We had correct solutions from Jack Bird and Jack Manku of Isleworth, Syon and Vicknan from Colester Royal Grammar and from Oswald, Jimmy and Derek of Doncaster Gardens Primary School who sent us this:

The Green light is divisible by $11$ according to the clue at the bottom of page 2 whose numbers are all divisible by $11$.

Red is divisble by $3$, since at the top of page 1 it says that the only single digit numbers that do not light red are $8,4,2,5,7$ and $1$ which leaves out $3, 6, 9$, therefore mutiples of $3$.

Purple being the hardest one took some time to figure out. According to the information on the bottom of page 1 and the information on page 2, out of the numbers from $1$ to $9$ none light it up purple, yet $10$ to $99$ do. Also, the 4 digit and 6 digit numbers light it up such as $1234$ and $111111$ therefore, even digit numbers for example in this case, $2,4$ and $6$ digit numbers.

Orange is lit up by square numbers, referring to the information on page 2, saying $4,16,25$ and $36$ all light orange up.

The smallest number that lights up all lights is $1089$ it is equal to $33$ squared, is divisible by $3$ and $11$, and is a $4$ digit number.

Benjamin from Maidehnead sent us a partial solution for the second machine:

Red is divisble by $5$

Green is divisble by $11$

Blue lights up when the tens digit is odd

Lydia from Wellington School worked on both machines:

For each of the problems I wrote down all the numbers for each colour that I knew. I then looked for the patterns between the numbers.

Using this method I found that, in the first problem:

square numbers turned on the orange light,

multiples of 3 turned on the red light,

numbers with an even number of digits switched on the purple,

and multiples of 11 the green.

The lowest number that is a multiple of 3 and 11, that is square and has an even number of digits is 1089, and so this is my answer.

For the second problem I used the same method. I soon discovered that:

multiples of 5 turned on the red light,

multiples of 11 turned on the green,

numbers with a digital root of 8 turn on the orange light,

and numbers that had an odd number of tens switched on the blue.

935 is the smallest number which will make all four colours light up.