a*+*b*i)|=*r* are drawn in an Argand diagram for each of the cases where *a*, *b* and *r* take a distinct value of either 1, 2 or 4.

Show that the six points A, B, C, D, E and F where two of the loci touch, but don’t cross, all lie on a straight line. Will this be the case if *a*, *b* and *r* take all possible distinct values from *any* set of three different numbers?

This problem is from one of forty *Rich tasks for Further Maths* which will appear in Integral this term. For more information visit integralmaths.org.

### April 2018

Easter egg

An egg is constructed as follows:

- Draw a circle of radius 1 at a point A.
- Mark diametrically opposite points on the circumference, B and C. Draw the circle with centre B through C and the circle with centre C through B.
- Join the two points of intersection, D and E, of these two circles and find one of the points of intersection, F, of this line with the original circle.
- Draw the largest circle with centre F that sits inside the circles with centres B and C.
- Join the arcs CB, BG, GH and HC.

What is the area of egg?

### March 2018

Star Cores

In a square with side length 1 the vertices are joined to the midpoints of the non-adjacent sides to make a star.

What is the area of the octagon created in the centre?

### February 2018

Quarter Master

Given an equilateral triangle, in how many different ways can you construct a shape that has an area that is a quarter of the original triangle, using just a straight edge and a pair of compasses?

An interactive version of this problem is available at: www.geogebra.org/m/ravcumBt.

### January 2018

Happy 2018

This New Year crosses over two years that can be written as the sum of the squares of two positive integers: 2017=9^{2}+44^{2} and 2018=13^{2}+43^{2}.

Will there ever be three years in a row that can be written as the sum of the squares of two positive integers?

Will there ever be four years in a row that can be written as the sum of the squares of two positive integers?

### December 2017

Secret Santa

A group of five friends want to take part in a Secret Santa gift exchange. They place all five of their names in a hat and each select one at random. The draw is a success if they each select another person's name. If one of them selects their own name the draw is a failure and must be repeated.

What is the probability that the draw is a success?

What happens to this probability as the number of people increases?

### November 2017

Ever Increasing Circles

- The largest circle that touches the origin and is inside the graph of
*y*=*x*^{2}is drawn. - The largest circle that touches the first circle and is inside the graph of
*y*=*x*^{2}is drawn. - The largest circle that touches the second circle and is inside the graph of
*y*=*x*^{2}is drawn.

What are the coordinates of the points A, B and C (the intersections of these circles with positive *y*-axis)?

What would happen if this pattern was continued?

### October 2017

Ritangle competition

**Preliminary question A**

The 7-digit number 3211000 is called self-descriptive since it contains three 0s, two 1s, one 2, one 3, zero 4s, zero 5s and zero 6s.

Find the two smallest self-descriptive numbers and add them together.

**Preliminary question B**

You are given nine rods of lengths 6, 7, 8, 10, 15, 17, 24, 25 and 26. You pick three at random. *p* is the probability that you can form a triangle with your rods. The choice (6,7,26) is a fail and so is (7, 10, 17). In addition, *q* is the probability that your three rods make a right-angle triangle.

What is *q*/*p*? Multiply your answer by 1000 and round to the nearest integer.

**Preliminary question C**

Two competing shops have a suit for sale, and both are asking the same price. Both shops have a sale; the first shop drops the price of the suit by £18, the second drops it by 18%. The following week, the first shop drops the prices of the suit by a further 21%, while the second shops takes off a further £21. After this second round of reductions, the two shops are again offering the the suit at the same price.

What was the original price of the suit in pounds?

**Preliminary question D**

A triangle ABC has a perimeter of *P*cm and an area of *Q*cm^{2}, where *P* = 2*Q*. Triangle DEF is similar to ABD. The sum of the perimeters of the two triangles in cm is equal numerically to the sum of their areas in cm^{2}. DEF has an area *k*
times larger than ABC.

What is *k*? Multiply your answer by 100 and round to the nearest integer.

**Please don’t share answers outside your team, others are having
fun finding them! Main competition starts on 9th November: integralmaths.org/ritangle**

### September 2017

Triangle in Adjacent Squares

Squares PQRS and QTUV, where QTUV is larger than PQRS, are drawn so that PQT is a straight line and QR lies along QV. The lengths of the sides of the squares are *a* and *b*. Find the area of the triangle PUR in terms of *a* and *b*.

In how many ways can you show this result?

### August 2017

Triangle on a hyperbola

The diagrams below show two triangles, ABC, with vertices on the curve

Show that, for any such triangle, the orthocentre, P, i.e. the point where the altitudes of the triangle meet, also lies on the curve