Lycee Francais SAMI Maths Club

This activity was created using content from this worksheet from Maths Craft NZ and this Mathigon Tesselation Activity.

A tiling consists of covering a surface (a plane) using copies of shapes, called tiles, without overlapping or leaving gaps.

For example if we use a square or an equilateral triangle as a tile we can create a tiling that will cover an infinite plane.

Only one other regular (all sides and angles the same) polygon can tile the plane, which polygon is this?

Can you prove that there are no others?

You can also use multiple shapes to tile the plane. For example:

You can also use irregular shapes to tile the plane, any triangle or quadrilateral works:

Exactly 15 different pentagons will work, for example:

Go to Mathigon and make a tiling using either one regular polygon, one irregular triangle or quadrilateral or multiple shapes. Use the “Polygons and Shapes” or Pentagon Tilings” menus. You can change colour at the bottom.

All the above tilings are periodic tilings which means they have translational symmetry.

Aperiodic Tiling

Go to Mathigon and make a tiling using either one regular polygon, one irregular polygon or multiple shapes.Use the “Polygons and Shapes” or Pentagon Tilings” menus. You can change the colour of the tiles at the bottom and you can quickly duplicate whole sections of your tiling by highlighting and clicking on the copy icon.

All the above tilings are periodic tilings which means they have translational symmetry.

From wikipedia:

Covering a flat surface (“the plane”) with some pattern of geometric shapes (“tiles”), with no overlaps or gaps, is called a tiling. The most familiar tilings, such as covering a floor with squares meeting edge-to-edge, are examples of periodic tilings. If a square tiling is shifted by the width of a tile, parallel to the sides of the tile, the result is the same pattern of tiles as before the shift. A shift (formally, a translation) that preserves the tiling in this way is called a period of the tiling. A tiling is called periodic when it has periods that shift the tiling in two different directions.

The tiles in the square tiling have only one shape, and it is common for other tilings to have only a finite number of shapes. These shapes are called prototiles, and a set of prototiles is said to admit a tiling or tile the plane if there is a tiling of the plane using only these shapes. That is, each tile in the tiling must be congruent to one of these prototiles.

A tiling that has no periods is non-periodic. A set of prototiles is said to be aperiodic if all of its tilings are non-periodic, and in this case its tilings are also called aperiodic tilings.

In 1961 Robert Berger constructed such a set, using more than 20,406 tiles. Later he found a much smaller set of 104, and Donald Knuth was able to reduce the number to 92. 

Then Raphael M. Robinson found this set of 6 tiles in 1971: 

In the mid 70s Roger Penrose found a two tile set that would work. This discovery eventually led to him suing a toilet paper company and to a new way to make non stick frying pans.

And finally in 2022 a retired systems printing engineer found a single tile – called an Einstein tile for the German for one stone.

A Penrose tiling uses only two shapes, with rules on how they are allowed to go together. These shapes are called kites and darts and in the mathigon version they have white lines that need to match up. This means for example that the dart cannot sit on the kite. Three possible ways to start a Penrose tiling are:

These are called the ‘star’, ‘sun’ and ‘ace’ respectively. 

Using the pictures above, can you calculate all the angles in the dart and the kite?

There are 4 other ways to arrange the tiles around a point. Can you find them all?

The solution is here

Oxford Online Maths club have done a nice video on this topic. Roger Penrose is a famous professor at Oxford and has a floor built in his honour with his tiles:

Here is an editable version of this page.

We found this activity on Nrich and it was originally in one of Brian Bolt’s books and developed by MEDIAN in their collection of interesting number resources.

This Daisy is special because you can make every number from 1 to 25.
You are only allowed to add neighbours (numbers touching each other) and you can
only use each number once in a sum.

1 = 1
2 = 2
3 = 1 + 2
4 = 4
5 = 5
6 = 2 + 4
7 = 1 + 2 + 4
8 = 5 + 1 + 2
9 = 4 + 5
10 = 7 + 1 + 2
11 = 5 + 4 + 2
12 = 5 + 4 + 2 + 1
13 = 7 + 6
14 = 7 + 6 + 1
15 = 7 + 6 + 2
16 = 7 + 6 + 2 + 1
17 = 7 + 5 + 4 + 1
18 = 7 + 5 + 4 + 2
19 = 7 + 5 + 4 + 2 + 1
20 = 5 + 7 + 6 + 2
21 = 5 + 7 + 6 + 2 + 1
22 = 4 + 5 + 7 + 6
23 = 5 + 7 + 6 + 1 + 4
24 = 5 + 7 + 6 + 2 + 4
25 = 5 + 7 + 6 + 2 + 4 + 1

Can you do better than this with a different set of numbers?
The challenge is to find six numbers to go in the Daisy from which you can make all the numbers from 1 to a number bigger than 25.

