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# Colour Wheels

##### Age 7 to 11 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.