Lycee Francais SAMI Maths Club

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!

Welcome back everyone!

Our first puzzle this year was to place 8 queens on a chessboard such that none of the queens could 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 H.B.Meyer’s website – try for 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? Here is a place to try this. 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!

There is a great numberphile video  on this puzzle, and all the answers for different size boards on wikipedia.

On a piece of paper, draw a triangle (any triangle -right angled, equilateral, isosceles, scalene). Colour one of the vertices red, the second blue, and the third green.

Next, take three dice – one red, one blue, and one green.

Now draw a point in the triangle. This point is the seed for the game. Then roll the die. Whichever die shows the highest number (throw again if there is a tie for the highest), draw a point halfway inbetween the seed and the appropriately colored vertex. Now do the same, using your new point as the seed for the next. After a few rolls you might have a drawing like this:

Now continue in this fashion for five rolls of the dice. Then rub out all the points except the most recent seed and the coloured points.

Now carry on but don’t erase any points.

Can you guess what pattern your points are going to make? You might be surprised!

You might want to use technology to help you find out. Here are some instructions for using geogebra.

The aim of this puzzle is to switch the red frogs with the blue frogs according to the following rules:

• The left set of frogs can only move right, the right set of frogs can only move left.
• Frogs can move forward one space, or move two spaces by jumping over another counter.
• The puzzle is solved when the two sets of frogs have switched positions.

How many moves does this puzzle take?

How about if we change the number of frogs? Can you find a rule for the number of moves it will take based on how many frogs there are?

NRich has an applet with nice graphics here, but you are allowed to go backwards in their version, so it is is best to use this one to help you count the moves.

For interested facilitators extra notes and the solutions are here.

Today we solved an intriguing homework riddle with the help of a Venn diagram.  We had clues such as

“Seven students in total have Maths homework but Miya isn’t one of them.”

and

“Rory has more homework than Aidan.”

Using all the clues, which you can find here, we were able to work out which students had which pieces of homework.

It was a nice challenge! The solution can be found here. This activity was originally found on the TES website.

We spent this maths club playing the highly addictive game Square it from Nrich.

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.

… will the 28th of July 2061 be?  (Next predicted sighting of Halley’s comet)

… was the 4th of April 1965? (Robert Downey Jr ‘s birth)

… was the 17th of July 1789? (Prise de la Bastille)

We puzzled over these two questions having to think about where to start, the number of days in each month, and the rules for leap years.

It was fun to try different strategies by hand, but then we used the mod function and the floor function in Zeller’s algorithm to find quickly the day of the week for any date.

Today we explored one of the most famous recursive sequences called The Logistic Map.

Try this activity leading on to exploring the sequence on Geogebra. There are Geogebra instructions here if you get stuck.

We were inspired for this activity by this amazing video from Ben Sparks on Numberphile. It explains how this activity links to one of the Feigenbaum constants, pseudo ramdom numbers and Chaos Theory. Definitely worth watching!

The first question of today is how many sides a piece of paper has. So the answer is 2 of course. Now we want to find a way of folding the paper in such a way that there is only one side. One way to verify this is to take a pen and colour the folded piece of paper. If you do not need to turn the piece of paper around, there is only one side. The method to do this is to twist the piece of paper by half a turn and you will obtain a möbius band.

This is all explained in this presentation from Suffolk maths, that also contains a brilliant exploration table. Your challenge is to fill in the first three lines and to come up with three other results to add to the table below.

Good luck!