You can check your ideas here:

Could you adapt the program above to search for a higher solution?

To write the program above we needed to find all the combinations from a general daisy like this:

For example you can have a+b or a+f but not a+d. Can you find all the combinations? There is a worksheet to fill in here (with the solution on the second page). An editable version is here.

Today we cracked codes on this site https://www.cipherchallenge.org/ set up by the University of Southampton.

This page is great for decrypting.

Challenge

Can you draw this curve smoothly using only a straight edge and a pencil?

You can do it by continuing this pattern:

You could try it on Python from this starting point. Note that the co-ordinates are not correct for the first segment yet!

You could even try to do it in all four quadrants:

Or at an angle:

There is an ancient art called cross stitching that uses these ideas.

We were inspired by this Numberphile video by Neil Sloane today.

Starting with 2 ones, what is the maximum number you can place on an infinite grid according to the rule:

“You can place a number if all the numbers immediately around it add up to itself”

Here is a way to get up to 4:

But then we are stuck as there is nowhere to put the 5. Can you do better than 4?

You can choose where to place the 2 ones to start with, for example:

This puzzle can be tried with any number of 1s to start with.

Imagine this diagram continuing …

If we split it into sections we have demonstrated that a(n)>=3n-3

Can you find a higher lower bound for a(n) by drawing a different pattern that would continue indefinitely?

Some solutions are here.

On the 10th of October we engaged in an activity in which thousands of students around the world were concurrently engaged.

First, pick a rectangle in the room (e.g. table, computer screen, piece of paper). Take a piece of string and make it the length of the perimeter of the rectangle plus one metre.

Now try and hold the string around the shape so that you have made a path of equal width around the shape. How wide is your path?

We did this practically and then resorted to pen and paper to find exact widths for different sized starting rectangles.

Amazingly we all found the same width …

We then tried the same puzzle but keeping the path the same width all the way round the shape:

Finally we tried with a circle

which led to us being able to understand the answer to this classic puzzle:

Imagine that the Earth is a perfect sphere and imagine we have a large piece of string tied around the equator.

We then add one meter to the length of the string so that now the string is hovering above the equator, still in a circle.

What is the size of the gap between the equator and the piece of string? Enough to fit a piece of paper under? A cat? A car?

For the solution see here.

We prepared for this session online with our maths club colleagues in Kenya and here is their brilliant report of their session on the same day.

1. Here are 10 straight lines and 17 squares. What are the dimensions of the biggest square?

2. Can you draw another shape that makes 17 squares?

3. Here are 9 straight lines and 20 squares. Can you find all of the squares?

For the following questions only use rectanglular shapes (including square shapes):

4. Can you draw another shape that makes 20 squares?

5. How many different ways can you make 5 squares?

6; How many different ways can you make 14 squares?

7. How many different ways can you make 15 squares?

By thinking about what happens when you add a row and what happens when you add a column can you …

8. Find the smallest number of straight lines needed to make 30 squares.

9. Find the smallest number of straight lines needed to make exactly 100 squares.

You are given two lives to guess my number (which is between 1 and 100 inclusive).

If you guess too high you lose a life.

For example …

I’m secretly thinking of the number 42. If you guess 50 you lose a life. If you then guess 45 you lose your final life and you have failed.

After guessing 50 and losing a life you would then have to guess 1, guess 2, guess 3 and so on until you correctly guess 42. So this would work but would take a long time.

What strategy should you use to minimise the number of guesses it takes to guess my number? And what is the worst case scenario for how many guesses it could take?

If you are familiar with the two eggs problem it is essentially the same puzzle!

Thanks to nrich for the following puzzle:

Have you met Happy Numbers? Start with any number you like and form a sequence by writing down the sum of the squares of the digits of each number to get the next number in the sequence. For example, from 31 you go to 10 because 3 squared plus 1 squared makes 10. Here is a ‘happy’ sequence:

31 -> 10 -> 1 -> 1 -> 1 … 

Numbers are called ‘happy’ when their sequences, sooner or later, give repeated 1’s. We say 1 is a fixed point.

