We had fun in maths club trying to estimate the number of popcorn kernels in this jar. We thought about optimal stacking, averaging guesses, lines of best fit and finally used a smaller sized container to refine our estimate.
You can see our thought processes here.
Click on the link below to try the fun puzzle above – can you turn all the lights out for any size grid?
Square numbers
Triangular numbers
Square Triangular Numbers
1 appears on both lists so is a square triangular number.
0 could also be described as a square number (0x0) and is technically the first triangular number so 0 is also a square triangular number.
Can you find any more square triangular numbers (numbers that appear on both lists)?
Solutions, notes and investigations are here.
Today we played a game where you have to take card and try and make 15 before your opponent. It relates to a game you have almost certainly played before. You could use playing cards to play and then discover the secret.
Here is a worksheet with all the rules and extensions to a hexagon puzzle, and the solutions to the hexagon are here.
We played a game posted on youtube by Mr Allen and extended it to different bases.
Set up
2-4 players. Using Ace -10 cards only.
Rules:
7 (or 6 if there are four of you) cards are placed face up for each player and two in the middle. The player who gets rid of their cards first wins.
You can put a card in the middle (on top of either pile) when one of your cards is an answer to an expression using the two cards in the middle. eg. if 6 and 2 are in the middle possible cards that can be played are: 8 (6+2) or 4 (6-2) or 3(6 2) or 2(6×2 – 12 is the answer but we use the last digit only).
Each player has to say the expression as they put the card down. eg 6 twos are 12 – placing the 2 or 6 minus 2 is 4. All players play on until no cards can be played. Then the dealer deals each player a card then put one on one of the two middle piles. Players then race to place cards until someone gets rid of all their cards!
Tasks
Play a few games of this and try some variations:
In the original game we used the last digit if the calculation was bigger than 10. This is equivalent to subtracting 10 until you have an answer that is the right size. This is called Mod 10. Mod 8 means that if you do e.g. 7 + 3 = 10 and get an answer that is too big you subtract 8 until it is the right size. 10 – 8 = 2.
More examples: 5 + 6 = 11 = 3
6 × 2 = 12 = 4
What is the best version (10,8 or 7) to play in terms of chances to play a card? You could use this worksheet to help you decide.
Task 1 – Warm up
To see how to draw circles, plot points and pick random numbers have a look at this code and adapt it to draw a face:
Task 2
Find the area of the square below
Find the area of the circle
Find the answer to (4 x area of circle / area of square)
Task 3
Adapt the app below to estimate pi. You are aiming for this:
Note that the formula for a circle is x**2+y**2=r**2
where ** on python is squared.
Suppose there was one of six toy animals inside your favorite box of cereal and you would like to collect them all.
You could be lucky and only buy six boxes, but how many boxes of cereal would you expect to have to buy on average?
With thanks to this website we carried out an experiment with dice to simulate this problem. We rolled a dice until we had thrown all the numbers 1 to 6 and recorded how many rolls it took.
Click here for a pdf with several blank tables to cut out and use.
We then used python to generate 1000 trials and compared our average to the theoretical average we calculated. Can you try and program this in the trinket below?
Or, for very advanced mathematicians, you could try and find the theoretical probability. Worksheet here.
Solution in python is here, and the theoretical probability can be found in this article.
I am 35 years old.
My friend is 53 years old.
35 and 53 are mirror birthdays.
We also looked at this puzzle in binary!
