# Competition

## Maths Challenge 2016/17

Throughout the year EMS will release a new challenge on the EMS website. The challenges are substantial problems and will require careful thought and plenty of time to solve.

Once you have a solution, email us your work with the subject heading: ‘Maths Challenge’.

Each month the winner’s solution will be published on the EMS website.** **You will not only be rewarded for correct solutions, but also accuracy of method and speed of response.

The deadline for solutions to challenges is the 30^{th} of each month.

Email your solutions to events@exeterms.ac.uk

*Good luck!*

### Maths Challenge January 2017

### Maths Challenge December 2016

Happy Christmas everyone! Why don’t you give the grey matter a work-out after one too many mince pies with this month’s challenge? Good luck!

### Maths Challenge Solution November 2016

How did you get on? Here’s the solution to November’s challenge. No rest for the wicked – have a go at December’s!

### Maths Challenge November 2016

Set theory is at the heart of mathematics but it is rarely discussed at schools or sixth forms. Once you know the rules and notation, set theory is actually fairly simple. Some research might be required to understand the symbols, and therefore translate the hieroglyphs into English.

Your task is to define a set S that obeys the rules below.

Hint to get started: look at hint 8 first.

### Maths Challenge Solution October 2016

**Congratulations to Ali Godjali who solved October’s puzzle!**

See below the response:

Reflect the point (3,9) with respect to the y-axis getting the point (-3,9). Similarly reflect the point (3,9) with respect to the line y = x getting the point (9,3). The answer is the distance between (-3,9) and (9,3) which is 6 sqrt(5). The justification is as follows:

Let A and B be the vertex of the triangle confined to the y-axis and the line y = x respectively. The perimeter of the triangle is the length of the path from (-3,9) to A to B to (9,3) since the distance from (3,9) to A is the same as the distance from (-3,9) to A and the distance from (3,9) to B is the same as the distance from (9,3) to B. So the smallest perimeter is the smallest length of path from (-3,9) to (9,3) that passes through the y-axis and the line y = x. The straight line joining those two points clearly satisfies the requirement.

### Maths Challenge October 2016

Please email your answers to events@exetermathematicsschool.ac.uk. Have fun!

## Maths Challenge 2015/16

### Maths Challenge June 2016

### Maths Challenge May 2016

May’s theme is numbers, it is intended to be quite light-hearted and a relaxing break in your revision schedule!

### To Add or Multiply?

Given two 2’s, “plus” can be changed to “times” without changing the result: 2 + 2 = 2 x 2.

The solution with 3 numbers is easy too: 1 + 2 + 3 = 1 x 2 x 3

### Challenge

Find the answer for 4 numbers and the three possible answers for 5 numbers.

### Bonus Challenge

Can you find multiple solutions for 6, 7, and more numbers.

### May Challenge 2

Continuing our monthly theme of numbers, here’s our next challenge for you. Sorry it’s a little late!

### Numbers subtracted from their reverse

- Find a three digit number that upon subtraction with its reverse, gives the same three digits in different order.
- There are multiple solutions for four digit numbers, can you find them all?

### Extension for programmers

- Can you write a program to find all solutions for different length numbers?
- Can you find solutions that work in different bases?

### Maths Challenge April 2016

**Squircle Perimeter Puzzle**

A circle is constructed so the circumference intersects the midpoint of the top edge and the two bottom vertices of a square. Which perimeter is larger, the square of the circle?

Some justification or the ratio will be needed to prove you aren’t guessing!

### Maths Challenge February 2016

Congratulations to all those who successfully attempted January’s Maths Challenge – a variety of mostly programming methods were employed and surprisingly, we had no entries that were approached mathematically – there is a nice link to recurring fractions – perhaps you could review this again.

For this months Maths Challenge, we have four mini cryptogram puzzles to solve:

### Maths Challenge January 2016

Find the smallest positive integer such that when its last digit is moved to the start of the number (example: 1234 becomes 4123) the resulting number is larger than, and is an integer multiple of, the original number. Numbers are written in standard decimal notation, with no leading zeros. The number you are looking for does not have to be 4 digits long like the example.

