Author: Emily Fleming

Bulgarian Solitaire

Take 7 playing cards.

Put them into any number of piles and arrange in descending order. For example:

Take one card from each pile to form a new pile.

Put the piles in descending order.

Keep repeating this process.

What do you notice?

Try all different combinations of 7 cards and try and represent your findings on paper. What is the longest sequence you can find?

Then try with 6 cards. What is different this time?

For more questions and solutions see our SAMI Maths Club app.

Discussion of Dice Problem

Here is an example of the experiment we did:

It looks like it only takes 4 rolls to have a 50% chance of getting a double … smaller than you might think! The more trials we do, the more accurate our results. You can run many trials in the applet below:

 

You can also use the applet above to try different number of sides on the dice. How many rolls do you think it would take to have a 50% chance to get a double on a 20 sided dice?

If you extend this question to 365 sides … it is the Birthday Problem! How many people do you have to have in a room for there to be over 50% chance of two people sharing a birthday? Try and discover the answer using the applet above, or see our card deck website for the maths behind the answer.

Dice problem

I have a 10-sided dice with sides numbered: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.

I roll the dice many times until a same number has come up twice.

Example: 3, 1, 5, 6, 8, 4, 5

5 has come up twice so I stop. It took 7 rolls for this to happen.

  1. How many times do I need roll the dice until I am certain to roll a same number twice?
  2. What is the smallest number of times I can roll the dice before I could get a same number twice?
  3. How many times do I need to roll the dice until there is a 50% chance of getting a same number twice….? Try to do an experiment to find out….

Experiment: Use your calculator (RanInt#(1,10)) to simulate a dice roll. Keep rolling until you get a same number twice. Repeat this experiment 50 times in total and complete the table below.

Or write a program in Python to do this for you.

(Using a simulation to learn about probabilities is called a Monte Carlo Simulation)

See here for a Python program and discussion of the solution and extensions …

Cross Stitching

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. Instead of stitching, we could make Christmas cards for the Bazar de Noël using Python and a 3D printer … let me know your designs at [email protected]

Latin Squares

Select the 16 Jacks, Queens, Kings and Aces from a pack of cards. Try to put them in a 4 by 4 square so that each rank (J, Q, K, A) and each suit (Clubs, Diamonds, Hearts, Spades) appears only once in each row and column.

This puzzle is an example of a Latin square. Latin squares are used in medical trials to ensure every participant is allocated to each treatment for the same time period to prove which is the best treatment. See these links for some more info – mathsisfun, wikipedia and nrich.

We are really missing having maths club in person, and we missed the maths camps in Africa in the summer … but we are so happy that circumstances have meant the launch of the Virtual Maths Camp. The puzzle above is going to appear on the 6 of spades in our card deck! Please check out the idea behind our maths club app  which can be accessed online here. We are very keen to translate our activities into French so that they can be used in countries like Togo. If you would like to help with translation, please get in touch with Emily Fleming at [email protected].