# 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

### 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?