Acceptable methods of solving may include algebra or writing a computer program.

Good luck!

### Maths Challenge December 2015

Today’s puzzle may take longer than last months.

Read the instructions carefully…Half the clues are missing but can be solved without them.

Good luck!

### Maths Challenge November 2015

### Maths Challenge October 2015

A car travels downhill at 72 mph, on the level at 63 mph, and uphill at only 56 mph.

The car takes 4 hours to travel from town A to town B.

The return trip takes 4 hours and 40 minutes.

**Find the distance between the two towns.**

### Maths Challenge September 2015

This puzzle was cut out of the Sunday Times some years ago, it is called “A Touching Story” by Victor Bryant.

The touch-pad on my family’s safe is as shown above. I have a code number, which consists of a four-figure number with four different non-zero digits, and to gain entry to the safe I have to touch those four digits in the correct order. The safe is also programmed to keep a note of who opens it, so we each have a different access code. My wife’s is double mine and my son’s is double my wife’s. Each of the codes consists of a four-figure number with four different non-zero digits. Furthermore, any two of our three codes have the property that their first digits are adjacent on the touch pad (ie, across, down or diagonally), their second digits are adjacent, their third digits are adjacent, and their fourth digits are also adjacent.

**What is my code?**

Further challenges

- What would Victors code be if each code consisted of a three-figure number?
- If each code had equal code lengths can you find solutions for different code lengths? Are they unique or are there multiple solutions?
- What is the longest code if it is required that no digit is repeated in a code? How long could each code be if this condition is removed?

## Maths Challenge 2014/15

### Maths Challenge July 2015

and here are six challenges:

1) Divide a square into three similar regions, **ALL THREE **of which are congruent;

2) Divide a square into three similar regions, **EXACTLY TWO **of which are congruent;

3) Divide a square into three similar regions, **NO TWO **of which are congruent;

4) Divide an equilateral triangle into three similar regions, **ALL THREE **of which are congruent;

5) Divide an equilateral triangle into three similar regions, **EXACTLY TWO **of which are congruent;

6) Divide an equilateral triangle into three similar regions, **NO TWO **of which are congruent.

** Which of these have multiple solutions? of these do any have an infinite number of solutions?**

### Maths Challenge June 2015

Divide the below grid into multiple partitions (there will be no gaps when complete), each of which contains one circle (black or grey) which is its centre of rotational symmetry.

One piece has been completed.

Once complete shade in the shapes which contain a grey dot for a bonus picture.

### Maths Challenge April 2015

## April 2015

An n-polygon in the plane is a polygon with n vertices connected by n line segments. So a 3-polygon is a triangle, a 4-polygon is a quadrilateral, and so on.

But suppose we draw polygons on a sphere. The line segments can be replaced by pieces of great circles (a great circle is the biggest circle you can draw on a sphere – on the Earth’s surface the equator is a great circle, and so are longitude lines from the North to South poles).

Consider a spherical triangle with one vertex at the North pole and two on the equator. The angle at the North pole is theta. **What fraction of the surface area of the sphere is inside the triangle?**

**What does a 2-polygon look like? Draw a 2-polygon with vertices at the North and South poles.**

**On the sphere 1-polygons exist.** **What does a 1-polygon look like? What fraction of the surface area of the sphere is inside the 1-polygon?**

### Maths Challenge February 2015

## February 2015

On the cold blustery morning of February 1^{st}, 2005, Mr and Mrs Miller, their two children, Polly and Dave, and Grandma and Grandpa, set off on their annual road trip from Evanston, Illinois to Punxsutawney, Pennsylvania, USA. The following day (Feb 2^{nd}) was Groundhog Day, an annual event that sees the Millers and many other families converge on Punxsutawney to experience the phenomenal weather forecasting powers of Punxsutawney Phil. Phi is a groundhog, and, in an age-old tradition dating back to 1887 (https://www.groundhog.org), if on the morning of Feb 2^{nd} Phil wakes up and sees his shadow, he predicts that there will be six more weeks of winter and returns to his burrow. However, if Phil does not see his shadow, he predicts an early spring.