25 -> 29 -> 85 -> 89 -> 145 -> 42 -> 20 -> 4 -> 16 -> 37 -> 58 -> 89 ->…

The number 25 is sad because, however long we go on with the sequence, it will never come to 1, it will just keep repeating the terms (89, 145, 42, 20, 4, 16, 37, 58) over and over again. We call this an 8-cycle or loop. Can you find some more happy numbers?

Can you use excel or python to find more?

We were delighted to welcome ex Lycee student Al Sergeevo back to Maths Club. He did a fantastic interactive presentation on graph theory culminating in using Euler’s formula to derive the number of faces on a dodecahedron.

Thanks to Mr Boswell for taking these notes to share.

Al left us with the puzzle – Given that a football is made up of hexagons and pentagons, how many faces does it have?

We played a two player strategy game today, using all the information from +plus magazine.

The game is played as follows. At the start of a game each player decides on their three colour sequence for the whole game. The cards are then turned over one at a time and placed in a line, until one of the chosen triples appears. The winning player takes the upturned cards, having won that “trick”. The game continues with the rest of the unused cards, with players collecting tricks as their triples come up, until all the cards in the pack have been used. The winner of the game is the player that has won the most tricks. An average game will consist of around 7 “tricks”.

e.g. Player 1 picks RRB, Player 2 picks RBB and see who wins.

The question we thought about was:

• Is it just a game of chance, or if you are choosing second, could you improve your chance of winning?

There is a related game called Penny’s game, using Heads and Tails of a coin toss instead of playing cards. In this version, you just play until one person has won a “trick”. We worked out some of the odds given in the first table in this article.

We ended up having to sum a geometric series! Great fun!

This game is thanks to David Bedford at the BCME Conference in Warwick.

One person writes down a polynomial with positive integer coefficients. Call it f(x) They then choose an integer that it bigger than all their coefficients. Call it n. They then calculate f(n) and give just n and f(n) to the second person.

The second person should be able to name that polynomial! How?

Example:

Given only n=8 and f(n) = 6855 how could you work out that the polynomial was

?

It might help to think about an example when n=10 first.

Here is a cheatsheet for when you have a strategy. Can you write some Python code to work out the polynomial for you?

We recreated Ben Sparks geogebra app which demonstrates how seeds would look being sowed at different fractions of a full turn.

Here is some polar graph paper to draw patterns by hand first.

You might like to try with 1/4, 1/5, 3/8, 5/9 and moving out by one circle each time you sow a seed.

On Geogebra you can set up a slider r and then use Sequence((0.5 i; i r*2 π),i,1,150)

Lights Out was an electronic game released by Tiger Electronics in 1995. The game consists of a 5 by 5 grid of lights. When the game starts, a random number or a stored pattern of these lights is switched on. Pressing any of the lights will toggle it and the adjacent lights. The goal of the puzzle is to switch all the lights off, preferably with as few button presses as possible.

Here, or below, you can play a 3 x 3 version on Geogebra (courtesy of Stephen Jull and Yin Su).

One method to solve – Light Chasing

“Light chasing” is a method similar to Gaussian elimination which always solves the puzzle (if a solution exists), although with the possibility of many redundant steps. In this approach, rows are manipulated one at a time starting with the top row. All the lights are disabled in the row by toggling the adjacent lights in the row directly below. The same method is then used on the consecutive rows up to the last one. The last row is solved separately, depending on its active lights.

Here is a worksheet showing all the possibilities (effectively) for the last row. Can you solve them all, and hence solve any 3×3 grid?

Here are links to play the 4×4 game and the 5×5 game.

One of our favourite chess puzzles …

Place 8 queens on a chessboard such that none of the queens can take any of the others. Above is a fail – only 5 queens are on the board and they are no places left to put any more.

Here is a website to try yourself to fit 8 queens on the board, it is possible!

We then looked at changing the size of the board from an 8 by 8 to smaller sizes, e.g. can you fit 5 queens on a 5 by 5 board? And if so, how many “unique” solutions are there? We defined a unique solution as being one that did not look like any others we had found when we rotated our paper or put it up to the light so it appeared flipped! Here are our solutions for 5×5, 6×6 and 8×8.

There is a great numberphile video  on this puzzle, and all the answers and stats you could want here.

And finally this great puzzle from Alex Bellos in the Guardian

Can you create a sudoku?

Think about

• What would be an efficient way to create a puzzle?
• How difficult would you like it to be?
• What is the maximum number of 3×3 squares you could remove and it still be solvable? Does it matter which squares?
• Could you design it so that it could be easily transformed into mutliple versions? How many versions could be created from your one puzzle?
  

