Today we worked on a really fun set of festive challenges from GCHQ. You can find all puzzles and challenges on their website.

Today we worked on problems such as …

I select 11 positive whole numbers at random. Prove that two of them must have the same last digit.

All the challenges (they get harder!) here.

Why do we use base 10? How does base 16 work?

Read this brilliant article: https://betterexplained.com/articles/numbers-and-bases/.

Here is a pdf with explanations adapted from Better Explained and all the activities below. Solutions to all the activities can be found here.

Decimal	15	20	32	47	50	170	171	141
Hexadecimal	F	14	20	2F	32	AA	AB	8D

Activity 1

Colour in all the hexagons that are multiples of 7 when converted to decimal numbers.

Activity 2

Fill in the table below. You will need to work out what the 3^rd column represents in hexadecimal numbers

Decimal				17	88	740
Hexadecimal	11B	FFF	1000

Activity 3

Find the sum of 3A5 + 2D1 by converting to decimal, doing the addition and converting back to hexadecimal.

Can you do it without converting to decimal numbers?

Activity 4

What’s great about binary?

It’s the simplest number system as it only uses two digits – 1 and 0. This means it is very easy to build in hardware. You just need things that can turn on or off (representing 1 and 0), so it is fundamental to all computers. For example, in a transistor, ‘0’ means no electricity is flowing, whilst ‘1’ means there is a flow of electricity.

How are binary and hexadecimals linked?

Because one block of four binary numbers represents one hexadecimal number it is easy for programmers to visualise numbers in hexadecimal. Programmers sometimes use words written in hexadecimal in their programs.

Activity 5

What word is this?

1101 1110 1010 1101 1011 1110 1110 1111

Activity 6

From https://nrich.maths.org/problems/base-puzzle

Find the missing number

10000, ? , 100, 31, 24, 22, 20, 17, 16, 15, 14, 13, 12, 11, 10

Today we played a brilliant game called The Mind where in teams of 2-4 you try to psychically play your cards in ascending order.

We then worked on this code to play collaboratively with a computer working on a perfect timing strategy.

How many different paths are there starting at 1 and finishing at 8 (throught adjacent hexagons only)?

You can only move to a higher number as you go.

We found a number of paths and then looked at a smaller number of hexagons (1,2,3, … etc) until we saw an interesting pattern! Notes/solution here.

A fun related problem can be found here.

We looked at a lovely way to encode secret messages from a fantastic book published by GCHQ.

Here are some instructions to code using this method on excel

Challenge

A country was used as the key instead of DOLPHIN.  Given the following encodings, can you guess which country?

CHINA- 0FJBD

SCOTLAND – T0WSYDBA

KENYA – Z9BLD

BRUNEI – NGQB9J

24 25 26 27 28 29 30 31 32 33 34 35

Take the sequence above. Cross off all the numbers in the middle that share a factor with either 24 or 35. You should be stuck with 29 and 31 not crossed out.

Can you choose a sequence where you can cross out all the numbers in the middle? It can start anywhere and be any length.

For our final maths club session of the year we played a fun version of Yahtzee from this great website called Games4Gains.

We looked at Math Pickle’s brilliant activity. Finishing up with working out this probability:

This activity was inspired by this numberphile video by Neil Sloane.

Here are the only distinct three ways you can draw two circles in a Venn diagram.

If we take the universal set to be the positive integers, we can come up with some sets that would fit for A and B for the three examples.

1. A = Odd numbers, B = Multiples of 4
2. A = Even numbers C= Prime numbers
3. A= Even numbers B=Multiples of 4

Note that these choices mean that there is at least one number in each region.

Now for the challenge …

Draw all the different Venn diagrams you can make with three circles, and find rules for them all.

Number of ways can be found here. A picture of them all is here.

Watch the numberphile video to find out how many Jonathan Wild found for 4 and 5 circles. It is a lot more! Nobody knows for 6 …