# 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 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

### July 2017

Conference Pairs by Zag

Diametrically opposite pairs of answers (such as 1ac & 10ac) each sum to a different square. In addition you are provided with 5 clues.

The total of the six square numbers when considered by a Roman taking Einstein to heart is relevant. No answer starts with a zero and all answers are unique.

**Across**

**1** Sum of another answer's digits

**4** Product of 2 consecutive primes

**9** Prime reverse(6dn)

**Down**

**5** Reverse(multiple(9ac))

**6** Divisor(4ac)

### June 2017

MEI Conference – Problem Solving

Problem solving features heavily amongst the 100 sessions on offer at the 2017 MEI conference. Here's a problem that will be discussed in the session 'Mathematical comprehension in the OCR(MEI) A level'

In each of the equilateral triangles above, the circles are touching each other and sides of the triangle. In which diagram is a greater proportion of the triangle being covered by circles?

Wherever your interest lies, there'll be problem solving ideas for KS3, GCSE, GCSE resit, Core Maths, A level Maths, Further Maths and using technology. To see details of many more sessions visit the conference website: conference.mei.org.uk

We hope to see you there!

### May 2017

In Memory of Malcolm Swan

It was with great sadness that MEI learnt of the death of Malcolm Swan towards the end of April. Malcolm was a hugely influential and highly respected figure in Maths Education, particularly through his work with the Shell Centre and the University of Nottingham, as well as being universally liked by everyone who came into contact with him. As a tribute to Malcolm we would like to highlight a small sample of some of his resources.

The “Language of Functions and Graphs” and “Problems with Patterns and Numbers” are now widely regarded as classic texts in supporting students developing their understanding of mathematics. They can be downloaded from mathshell.com.

Malcolm also worked on the “Improving Learning in Mathematics” materials, developed for the DfES Standards Unit. The full materials are available from the STEM Learning resources at stem.org.uk/elibrary/collection/2933. MEI has recently reworked the Traffic program that accompanied these into a web-based version using GeoGebra – this can be found at geogebra.org/m/AjWXqFVM.

### April 2017

Multiple Choice

The lowest common multiple of 3 and 4 is 12. The sum of 3 and 4 is 7, which is not a factor of 12.

The lowest common multiple of 14 and 72 is 504. The sum of 14 and 72 is 86, which is not a factor of 504.

Is it possible to choose two positive integers such that their sum is a factor of their lowest common multiple?

### March 2017

Twin Prime Days

1st March 2017 and 3rd March 2017, when written in the ISO basic format are 20170301 and 20170303. 20170301 and 20170303 is a pair of twin primes.

Is the product of twin primes (except 3 and 5) always 1 less than a multiple of 36?

### February 2017

A Surprise Omission

When the product of all the factorials from 1! to 100! is divided by *n*! (where 1≤*n*≤100) the result is a square number. What is the value of *n*?

For what other values of *m* is it possible to divide the product of all the factorials from 1! to *m*! by just one of these factorials so that it is a square number?

### January 2017

Happy 2017

2017 is a prime number, *p*, where (*p*+1)/2 and (*p*+2)/3 are also prime: 2018/2=1009 and 2019/3=673. Unfortunately (*p*+3)/4 is not prime: 2020/4=505.

Find a prime number *p* where (*p*+1)/2, (*p*+2)/3 and (*p*+3)/4 are also prime.

### December 2016

Sprouts

Sprouts is a pencil and paper game with the following rules:

- Mark a number of dots anywhere on a sheet of paper.
- Each player in turn draws a line joining a dot either to itself or to another dot and places a new dot on this line.
- No line may cross either itself or any other line.
- No dot may have more than three lines leaving it.
- The last player able to make a legal move wins the game.

The first two moves from a game of Sprouts with 3 dots are shown below.

Sprouts was devised by by mathematicians John Conway and Michael Paterson. For more details see: en.wikipedia.org/wiki/Sprouts_(game)

### November 2016

Ritangle Competition - Preliminary Question

Given a positive integer *n* we say s(*n*) is the sum of all the factors of *n* not including *n* itself.