Part way into their journey, Mr and Mrs Miller were reminiscing about years gone-by. They were trying to remember the day of last year’s event. Mr Miller was sure it was a Friday (he recalls that the following day, a Saturday, was the day that the Pittsburgh Penguins won the National Hockey League). Mrs Miller was sure it was a Saturday (she remembers that it was the year that the children would have missed school on the following Monday had it not been for a very nice car repair man who towed the Millers home on Sunday after their car refused to start — it being a Sunday, they had great difficulty in finding a garage open to repair their car). To settle the issue they decided to ask the children.

“Children, on what day did last year’s Groundhog day fall?”, Mrs Miller asked

“Thursday”, replied Polly.

“No, it was Wednesday”, Dave disagreed.

Mum and Dad frowned. In search of another memorable event from which they could work backwards, Dad asked,

“OK, children. On what day did Valentine’s day fall last year?”

“I remember”, replied Dave, “Phil predicted 6 more weeks of winter and he was correct. It snowed on Valentine’s day and I went sledging with Lucy. It was a Saturday.”

“No, no”, said Polly. “Valentines day was a warm, sunny, school day. I remember walking along the lakeside in the evening with Jamie. It was a Friday.”

Grandma groaned.

Grandpa had a thought:

“I know. The answer is easy if anyone remember on what day the last day of February fell last year.”

“It was a Monday”, replied Dave

“No, it was a Sunday”, claimed Polly

“Goodness me”, sighed knowledgeable Grandma. “Children, you have each given 1 correct answer and 2 wrong ones”.

**Answer the following:**

- On what day was Groundhog day in 2004?
- What do you think was the logical argument behind Grandad’s thought?
- On what day did the conversation take place?

Note: Marks for this quiz will be gained for logical reasoning and correct mathematics (and not for simply stating the correct answer – which can be obtained by looking in an old diary!). You must therefore make sure that you clearly explain how you arrived at your answers.

### Maths Challenge January 2015

## January 2015

A paperweight is made from a glass cube of side 2 units by first shearing off the eight tetrahedral corners, which touch at the midpoints of the edges of the cube.

The remaining inner core of the cube is discarded and replaced by a sphere. The eight comer pieces are now stuck onto the sphere so that they have the same positions relative to each other as they did originally.

**WHAT IS THE DIAMETER OF THE SPHERE?**

### Maths Challenge December 2014

## December 2014

Santa is beginning to think about the best way to organize his reindeer into a team. He has eight reindeer: Dasher, Dancer, Prancer, Vixen, Comet, Cupid, Donner and Blitzen.

He wants to arrange them into four rows of reindeer, with two reindeer in each row. Unfortunately, the reindeer are picky. Donner and Vixen hate each other and cannot be in the same row. Comet and Vixen must be in the same column. Donner and Blitzen are good friends and must *either* be in the same row or they must be in the same column with one directly in front of the other.

**IN HOW MANY WAYS CAN SANTA TEAM UP HIS REINDEER? **

**IN HOW MANY WAYS CAN SANTA TEAM UP HIS REINDEER?**

### If this is too easy, an extra challenge would be to do the same question, but for a team of 4n reindeer (for some whole number n > 1)!

## Monthly Maths Challenge 2013/14

### Maths Challenge April 2014

## April 2014

STEREO consists of two spacecraft – one ahead of Earth in its orbit, the other trailing behind. With this new pair of viewpoints, scientists are able to see the structure and evolution of solar storms as they blast from the Sun and move out through space. From the geometry, astronomers can accurately determine their speeds, distances, shapes and other properties. By studying the separate ‘stereo’ images, astronomers can determine the speed and direction of the cloud before it reaches Earth.

