Lycee Francais SAMI Maths Club

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.

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.