# 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 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?