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# Charlie's Delightful Machine

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Age 11 to 16

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- Problem
- Getting Started
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Vinay from Ripley Valley State Secondary College in Australia, Leala from Notredame in the UK, Tyler from Burnside Primary School in Australia and R from the UK worked out the rules for specific versions of the game. Every time you play, the rules are different, so people's answers are different. Here are some examples of the sets of rules they found.

Tyler:

Blue light turned on by repeating pattern of adding 2 starting from 1 so the pattern is 1,3,5,7,9,11 and so on.

Yellow is every 3 starting from 2, so 2, 5 8, 11, 14, 17,20 etc.

Green is every 9 starting from 2, so 2, 11, 20, 29, 38, 47 etc

Red is every 8 starting from 5, so 5, 13, 21, 29, 37, 45 etc.

Leala:

Yellow- multiples of 3

Red- going up in 8 them minus 2 (8$n$-2)

Blue- going up in 8 then minus 5 (8$n$-5)

Green- going up in 6 and then minus 2 (6$n$-2)

Shaunak from Ganit Manthan, Vicharvatika in India explained how to find the rules. You can read Shaunak's method below, or watch Shaunak's video by clicking here.

Here is a strategy to work out the rules controlling the lights:

1: Find the rules for the Yellow light - Start by keeping zero in the box. Keep on increasing the number in the box by one. Note down the first 5 numbers that satisfy the rule for the Yellow light.

Next, find the difference between any two consecutive numbers in this list. Call this difference $n.$

Then, find the remainder obtained by dividing any number from this list [by $n$]. Call this number $b.$

The rule is - Numbers $b$ more than [a multiple of] $n.$

So, any number that can be expressed as $a\times n + b,$ where $n$ and $b$ are those computed above, and a is any number, satisfies the condition for the Yellow light.

In a similar manner, one can find the rules for the Red, Blue and Green lights. These three also follow the $a\times n + b$ property.

Example:

I started with 0. The first five numbers that satisfy the rule for the Yellow light are - 2, 5, 8, 11, and 14.

Now, I need to find $n,$ which is the difference between two consecutive numbers in this list. 2 and 5 are consecutive numbers in this list, so

$n$ = 5 - 2 = 3

Next, I should find $b$, which is the remainder obtained by dividing $n$ by any number in this list. Because 11 is a number in the list, and $n$ is 3, so the remainder obtained is 2 (from 11$\div$3).

The rule is - Numbers 2 more than multiples of 3.

So, any number that can be expressed as 3$a$ + 2, satisfies the above rule.

If $a$ is 343, I will get a number larger than 999, or a 4-digit number.

Let me put $a$ as 343 and see if the Yellow light gets switched on...Yes! It worked!

Using the same strategy one can find the rules and properties for Red, Blue and Green lights.