Use the diagram, (angles and distances not drawn to the same scale of the ‘givens’ below) to answer the following question. The two STEREO satellites are located at points A and B, with Earth located at Point E and the sun located at Point S, which is the center of a circle with a radius ES of 1.0 Astronomical unit (150 million kilometers). Suppose that the two satellites spot a Coronal Mass Ejection (CME) cloud at Point C. Satellite A measures its angle from the sun mSAC as 45 degrees while Satellite B measures the corresponding angle to be mSBC=50 degrees. As a ** clue**, the astronomers know the ejection angle of the CME, mESC= 14 degrees, but in fact they didn’t need to know this in order to solve the problem below!

**Problem 1 **– The astronomers want to know the distance that the CME is from Earth, which is the length of the segment EC. The also want to know the approach angle, mSEC. Use either a scaled construction (easy: using compass, protractor and millimeter ruler) or geometric calculation (difficult: using trigonometric identities; challenge yourself: without using the clue) to determine EC from the available data.

**Givens from satellite orbits:**

SB = SA = SE = 150 million km, AE = 136 million km, BE = 122 million km

mASE = 54 degrees, mBSE = 48 degrees

mEAS = 63 degrees, mEBS = 66 degrees mAEB = 129 degrees

Find the measures of all of the angles and segment lengths in the above diagram rounded to the nearest integer.** **

**Problem 2 **– If the CME was traveling at 2 million km/hour, how long did it take to reach the distance indicated by the length of segment SC?

EMS April 2014 Maths Question, courtesy of Dr Claire Foullon, University of Exeter: https://emps.exeter.ac.uk/mathematics/staff/cf337

### Maths Challenge March 2014

## March 2014

In preparation for the spring equinox on March 20, 2014, when day and night are of equal length, the EMS Space Agency decide to reconfigure their ground and space-based instruments.

At 06:00 GMT on March 20, at a distance of a few thousand kilometres above London, EMS put into orbit 4 satellites. The first orbits the Earth in 16 hours, the second in 12 hours, the third in 8 hours and the fourth in 4 hours. At what time will it be when they all meet at above London again?

In addition, EMS decides to reposition their 24 ground-based radio telescopes. These are currently distributed unevenly at 4 different European observatories (locations A, B, C and D, say). However, in order to accurately interpret the signal during the equinox, EMS needs an equal number of instruments at each location. In the new configuration there are half as many as were originally at A, twice as many as originally at B, 5 more than originally at C, and 2 fewer than originally at D. How many telescopes were at each of the observatories before the reconfiguration?

### Maths Challenge February 2014

## February 2014

If S is a 2014 x 2014 square split into unit squares, a diagonal of S will pass through the interior of 2014 unit squares.

If R is a 2014 x 2015 rectangle split into unit squares, a diagonal of R will pass through the interior of how many unit squares?

Can you propose a general formula for the number of the unit squares crossed by the diagonal of a rectangle R, where R is a N x M rectangle split into unit squares and N and M are coprime, that is their greatest common divisor equals 1?

Can you extend this to consider unit squares which are not co-prime?

### Maths Challenge January 2014

## January 2014

Jack Frost has a collection of his favourite 600 snowflakes. One day, he returned home to find that a burglar had broken into his house and stolen some of the snowflakes. Though devastated by the theft, Jack found comfort in the thought that he could recover some of his losses through his insurance company, Fairy Mutual. First he would have to count how many snowflakes had been stolen.

Jack worried that he might make an error if he tried to count the snowflakes one-by-one. Instead, he decided to make many equal-sized piles of snowflakes. At first, he made many piles with 20 snowflakes in each pile. When doing this, he found that there were 8 snowflakes left over. Next, he made many piles containing 31 snowflakes each. This time, there were 5 snowflakes left over.

How many snowflakes had the burglar stolen?

### Maths Challenge December 2013

December 2013

Santa and Mrs Claus are decorating their Christmas tree and this year it really does look quite spectacular. This is partly because they didn’t go overboard as is previous years. In actual fact, they each have some decorations left over.

Santa then declares: ‘If you give me 5 decorations, I will have n times as many as you’.

