Today we looked at this “playable post on the nature of society”. It is a great activity to make you think about how you can positively contribute to society becoming less segregated.
We tried to work out the maths behind the applet. Where does 95% come from in the screenshot from above?? In the javascript for the applet you can see that a measure of “sameness” is used which involves the fraction of neighbours that are the same as you. Can you create/recreate the algorithm used to find the 95%?
91 card game
This card game can be played with two or more players. The goal of the game is to collect the most number of points possible. You do this by biding on the diamonds, using your cards.
The game starts with each player having one complete suit, for instance, one player could have the spades. You can use more than one pack if there are more than three players.
The diamonds are mixed up and placed in the middle. These diamonds are turned over one by one and each player throws a card depending on the value of the diamond card. The values for each card are as follows:
King – 13 Queen – 12 Jack – 11 and the rest of the cards follow as such, Ace being worth 1 point.
The player who’s card has the highest value collects the diamond card. In the picture above, the person who played the King of Clubs would win the Jack of Diamonds.
If there is a tie and two or more people play cards of the same value, then another diamond is added to the middle and all the players bid again for these cards. If the final card played by each player is a tie then no one wins the diamonds in the middle.
The player who wins is the player who collects the most points out of the 91 available after all cards have been played.
Can you find a winning strategy to be able to collect the most points every time?
Another great puzzle from MathPickle … can you find the winning starting position? What happens somewhere between 20 and 30?
Polypad can be used to display some ideas on the computer, here is a screenshot of one of our combinations:
Taxicab geometry is a form of geometry, where the distance between two points A and B is not the length of the line segment AB but the sum of the shortest horizontal and vertical distances between the two points. Example:
The first challenge is to try to find what a midpoint would be in taxicab geometry. Here is an example to help:
Since the distance between A to the midpoint is the same as the distance between the point B and the midpoint, the midpoint is at the same distance from A and B. Can you spot any more midpoints, if there are any? Can you pick two different points that do not have a midpoint?
The second task is to find what a perpendicular bisector looks like on taxicab geometry.
The third task is to try to draw a circle in the taxicab geometry.
Next, try to draw an equilateral triangle and a rhombus.
If you can do all that, try to draw other geometrical shapes you know on taxicab geometry and post them in the comments.
For more info have a look here.
This week we explored a brilliant puzzle on MathPickle.
The link above will send you a series of slides explaining the puzzle, and here is a the first page of trampoline puzzles to try.
A cryptarithm is a type of mathematical puzzle in which the digits in numerical calculations are replaced by letters of the alphabet.
Alphametics are cryptarithms in which letters form meaningful words, often in meaningful phrases. There are only a few simple rules for these puzzles:
1. The same letter always stands for the same digit, and the same digit is always represented by the same letter. So if a P stands for a 2, every P in the puzzle is a 2. If a 3 is represented by a K, every 3 in the puzzle will be a K.
2.The digit zero is not allowed to appear as the left-most digit in any of the numbers in the puzzle. For example, if the word FOOD represents an addend in a puzzle, the F may not be a zero.
3.Most alphametics have unique solutions (there is only one possible answer).
One of our favourite websites is brilliant.org, so today we spent time on it solving different puzzles. You have to register an account, but then you have access to great mathematics and other science questions.
Here is one of the puzzles we worked on:
As it was Pi Day on Tuesday (14th of March is 3.14 in American style date!) we thought we would try and estimate the value of Pi by simulating throwing darts randomly at a square dartboard like below and seeing how many land inside the circle.
One way to do this is to generate a random point by typing the following in the input bar.
(RandomUniform[-1, 1], RandomUniform[-1, 1])
This will generate a point with x coordinate inbetween -1 and 1 and y coordinate inbetween -1 and 1.
Then right click on the point and select “Trace On”.
In the View menu there is a command called “Recompute All Objects”. The shortcut for this is to hold down Ctrl and press R.
You should see your random point going to different places in the screen. You can do this say 80 times.
Alternatively, you can create 80 points on a spreadsheet with the Random Uniform command and then “Create List of Points”. See here for a working applet with this method.
Either way you can now estimate the value of Pi!!!
Count the number of points inside the circle
Divide this by 80 to get the proportion of points that are inside the circle.
Area of the circle is π * 1² = π
Area of square is 2*2=4
So this proportion should be π/4.
So if we multiply our proportion by 4 we get an estimate for π.
The more points you use the better the estimate will be.
Good luck, and happy Pi Day!
Think of an positive integer.
If it is equal to 1, stop.
If it is even, divide it by 2.
If it is odd, multiply it by 3 and add 1.
With the new number you get, repeat the instructions above.
If you picked the number 6, your sequence would look like this:
3, 10, 5, 16, 8, 4, 2, 1
Did your sequence end up at 1? Mathematicians have guessed (hypothesised) that all starting choices will eventually end up at 1, but no one has been able to prove this. If you can find just one example that disproves this theory then you could claim a prize of 120 million Japenese Yen! Here is a fun example of a counterexample that took a long time to find.
Try a few starting numbers by hand. Then you could try and use the power of a spreadsheet such as Geogebra, or by writing some code on Python and let the computer do the hard work for you!
Open up Geogebra.
You will need to use the Mod command:
Mod[ <Dividend Number>, <Divisor Number> ]
to check if the number is even.
and the If command:
If[ <Condition>, <Then>, <Else> ]
to choose what to do if it is even and what to do if it is odd.
Put together, here is the formula you should put into cell A2, once you have put a starting number in A1.
=If[Mod[A1, 2] == 0, A1 / 2, A1*3 + 1]
Then just hover your mouse pointer in the bottom right corner and click the left button and hold down and drag down lots of cells. You should see the sequence appear.
If you want to be super clever you could try and combine two If statements so that if the cell was equal to 1 it would stop calculating and just say “STOP”.
You could also try this activity on Python, click on the menu bar on the top left to make it full screen.
Good luck!
We started off by playing the card game Dobble. If you haven’t seen it before, the game consists of a set of cards like the one above with 8 symbols on each card. You compete in a small group of people to be the first to spot a common symbol with your topmost card and a card in the middle.
The question is, what is the maximum number of cards you could have in the Dobble pack so that there is always exactly one identical symbol between any two given cards, and this identical symbol is not the same for all the cards (that would be a boring game!).
This is a hard question to answer straight away, try first to create a “Dobble” set of cards with just 2 symbols per card.
Then try and create a set with 3 symbols per card.
Any patterns you spot should help you to answer the hard question!
Here is a nice blog on the maths behind Dobble.