Here is a fascinating article on how one website generates their sudokus and a paper proving the minimum number of clues needed.

We played a game posted on youtube by Mr Allen.

Here are his instructions:

“This is a card game I made up where you do basic mental calculations and race your opponent ( 2-4 players). Using Ace -10 cards only. Rules: 7 cards are placed face up for each player and two in the middle. The player who gets rid of their cards first wins. You can put a card in the middle (on top of either pile) when one of your cards is an answer to an expression using the two cards in the middle. eg. if 6 and 2 are in the middle possible cards that can be played are 8 (6+2) or 4 (6-2) or 3 (6 divided by 2) or 2 (6×2) – 12 is the answer but we use the last digit only. Each player has to say the expression as they put the card down. eg 6 twos are 12 – placing the 2 or 6 minus/take/sub 2 is 4. all players play on until no cards can be played. Then the dealer deals each player a card then put one on one of the two middle piles. Players then race to place cards until someone gets rid of all their cards! (4 players start with 6 cards)”

We played a few games of this and made up our own versions:

1. Instead of a race we played in turn – competitively or collaboratively
2. We removed the 9s and 10s and played in mod 8

Suppose there was one of six toy animals inside your favorite box of cereal and you would like to collect them all.

You could be lucky and only buy six boxes, but how many boxes of cereal would you expect to have to buy on average?

With thanks to this website we carried out an experiment with dice to simulate this problem:

Click here for a pdf with several blank tables to cut out and use.

We then used python to generate 1000 trials and compared our average to the theoretical average we calculated.

Here is a series of puzzles with red and blue hats that will challenge your logical skills.

This activity comes from the great website mathsisfun.com.

Have you ever coloured in a pattern and wondered how many colours you need to use?

There is only one rule

Two sections that share a common edge cannot be colored the same.

Having a common corner is OK, just not an edge.

Let’s start with a simple pattern like a group of nine squares:

What is the minimum colours you need to colour the pattern of nine squares?

A Little More Complicated

How about this one?

How many colours do you need this time?

Even More Complicated

Let’s try another:

How many colours do you need this time?

Nine? Eight? Seven? Six? Five? Four?

How about this one?

Maps

Things get more interesting if we want to colour a map.

Here is a map of Africa, showing six countries and how they border on each other:

Try colouring in the map and see what is the fewest number of colours you need.

Extension

Can you draw a map with 3 countries such that every country has exactly two neighbours, and then colour it.

Can you draw a map with 4 countries such that every country has exactly two neighbours, and then colour it.

Can you draw a map with 6 countries such that every country has exactly four neighbours, and then colour it.

Can you draw a map with 12 countries such that every country has exactly five neighbours, and then colour it.

Further Reading

The proof of the four colour theorem was famously tricky, and comes from graph theory, where mathematicians investigate an equivalent problem of colouring vertices of a network so that no edge has endpoints the same colour. The original four-colour proof was attempted by Alfred Kempe in 1879, but unfortunately Percy John Heawood found an error 11 years later. However his work was not useless, as Percy was able to prove the five-colour theorem (that one can colour a map with no two adjacent regions the sample colour using at most 5 colours) based on Kempe’s work. The four colour theorem was finally proved in 1976 by Kenneth Appel, Wolfgang Haken, and John Koch using a computer to check it. This was the first major theorem to be proved using a computer. They checked around 1500 configurations using about 1200 hours of computer time. Some people were sceptical about a proof using a computer but independent verification soon convinced everyone that the four colour theorem had finally been proved.

You can read more here.

We played an intriuging game from Alan Parr via Nrich called Femto.

So many mathematical questions to think about while playing. Is there a winning strategy for each hand you are dealt?

See here for solutions to tasks 1 and 2 and here for a brilliant video about Task 3 from Numberphile.

Today we tried the puzzles from Alex Bellos’ excellent column in the Guardian.

Start with these cells.

The next iteration would be the image below, because all cells survived and three new ones were born.

Can you draw the successive iterations?

You can check your patterns here: https://bitstorm.org/gameoflife/ .

Challenges:

Can you create a starting shape that stays the same?

Can you find a cycle of length 2,3,4 and 5?

You might like to read the article from Plus Maths or even try this question from the 1996 British Informatics Olympiad.

Cribbage

We played a simplified version of Cribbage, and looked at some of the interesting maths behind it.