Mrs Claus retorts: ‘If you give me n decorations, I will have 5 times as many as you’.

As you may expect from this remarkable couple, both Santa and Mrs Claus are telling the truth.

Your challenge is to discover what the possible values of n are. Please bear in mind that, despite their many magical qualities, it makes no sense for either of them to have either a negative or fractional number of decorations.

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## Year 6 Mathematics Poster Competition

This is an annual competition, run every year in the Summer term. Details of the 2017 competition will be posted here shortly.

### Year 6 Mathematics Poster Competition 2016

**Mathematics can be creative, beautiful, surprising and interesting.** EMS are inviting your students to celebrate this by entering our inaugural Mathematics Poster Competition.

The competition is open to students in year 6 from across Cornwall, Devon, Dorset and Somerset. Students may enter as individuals or as part of a small team (maximum of 4 students per team). Schools may enter more than one team should they wish to do so.

The theme for this year’s competition is: **‘Magic Squares’: **further details with specific requirements can be found below. For more information and an entrance form, please contact: events@exetermathematicsschool.ac.uk

The winner of the Mathematics Poster Competition will receive a mathematical prize as well as a trophy. In addition, their poster will be produced professionally and sent to primary schools across Cornwall, Devon, Dorset and Somerset.

**Please send in completed posters, along with an entrance form (one per poster) by Wednesday, 22 ^{nd} June. **Completed posters should be sent for the attention of: Poster Competition, Exeter Mathematics School, Rougemont House, Castle Street, EXETER EX4 3PU

**.**

**Task:**

- Produce an A3 poster with the title
**‘Magic Squares’**

**Your Poster Should Include:**

- Your answer to the questions given below
- The results of your research
- Some ideas of your own

**Your Poster Will Be Given Marks For: **

- Mathematical content
- Creativity
- Overall presentation

_______________________________________________________________________________________

## KINGS-EXETER Annual Mathematics Competition

The Kings-Exeter competition is not being run in the 16/17 academic year. See below for info from previous years.

### Kings-Exeter Mathematics Competition 2015

Following the success of previous years, Exeter Mathematics School (EMS) will be organising a mathematics video competition in collaboration with King’s College London Mathematics School (KCLMS).

The competition is open to years 10 and 11 students with teams of 3 – 4 students and schools may enter more than one team should you wish. Winning teams from entries submitted to both EMS and KCLMS will be invited to an award ceremony in London, where each student will be presented with a graphing calculator, and will be featured on our website and social media pages.

Further information has been sent to Heads of Mathematics. Teachers must register their team’s interest by the Thursday, 1st October.

### Kings-Exeter Mathematics Competition 2014

Following the success of last year’s competition, the Kings-Exeter Mathematics Competition returns. This competition is open to teams of 3 or 4 students from year 10 and year 11. This year our regional champions will meet the London winners at the MET Office in Exeter where they will enjoy a day of enrichment activities and will receive their prizes.

Further information has been sent to Heads of Mathematics. Teachers must register their team’s interest by the 30th September.

### Kings-Exeter Mathematics Competition 2013

### Do you enjoy Mathematics?

### Are you creative?

### Want to share your passion with others?

### This could be the competition for you!

The Kings-Exeter Mathematics Competition offers young mathematicians the opportunity to share their love of the subject in an interesting and engaging manner. By taking part, students will develop their ability to communicate and explain mathematics whilst expressing their creativity. The winning team from the South West will meet with their counterparts from London at an awards ceremony, where they will present their work to the Minister for Education.

### Eligibility

The competition is open to teams of pupils from Years 10 and/or 11. Teams should be made up of 3 or 4 pupils. Schools may enter more than one team.

### Registration

Schools must register interest by 30^{th} September 2013, and inform us of how many teams the school intends to enter. Please send this expression of intent to enquiries@exetermathematicsschool.ac.uk with Maths Competition in the subject line.

This email may also be used for any queries you have about the competition.

Good luck!

*The Kings-Exeter Team*

## PDF Download## Competition Instructions |

## PDF Download## The Problems |