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We found this activity on Nrich and it was originally in one of Brian Bolt’s books and developed by MEDIAN in their collection of interesting number resources.

This Daisy is special because you can make every number from 1 to 25.
You are only allowed to add neighbours (numbers touching each other) and you can
only use each number once in a sum.

1 = 1
2 = 2
3 = 1 + 2
4 = 4
5 = 5
6 = 2 + 4
7 = 1 + 2 + 4
8 = 5 + 1 + 2
9 = 4 + 5
10 = 7 + 1 + 2
11 = 5 + 4 + 2
12 = 5 + 4 + 2 + 1
13 = 7 + 6
14 = 7 + 6 + 1
15 = 7 + 6 + 2
16 = 7 + 6 + 2 + 1
17 = 7 + 5 + 4 + 1
18 = 7 + 5 + 4 + 2
19 = 7 + 5 + 4 + 2 + 1
20 = 5 + 7 + 6 + 2
21 = 5 + 7 + 6 + 2 + 1
22 = 4 + 5 + 7 + 6
23 = 5 + 7 + 6 + 1 + 4
24 = 5 + 7 + 6 + 2 + 4
25 = 5 + 7 + 6 + 2 + 4 + 1

Can you do better than this with a different set of numbers?
The challenge is to find six numbers to go in the Daisy from which you can make all the numbers from 1 to a number bigger than 25.

You can check your ideas here:

Could you adapt the program above to search for a higher solution?

To write the program above we needed to find all the combinations from a general daisy like this:

For example you can have a+b or a+f but not a+d. Can you find all the combinations? There is a worksheet to fill in here (with the solution on the second page). An editable version is here.

Today we cracked codes on this site https://www.cipherchallenge.org/ set up by the University of Southampton.

This page is great for decrypting.

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.

We were inspired by this Numberphile video by Neil Sloane today.

Starting with 2 ones, what is the maximum number you can place on an infinite grid according to the rule:

“You can place a number if all the numbers immediately around it add up to itself”

Here is a way to get up to 4:

But then we are stuck as there is nowhere to put the 5. Can you do better than 4?

You can choose where to place the 2 ones to start with, for example:

This puzzle can be tried with any number of 1s to start with.

Imagine this diagram continuing …

If we split it into sections we have demonstrated that a(n)>=3n-3

Can you find a higher lower bound for a(n) by drawing a different pattern that would continue indefinitely?

Some solutions are here.

On the 10th of October we engaged in an activity in which thousands of students around the world were concurrently engaged.

First, pick a rectangle in the room (e.g. table, computer screen, piece of paper). Take a piece of string and make it the length of the perimeter of the rectangle plus one metre.

Now try and hold the string around the shape so that you have made a path of equal width around the shape. How wide is your path?

We did this practically and then resorted to pen and paper to find exact widths for different sized starting rectangles.

Amazingly we all found the same width …

We then tried the same puzzle but keeping the path the same width all the way round the shape:

Finally we tried with a circle

which led to us being able to understand the answer to this classic puzzle:

Imagine that the Earth is a perfect sphere and imagine we have a large piece of string tied around the equator.

We then add one meter to the length of the string so that now the string is hovering above the equator, still in a circle.

What is the size of the gap between the equator and the piece of string? Enough to fit a piece of paper under? A cat? A car?

For the solution see here.

We prepared for this session online with our maths club colleagues in Kenya and here is their brilliant report of their session on the same day.

1. Here are 10 straight lines and 17 squares. What are the dimensions of the biggest square?

2. Can you draw another shape that makes 17 squares?

3. Here are 9 straight lines and 20 squares. Can you find all of the squares?

For the following questions only use rectanglular shapes (including square shapes):

4. Can you draw another shape that makes 20 squares?

5. How many different ways can you make 5 squares?

6; How many different ways can you make 14 squares?

7. How many different ways can you make 15 squares?

By thinking about what happens when you add a row and what happens when you add a column can you …

8. Find the smallest number of straight lines needed to make 30 squares.

9. Find the smallest number of straight lines needed to make exactly 100 squares.

You are given two lives to guess my number (which is between 1 and 100 inclusive).

If you guess too high you lose a life.

For example …

I’m secretly thinking of the number 42. If you guess 50 you lose a life. If you then guess 45 you lose your final life and you have failed.

After guessing 50 and losing a life you would then have to guess 1, guess 2, guess 3 and so on until you correctly guess 42. So this would work but would take a long time.

What strategy should you use to minimise the number of guesses it takes to guess my number? And what is the worst case scenario for how many guesses it could take?

If you are familiar with the two eggs problem it is essentially the same puzzle!

Thanks to nrich for the following puzzle:

Have you met Happy Numbers? Start with any number you like and form a sequence by writing down the sum of the squares of the digits of each number to get the next number in the sequence. For example, from 31 you go to 10 because 3 squared plus 1 squared makes 10. Here is a ‘happy’ sequence:

31 -> 10 -> 1 -> 1 -> 1 … 

Numbers are called ‘happy’ when their sequences, sooner or later, give repeated 1’s. We say 1 is a fixed point.

25 -> 29 -> 85 -> 89 -> 145 -> 42 -> 20 -> 4 -> 16 -> 37 -> 58 -> 89 ->…

The number 25 is sad because, however long we go on with the sequence, it will never come to 1, it will just keep repeating the terms (89, 145, 42, 20, 4, 16, 37, 58) over and over again. We call this an 8-cycle or loop. Can you find some more happy numbers?

Can you use excel or python to find more?

We were delighted to welcome ex Lycee student Al Sergeevo back to Maths Club. He did a fantastic interactive presentation on graph theory culminating in using Euler’s formula to derive the number of faces on a dodecahedron.

Thanks to Mr Boswell for taking these notes to share.

Al left us with the puzzle – Given that a football is made up of hexagons and pentagons, how many faces does it have?

We played a two player strategy game today, using all the information from +plus magazine.

The game is played as follows. At the start of a game each player decides on their three colour sequence for the whole game. The cards are then turned over one at a time and placed in a line, until one of the chosen triples appears. The winning player takes the upturned cards, having won that “trick”. The game continues with the rest of the unused cards, with players collecting tricks as their triples come up, until all the cards in the pack have been used. The winner of the game is the player that has won the most tricks. An average game will consist of around 7 “tricks”.

e.g. Player 1 picks RRB, Player 2 picks RBB and see who wins.

The question we thought about was:

• Is it just a game of chance, or if you are choosing second, could you improve your chance of winning?

There is a related game called Penny’s game, using Heads and Tails of a coin toss instead of playing cards. In this version, you just play until one person has won a “trick”. We worked out some of the odds given in the first table in this article.

We ended up having to sum a geometric series! Great fun!