Lycee Francais SAMI Maths Club

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.

This week we solved a tricky puzzle involving swords and trying to sit in the right position of a circle to survive!

A powerpoint, created at the Ghana Maths Camp in 2015, explaining the puzzle and with solution is here.

Slaying the dragon

We looked at some fun card tricks with maths behind them.

This highly addictive shape game is brilliant fun and has some interesting mathematical ideas to explore. We started with drawing all the one block, two block, three block and four block shapes we could think of. No repeats (reflections, rotations) were allowed – for example these two shapes of four blocks are effectively the same:

There are 9 different shapes, and the game contains each of these possibilities. There are 7 dice which you throw to generate where to put 7 wooden pieces. For example the game could start like this:

The values on the dice are:

Once you have the wooden pieces in place, it is a two player race to fit all the coloured pieces on the grid. Given the dice configuration above, how many possible games are there to play? The makers of the game have ensured all these games are possible in at least one way.

After we played a few times, we tried to make up an impossible configuration of wooden pieces (ignoring the dice). Here is a trivially impossible one (because there is only one blue one block piece):

But can you make up one that looks like it could work but then doesn’t? Ghazi even found a way of generating impossible solutions and proving they were impossible.

Puzzle 1 – The classic

You are on one side of a river, and with you, there is a wolf, a goat and a cabbage. You have one boat, and can only take one living thing at a time. The goat cannot be left alone with the cabbage and the wolf cannot be left alone with the goat. How many journeys must you do in minimum to get all the objects to the other side of the river? In how many different ways can you do it?

See this video for a brilliant extension leading to an explanation of Vertex Cover.

Puzzle 2 – The answer is smaller than you think … 

There are 4 people (A, B, C and D) who want to cross a bridge in night.

1. A takes 1 minute to cross the bridge.
2. B takes 2 minutes to cross the bridge.
3. C takes 7 minutes to cross the bridge.
4. D takes 10 minutes to cross the bridge.

There is only one torch with them and the bridge cannot be crossed without the torch. There cannot be more than two persons on the bridge at any time, and when two people cross the bridge together, they must move at the slower person’s pace.

What is the quickest way for all four to cross the bridge?

Puzzle 3 – Lions and Wildebeests

3 lions and 3 wildebeests need to cross a river using a raft.

The raft needs at least one animal to paddle it across the river, and it can hold at most two animals.

If the lions ever outnumber the wildebeest on either side of the river (including the animals in the boat it’s on that side) they will eat the wildebeest.

The animals can’t just swim across.

How many crossings does it take to get the animals across the river?

For the solution see this nice video.