Month: October 2021

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]