Take an A4 piece of paper.

Fold it a few times.

Then punch a hole through the paper.

Before you unfold your paper, can you draw on a fresh piece of paper where you think the holes will be?

Then can you fold a piece of paper and punch it to make the holes look like this:

Then can you fold and cut a piece of paper to make it wider then it was originally? Can you cut in such a way that you can step through it?

Similar ideas can be found here. Solution to stepping through it here.

On dotted paper we explored circles of different radius, counting the dots on or inside the circle. Brilliantly the number of dots is very close to the area of the circle.

We wrote this program in Python to check the ratio of dots to area for different size circles.

Challenge 1
Imagine you have an endless supply of 5p and 7p coins. You could make exactly 20p with four 5p coins. Or you could make exactly 19p with two 7p coins and a 5p coin.

What is the biggest amount that you cannot make? Can you explain why you can make all amounts after this one?

Challenge 2
The biggest number you cannot make given coins of value x and y is called the Frobenius number.

Can you think of a pair of numbers that wouldn’t have a Frobenius number?

What must be true about the two numbers for them to have a Frobenius number?

Challenge 3
Can you come up with a formula for the Frobenius number, if it exists, for two numbers x and y?

Solutions here.

Mr D. R. Kaprekar was an Indian school maths teacher who loved playing with numbers. See if you can follow these steps to find out what he discovered …

Choose a four digit number where the digits are not all the same (that is not 1111, 2222,…).

Rearrange the digits to get the largest and smallest numbers these digits can make.

Finally, subtract the smallest number from the largest to get a new number, and carry on repeating the operation for each new number.

What do you notice?

Does something similar happen for 3 digit numbers? Can you prove it?

Here are pdfs of the problem in English and French.

Try it out here

We worked out the magic/maths behind card tricks 1 and 4 from this set of tricks.

In the 501 game of darts players take turns at throwing 3 darts to reduce their score to zero.

A “checkout” refers to the process of finishing a game by reducing a player’s score to exactly zero, by hitting a double or the bullseye (50 points) with the final dart.

For example, if a player has 40 remaining, they can hit the double 20 (D20) to win.

Here is an example of a 3 dart checkout:

120 : T20 20 D20    (treble 20, single 20, double 20)

1. The maximum checkout is 170. How can you make this?
2. For which numbers between 140 to 170 can you find a three dart checkout?

140		150		160
141		151		161
142		152		162
143		153		163
144		154		164
145		155		165
146		156		166
147		157		167
148		158		168
149		159		169

Most darts players like to aim to finish on D20, D18, D16 or maybe Bullseye.

1. A player has 94 left with three darts. They aim for T18
2. What is their checkout if they hit T18?
3. What is their next dart if they hit a single 18 instead?

2. Imagine you have three darts. Would you rather a score of 32 or 30 left? Think about what happens if you just miss D16 and get single 16 left versus just missing D15 and getting single 15.

3. What checkouts can you find for three darts on 107? What would be the best option do you think?

Extra reading

Logic behind checkouts:

Checkouts Explained

Checkout game:

Darts Math Quiz application

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.