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?

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.

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!**

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?

Sums of three cubes

Both 11 and 12 can be written as the sum of the cubes of three integers:

11 = 3^{3} + (−2)^{3} + (−2)^{3}

12 = 7^{3} + 10^{3} + (−11)^{3}

Which of the numbers from 1-100 can be written as the sum of the cubes of three integers?

MEI Conference 2019 – A couple of taster problems

The 2019 MEI Conference takes place at the University of Bath on 27-29 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 2018 Conference sessions Using graphing technology for teaching calculus and Modelling and hypothesis testing with the Normal Distribution.

Which of the following is true:

- e
^{π}<*π*^{e} - e
^{π}=*π*^{e} - e
^{π}>*π*^{e}

Choose either adult men or adult women. How tall are the tallest and shortest that you are likely to meet? Use this to estimate mean and standard deviation.

Calculator Crunch

*Mathematics in Education and Industry* contains the following number of letters in each word:

11, 2, 9, 3, 8.

Using just one pair of brackets, find the difference between the largest and smallest possible values of the expression:

11 × 2 + 9 ÷ 3 − 8

Calculators are allowed!

This is an example of the type of problem that will appear in *Calculator Crunch*: a new fun, free, summer challenge to help Year 6s get ‘calculator-ready’, and provide extra practice for Year 7s. For more details see
mei.org.uk/Primary-KS2-3-Transition.

April fool: 1=−1

Where is the flaw in the following argument?

Regional differences

The parabolas y=*x*^{²}+2*x*−1 and y=*x*^{²}−*x*+1 split the plane into four regions:

What is the maximum number of regions that two parabolas of the form *y*=*ax*^{²}+*bx*+*c* can split the plane into?

What is the maximum number of regions that three parabolas of this form, or four parabolas of this form, can split the plane into?

Thinking equilaterally

Two equilateral triangles share a common vertex. Show that the lengths marked *a* and *b* are equal for any such arrangement.

Investigate the ratio *a*:*b* for other pairs of regular polygons.

Happy 2019

Some problems about the number 2019:

- The number 2019, its double 4038, and its triple 6057 contain all 10 digits. What is the next number with this property?
- The number 2019 can be written as the sum of the first
*n*perfect powers (integers of the form*a*where^{m}*a*>0 and*m*≥2). What is the value of*n*? - 2019 is the smallest number that can be written as the sum of the squares of 3 primes,
*p*^{2}+*q*^{2}+*r*^{2}, in*n*different ways (where the order doesn't matter). What is the value of*n*?