Rules

Game for up to 8 players. Each person is dealt six cards. Players choose four cards to keep. Then one card is turned up in the centre of the table and counts as part of each player’s hand. Ace is considered the low card, and king high. The scoring is as follows:

Fifteens. Each card is assigned a value. Ace through 10 are the face value of the card, and jack, queen and king have value 10. Each combination that totals fifteen is awarded 2 points.

Pairs. Each pair of cards, ace through king, is awarded 2 points.

Runs. Each run of three or more cards is awarded the number of points equal to the length of the run – a run of three is worth 3 points, a run of four, 4 points, and a run of five, 5 points. In this instance runs are not counted in multiple ways. For example, A ,2, 3, 4, 5 is not counted as one run of five, two runs of four and three runs of three, but only as a single run of five.

The person with the highest score after four rounds is the winner.

Example hand (scores 16 points)

Questions to think about:

What do you think is the minimum and maximum scores possible?

Can you find the maximum hand?

Are there any impossible numbers inbetween?

Are some points totals more common than others?  How could we know for sure?

See this page for some of the answers to these questions.

Here are the full version rules of Cribbage

Taxicab geometry is a form of geometry, where the distance between two points A and B is not the length of the line segment AB but the sum of the shortest horizontal and vertical distances between the two points. Imagine you are in a taxi in New York – you can’t go through buildings! Example:

The first challenge is to try to find what a midpoint would be in taxicab geometry. Here is an example to help:

Since the distance between A to the midpoint is the same as the distance between the point B and the midpoint, the midpoint is at the same distance from A and B. Can you spot any more midpoints, if there are any? Can you pick two different points that do not have a midpoint?

The second task is to find what a perpendicular bisector looks like on taxicab geometry.

The third task is to try to draw a circle in the taxicab geometry.

Next, try to draw an equilateral triangle and a rhombus.

For more info have a look here.

Bletchley Park inspired activity …

Task 1

Go to
https://billtuttememorial.org.uk/codebreaking/teleprinter-code/

Read about teleprinter code and the rules of addition (up to A+C =F and F+C=A)

Task 2

In the alphabet in the link above, I and N are incorrect.

Can you use their addition table to work out what I and N should be? Remember that same symbols added make a dot, and different symbols make a cross.

Here is an addition table in alphabetical order.

Task 3

Code HELLO with the key ANQPC.
How would you get back to HELLO?

Solution here

Task 4

Read about the Tiltman break here

Task 5

Try it out for yourself!!

Above are two messages sent with the same key. One has been abbreviated after the operator was asked to send it again. Your crib is that it starts MESSAGE NUMBER (of course with a 9 in the middle!). You also know it is a weather report.  

If you add together the two messages letter by letter (using the table) you will end up with the two messages added together, because for:

Message1 +key + Message2 + key

the keys will cancel out and it will be Message1 + Message2

So if you can guess it starts MESSAGE9NUMBER then you can add this to the sum of the two messages and as they start to be different you can work out each one …

We started off by playing the card game Dobble. If you haven’t seen it before, the game consists of a set of cards like the one above with 8 symbols on each card. You compete in a small group of people to be the first to spot a common symbol with your topmost card and a card in the middle.

The question is, what is the maximum number of cards you could have in the Dobble pack so that there is always exactly one identical symbol between any two given cards, and this identical symbol is not the same for all the cards (that would be a boring game!).

We used these questions to guide us to the answer …

1   Is Dobble a game of chance (stochastic game) or a speed game?

2   Take two random cards from the deck. How many symbols do they have in common? Is this number the same for any two cards?

3   Take two random symbols. Can you find a card which has both of these symbols ? Could there be another card with these two symbols? 

4   Now try and create a mini dobble game following these rules. (It will be easier to use letters or numbers rather than fancy symbols!):

• each card contains 3 symbols
• each pair of cards share exactly one symbol
• each pair of numbers appear together on some card

How many symbols did you need ? 

How many cards have a particular symbol on them ?

5   Now try to calculate the number of cards in the game of Dobble given that there are 8 symbols per card.

Here is a great article on the maths behind Dobble.

So many patterns to explore within this special triangle … today we focused on odd and even numbers and the shape this reveals.

See this interactive website from Mathigon or try out yourself on excel using Conditional Formatting or try out this seemingly unrelated activity on Geogebra.

We solved puzzles today inspired by the work of Fields Medalists chosen by Alex Bellos.