# 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

### July 2020

Function of a function

The function f(*x*)=(*x*+1)/(*x*+2).

Find f^{2}(*x*), f^{3}(*x*), f^{4}(*x*)..., where f^{2}(*x*)=f(f(*x*)).

### June 2020

Area of an arbelos

An arbelos is the shape created inside a semi-circle with diameter AB when two further semi-circles are drawn to a point C on AB so that their diameters are AC and BC. The arbelos is the area inside the larger semi-circle but outside the two smaller semi-circles: shown as the blue region below.

Show that the area of an arbelos is the same as the area of a circle with diameter CD where D is the point on the larger semi-circle directly above C: shown as the red circle in the picture above.

### May 2020

Walk on by

Two old women started at sunrise and each walked at a constant velocity. One went from A to B and the other from B to A. They met at noon and, continuing with no stop, arrived respectively at B at 4 p.m. and at A at 9 p.m. At what time was the sunrise on that day?

### April 2020

9 dots, 10 circles

In the 3×3 square grid of dots shown the dots are spaced one unit apart (vertically and horizontal).

Construct circles with radius √1, √2, √3, √4, √5, √6, √7, √8, √9 and √10 with a pair of compasses using just these points and any intersection points of circles created.

You can try an interactive version of this problem at: geogebra.org/m/wfegkq6f

This is adapted from a problem by Ed Southall: twitter.com/edsouthall

### March 2020

Nowt taken out

What is the sum of all the three-digit numbers that don't include a zero? For example 242 is included in the sum but 202 and 22 are not.

What is the sum for four-digit numbers? What is the sum for *n*-digit numbers?

### February 2020

Dotty squares

‘Dotty squares’ are squares whose vertices are on the dots of a grid spaced 1 unit apart. The two dotty squares shown have areas 4 and 10.

Show that it is not possible to draw a dotty square whose area is 3, 7, 11, ... or any number of the form 4*n*+3 (where *n* is an integer).

### January 2020

Autobiographical numbers

An autobiographical number is a number whose digits describe itself starting with the first digit giving the number of zeros, the next digit giving the number of ones, and so on. 2020 is an autobiographical number as it has 2 zeros, 0 ones, 2 twos and 0 threes.

What other numbers are autobiographical?

### December 2019

Santa's lost spreadsheet

Santa is delivering to children on Christmas Eve; if they’ve been nice over the year they will receive presents, if they’ve been naughty they’ll receive a lump of coal. There are 49 children living in Tinsel Town but Santa’s spreadsheet has lost the data for them. Santa decides to toss a fair coin for each child and deliver presents if it lands heads and a piece of coal if it lands tails.

Of the 49 children 25 receive presents and 24 receive a piece of coal. What is the probability that Santa’s last delivery in Tinsel Town is a piece of coal?

### November 2019

Maths Week England - MEI Desmos Maths Art Competition

As part of Maths Week England 2019 MEI is running a Desmos Maths Art competition. Students can submit entries of their best Desmos Art by Friday 22nd November and the best two in each of three age categories will win a Desmos T-shirt for themselves and their teacher as well as a pizza party for their classmates.

For more details see: mei.org.uk/competitions.

### October 2019

Ritangle competition

Ritangle is a competition for teams of students of A level Mathematics, the International Baccalaureate and Scottish Highers: integralmaths.org/ritangle. The first five questions will be released on the following Mondays: 7th, 14th, 21st October; 4th and 11th November. The other 20 questions will then be released daily (on weekdays) from Tuesday November 12th, with question 25 released on Monday 9th December. Correct answers to these questions are needed to solve the final question, released on Tuesday 10th December.

Registration opens on Monday 7th October.

**Please don’t share answers outside your team, others are having
fun finding them!**

### September 2019

Midpoints of the intersection of a line and a parabola

M is the midpoint of the points of intersection of *y*=*x*^{2} and *y*=2*x*+*c* (as *c* varies). Why does M move as it does?