### Pebbles

Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

### It Figures

Suppose we allow ourselves to use three numbers less than 10 and multiply them together. How many different products can you find? How do you know you've got them all?

### Bracelets

Investigate the different shaped bracelets you could make from 18 different spherical beads. How do they compare if you use 24 beads?

# Colour Wheels

##### Stage: 2 Challenge Level:

We had a good number of solutions sent in. Some sent in just the first part and that's fine - thank you. Others went through to the end.

Elijah who is home educated sent in the following.

RGBRGB wheel
A multiple of $3$ is blue. (You can quickly see if a number is multiple of $3$ by adding its digits and seeing if that sum is a multiple of $3$.)
$1$ more than a multiple of $3$ is red.
$1$ less than (or $2$ more than) a multiple of $3$ is green.
So:
$18$th is blue
$19$th is red
$31$st is red
$59$th is green
$299$th is green
$3311$th is green
$96312$th is blue

BBYG wheel
A multiple of $4$ is green.
So $24$th is green.
$49$ is $1$ more than $48$, which is a multiple of $4$, so $49$th is blue.
$100$th is green.

BYBR wheel
A multiple of $4$ is red.
So $24$th is red.
$49$ is $1$ more than $48$, which is a multiple of $4$, so $49$th is blue.
$100$th is red.

RRRBBY wheel
A multiple of $6$ is yellow.
So $24$th is yellow.
$49$ is $1$ more than $48$, which is a multiple of $6$, so $49$th is red.
$100$ is one more than $99$.
$99$ is an odd multiple of $3$.
Odd multiples of $3$ are red.
Blue comes after red, so $100$th is blue.

RRBRR wheel
There are blue marks at these places:
$3, 8, 13, 18, 23, 28, 33, 38$ ....
If the number ends with $3$ or $8$ there will be a blue mark.
Everything else is red.
So $24$th, $49$th and $100$th are red.

A wheel with six markings
The $6$th colour marking will be on the multiples of $6$.
$102$ is a multiple of $6$.
$100$ is $2$ less than $102$, so red must be $2$ back from the $6$th marking.
So if you want the $100$th mark to be red, the red must be the $4$th colour marking on the wheel.
Other wheels to make a red $100$th mark
How many colour markings: Position of red colour marking:
$4$ ---------------------------------$4$th
$5$ ---------------------------------$5$th
$7$ ---------------------------------$2$nd
$8$ ---------------------------------$4$th
$9$ ---------------------------------$1$st
$10$ -------------------------------$10$th

Emma from Bradon Forest School in the U.K. Sent in a similar solution but these were her explanations for the RRBRR wheel.

RRBRR:
Because red appears on every $5$th colour, the pattern is $5, 10, 15, 20$.
The last red of every spin is in the $5$ times table.
As for the other colours you find out how many $5$s there are in the number and then add however many you need to get the number you want. If you need to add $1, 2$ or $4$ then the mark will be red but if you need to add $3$ then the mark will be blue.
E.g to find the $20$th colour you find out how many $5$s are in $20$ which is $4$. $4$ times $5$ = $20$. You don't need to add anything so the mark is red.
The $100$th mark: You divide $100$ by $6$ which = $16$r$4$. $16$ means the wheel would have to spin $16$ times to get to $100$. r$4$ means that red should be the $4$th spoke on it so it can be the $100$th mark.

Well done to the many of you for the ideas you shared and the solutions you sent in - keep them coming.