Thus s(6) = 1 + 2 + 3 = 6; s(7) = 1; s(8) = 1 + 2 + 4 =7; s(9) = 1 + 3 = 4.

It is easy to find even numbers *n* so the s(*n*) > *n*,

for example s(12) = 1 + 2 + 3 + 4 + 6 = 16.

It's harder to find odd numbers where s(*n*) > *n* but it is possible,

for example, s(1575) = 1649 > 1575.

Find the odd number smaller than 1575 so that s(*n*) > *n*.

This is one of the preliminary questions for the Integral Ritangle competition starting on 9th November. Ritangle is a competition for teams of students of A level Mathematics. For more details about the competition see: integralmaths.org/ritangle/.

### October 2016

Tan lines

Tangents to the graphs of *y*=tan*x* and *y*=cos*x* are drawn at the point of intersection between *x*=0 and *x*=π/2. Parallel lines are drawn to these tangents at the other tangent's point of intersection with the *x*-axis. What is special about the quadrilateral created?

### September 2016

An unexpected fraction?

Start with any parallelogram. Mark the midpoint of each side. Join these midpoints to the vertex two places clockwise around the parallelogram. What fraction of the original parallelogram is the new quadrilateral?

What would happen if you started with any convex quadrilateral?

### August 2016

Square sum of squares

1^{2} + 2^{2} = 5, which is not a square.

1^{2} + 2^{2} + 3^{2} = 14, which is not a square.

What is the smallest positive integer value of *n*, *n*>1, such that 1^{2} + 2^{2} + ... + *n*^{2} is a square number?

Are there any larger possible values of *n*?

### July 2016

Perpendicular parabolas

If the curves *y* = (*x* – *p*)^{2} + *q* and *x* = (*y* – *r*)^{2} + *s* have four points of intersection will these four points always lie on a circle?

### June 2016

MEI Conference taster - Odd and distinct partitions

O(*n*) is the number of ways of writing *n* as the sum of odd positive integers.

e.g. O(6)= 4: {5+1, 3+3, 3+1+1, 1+1+1+1+1+1}

D(*n*) is the number of ways of writing *n*
as the sum of distinct positive integers.

e.g. D(6) = 4: {6, 5+1, 4+2, 3+2+1}

Does O(*n*)=D(*n*) for all natural numbers?

This problem is taken from the 2015 MEI Conference session *Desert Island Mathematics*. To see details of this year's sessions visit the conference website: conference.mei.org.uk

### May 2016

MEI Conference - Sessions about famous mathematicians

The 2015 MEI conference featured a strand of sessions about 12 famous mathematicians. The following problem is from the session about John Conway.

There are only three numbers (>1) that can be written as the sum of fourth powers of their digits:

1634 = 1^{4} + 6^{4} +3^{4} +4^{4}

8208 = 8^{4} + 2^{4} + 0^{4} + 8^{4}

9474 = 9^{4} + 4^{4} + 7^{4} + 4^{4}

Find the smallest number (>1) that can be written as the sum of fifth powers of its digits.

This year the 2016 Conference will feature a strand of sessions about a different set of 12 famous mathematicians. To see details of these, and other sessions, visit the conference website: conference.mei.org.uk

### April 2016

Are you sure?

What’s the area?

Now read the research: Drawing attention to a lack of attention

### March 2016

Cube Slice

A cube is sliced vertically along the line shown in the diagram and the smaller part is thrown away.

The remaining prism is going to be sliced vertically downwards again by a line going through corner D.

Where would the slice have to be to split it into two equal volumes?

This problem is taken from the FMSP GCSE Problem Solving Materials.

### February 2016

50, 60, 70, ... ?

Find the size of the angle α.

### January 2016

Happy 2016

2016 is a triangular number.

The first three triangular numbers are: 1, 3, 6. The first three pentagonal numbers are: 1, 5, 12.

The pentagonal numbers 1, 5 and 12 are all one third of a triangular number.

Are all pentagonal numbers one third of a triangular number?