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

Find all the different quadrilaterals you can make by joining four dots on a 9-dot grid.

Different in this case means not congruent – i.e. none of your quadrilaterals should be able to be formed by rotating, translating or reflecting one of your other quadrilaterals.

Here is a couple to get you started:

How many can you find?

Some of these puzzles are taken from a nice book by Alex Bellos called Can you solve my problems? Alex Bellos does a fun puzzle blog on the Guardian every two weeks.

The Four Stacks

Start with eight coins in a row, and create four stacks of two coins in four moves. A move consists of moving one coin to the left or right by hopping over two coins and landing on the third one along. You can hop over single coins or stacks.

Tait’s Teaser

The aim of the puzzle is to start with two different types of coin (or heads and tails of the same coin) in the arrangement above and get to the arrangement below in as few ‘moves’ as possible.

A move consists of moving two adjacent coins at the same time. You can move them anywhere in the same line, but you can’t switch the two coins around as you do so.

The answer is not five!

Frogs and Toads

Place six coins as shown above. The white ones represent toads and the grey ones are frogs. Frogs and toads can only move by hopping over one other frog or toad to an empty space. Toads can move to the right, and frogs to the left. Can you rearrange them into the position below?

See here for an interactive version of this puzzle created by NRich which allows you to change the number of frogs and toads.

Star Puzzle

Draw a star and try to place 9 coins on any of the black vertices. You can place coins by starting from a vertex which is empty, moving in a straight line and counting 1,2,3. Number 1 is the vertex you start on, number 2 may or may not have a coin on it, and number 3 is where you place your coin.

Two possible opening moves are shown below

Here is an attempt that has gone wrong! 7 coins have been placed but there is no way to place any more, because you must start on an empty vertex.

Here is a great little applet coded by Etienne Royer-Gray to play with:

The first challenge today was to cut a square out of a piece of paper using just one straight cut with a pair of scissors.

The picture above gives you a hint how to fold it first if you look closely!

Next challenge is to cut out an equilateral triangle by doing some careful folding and just one straight cut.

Here are instructions for making the letters of your name just using one cut!  Click on “Folding Steps” for step by step instructions.

Paper folding and cutting can be taken very far – you can try to make all the number diamond playing cards in this way!!

For hints on this challenge and more activities see this page from the Wild Maths website.

Happy New Year! Hope you enjoyed your extra second before hand! Try this game online aptly called Approaching Midnight.

Then try Got It and Escape Vortex.

You may able to use similar strategies for all three!

This week we’ve started on a series of six James Bond themed Christmassy maths puzzles.  All taken from this amazing website. Puzzles include Licence to Fill, Safe Cracker and Code Breaker.

On a special farm chickens live in circular fields, separated by fence posts and straight pieces of fencing

Is there a pattern between the number of fence posts and the maximum number of chickens that can be kept?

Are you sure?  How do you know?

Can you explain why?

Hint: The answer when you have 6 posts is not what you might think!

When you have had a play around with this problem you might like to read this article or check out this solution.

Exploring the Koch Snowflake (it’s getting cold after all!)

Draw the biggest equilateral triangle you can on a page

On each line/side, cut it in three parts and remove the middle part

Add two sides of a triangle in the gap.

Repeat like so:

What develops is a fractal. If you carry on this pattern for ever the area is bounded but the perimeter is infinite! You can learn how to make the Koch snowflake on Python here.

This site gives the code for a Seirpinski Triangle. You can try it on Trinket here.

Numberphile have an amazing collection of maths videos, one relating to fractals is here. And here is a great TED talk by Benoit Mandelbrot.

Logo is a language that was written in 1967, but is still interesting to work with nearly 50 years later!

There is an online version to use or Windows users can download a free copy of FMSlogo. A good free version for Mac users is ACSlogo.

THe basic LOGO commands are:

FD forward          BK backward

LT left                    RT right

PU pen up             PD pen down

HT hide turtle      ST show turtle

CS clear screen    CT clear text

The following challenges are adapted from the brilliant monthly puzzle website NRich, try the full set of puzzles after.

Challenge 1

Try drawing these shape on LOGO

Challenge 2

Draw regular polygons just by using the repeat command.

e.g.  REPEAT 5 [FD 30 RT 72]

Challenge 3

Now trying using nested repeats to create shapes like this!

Today we looked at strategies for solving a tricky version of a popular puzzle. Usual Sudoku rules apply – numbers 1 to 9 in every row, column and box – but in this version you don’t get any starting numbers, just the total sum of the numbers in each dotted box.

Hint: The numbers 1-9 add up to 45, see if you can use that fact and the bottom left square to calculate this number (see how one of the dotted boxes overlaps the squares):

12 + 7 + 16 = 35.  This helps a lot!