Today we played a game where everyone simultaneously guesses a number between 0 and 100. The person who is closest to the average (mean) wins. What number would you choose?
The game is played in multiple rounds. It is interesting to see how strategies change and how the mean changes over time.
One fun variation is that the winner is the person who is closed to 2/3 of the mean – an analysis is here.
We played a game that John H. Conway he co-invented with Michael S. Paterson while they were both at Cambridge.
It is called Sprouts, and the rules are summarised by Nrich here.
You can try and discover some of the maths behind it by working through this article. We managed to prove the maximum and minimum number of moves for a game starting with three points.
We started with this warm up puzzle from Nrich:
The interactive version is here.
We then used this worksheet to explore how a disease could spread through a network. This was adapted from this set of resources from Nrich.
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.
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.
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.
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 …
We had a great session today working together with the Kongali Community Library Maths Camp on modulo arithmetic.
Here are the slides (as a pdf), the 1st and 2nd explorations and a fantastic geogebra app.
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]
