Lycee Francais SAMI Maths Club

Click on the link below to try the fun puzzle above – can you turn all the lights out for any size grid?

https://www.geogebra.org/m/xkcxt8bt

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:

1. Instead of a race you could play in turn – competitively or collaboratively
2. Remove the 9s and 10s and play in mod 8
3. Also remove the 8s and play in mod 7

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.

1. Will we have another mirror birthday?
2. Have we already had a mirror birthday?
3. Can any 2 people have mirror birthdays at some point in their lives?
4. What other ages could I mirror with?
5. Who could you have a mirror birthday with?

We also looked at this puzzle in binary!

Puzzles with dominos, trominos and pentominos can be found here and there is this great applet if you don’t want to cut out all the shapes on paper!

Today we solved a series of puzzles about finding the fastest horse from a group, inspired by this website.

You can find the challenges and answers here.

Inspired by this fantastic numberphile video with Ben Sparks we are explored sequences of aliquot sums. An aliquot sum is the sum of proper factors of numbers.

The number 15 has factors 1, 3, 5, 15 but the proper factors don’t include the number itself so 15 has proper factors 1, 3, 5. Adding these together gives an aliqout sum of 1+3+5 = 8.

Now we find the aliquot sum of 8, which is 1+2+4 = 7. Then find the aliquot sum of 7, which is just 1 as 7 is a prime number. If we get to 1 we stop.

So 15 – 8 – 7 – 1.

Try with different starting numbers, do all numbers go to 1?

Aliquot 1 – Program to find the aliquot sum of a number

Aliquot 2 – Program to find the aliquot sequence of a starting number

Aliquot 1

Use the computer to find the sum of the factors of 220.

What is the sum of the factors of the answer?

Which one of the following pairs of numbers behaves in the same way?

607, 917

624, 1112

1184, 1210

1554, 2094

Aliquot 2

Can you find a sequence that goes up and then down?

Can you find a sequence that goes up and then down multiple times? Try 138 …

The number 276 is special as it is the smallest number that we don’t know whether or not it ends up at 1. This is an unsolved mathematics problem.