Secret Santa

A group of five friends want to take part in a Secret Santa gift exchange. They place all five of their names in a hat and each select one at random. The draw is a success if they each select another person's name. If one of them selects their own name the draw is a failure and must be repeated.

What is the probability that the draw is a success?

What happens to this probability as the number of people increases?

Ever Increasing Circles

- The largest circle that touches the origin and is inside the graph of
*y*=*x*^{2}is drawn. - The largest circle that touches the first circle and is inside the graph of
*y*=*x*^{2}is drawn. - The largest circle that touches the second circle and is inside the graph of
*y*=*x*^{2}is drawn.

What are the coordinates of the points A, B and C (the intersections of these circles with positive *y*-axis)?

What would happen if this pattern was continued?

Ritangle competition

**Preliminary question A**

The 7-digit number 3211000 is called self-descriptive since it contains three 0s, two 1s, one 2, one 3, zero 4s, zero 5s and zero 6s.

Find the two smallest self-descriptive numbers and add them together.

**Preliminary question B**

You are given nine rods of lengths 6, 7, 8, 10, 15, 17, 24, 25 and 26. You pick three at random. *p* is the probability that you can form a triangle with your rods. The choice (6,7,26) is a fail and so is (7, 10, 17). In addition, *q* is the probability that your three rods make a right-angle triangle.

What is *q*/*p*? Multiply your answer by 1000 and round to the nearest integer.

**Preliminary question C**

Two competing shops have a suit for sale, and both are asking the same price. Both shops have a sale; the first shop drops the price of the suit by £18, the second drops it by 18%. The following week, the first shop drops the prices of the suit by a further 21%, while the second shops takes off a further £21. After this second round of reductions, the two shops are again offering the the suit at the same price.

What was the original price of the suit in pounds?

**Preliminary question D**

A triangle ABC has a perimeter of *P*cm and an area of *Q*cm^{2}, where *P* = 2*Q*. Triangle DEF is similar to ABD. The sum of the perimeters of the two triangles in cm is equal numerically to the sum of their areas in cm^{2}. DEF has an area *k*
times larger than ABC.

What is *k*? Multiply your answer by 100 and round to the nearest integer.

**Please don’t share answers outside your team, others are having
fun finding them! Main competition starts on 9th November: integralmaths.org/ritangle**

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?

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?

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)

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.www.duigongzhanghao.com

We hope to see you there!

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.

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?

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?

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?

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.