Thank you to everybody who sent in their ideas about this problem. Helly looked at both positive and negative numbers to find the rule (this picture can be clicked on to enlarge it):
 

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I wonder at which numbers all four lights would switch on with these rules?

Yuna from QUEST Tokyo in Japan found some interesting rules for their green and blue lights:

The yellow light has a rule of multiples of 5.
The red light has the rule of multiples of 10.
For the green light, the rule is that it lights up for prime numbers.
For the blue light, it lights up for numbers which skip numbers with gaps increasing by 1 each time.

Etienne from QUEST Tokyo explained why all four of their lights would never light up at the same time:

Yellow lights up for multiples of 2 (even numbers).
Red lights up for multiples of 7.
Blue lights up for prime numbers.
Green lights up for odd numbers.

It is not possible for all the lights to light up at the same time because a integer cannot be both odd and even.

The children from Earls Colne Primary in England looked at numbers that would light up none of the lights, as well as numbers that would light up all four lights:

Yellow = 4 times table

Red = Prime numbers

Blue = Triangular numbers

Green = 6 times table

Numbers to light no lights = 99, 27, 57, 82, 121, 9 and 14

Numbers to light all the lights = It is not possible to light all the lights because a prime number can not be in the 4 or the 6 times table.

I hadn't thought of looking for numbers which light none of the lights - this is a really interesting idea. I wonder if the numbers you've listed have anything in common?

Joshua from Pristine Private School in the UAE used their knowledge of prime numbers to explain why not all of the lights would light up at the same time. This picture can be clicked on to enlarge it:
 

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This is interesting, Joshua. It looks like your green light will normally only light up by itself. I wonder when it will light up with one of the other lights?

Phyllis from the British School of Beijing in Shunyi, China had four very similar rules for their lights. They explained:

1. every four numbers the yellow light will light up.
2. every eight numbers the red light will light up.
3. every five numbers the blue light will light up.
4. every nine numbers the green light will light up.
5. It is possible to light up all four lights at the same time because we just need to have the multiple the 4,8,5 and 9 it smallest number will be 360.

How might Phyllis have worked this out?

Yuto S, Joshua, Daichi and Tom from St. Mary's International in Tokyo, Japan sent in several diagrams explaining their solutions to this problem. Here is one of their explanations:

A yellow check means the number is a multiple of 5. A red check means the number is an even number. A blue check is a number divisible by 9. A green check means the number is a square number.

If there was no limit to how much the number could go, there would be a number that all 4 lights light up. For instance, 900 would make all the lights light up as it is a square number, an even number, a multiple of 5, and a multiple of 9

You might like to take a look at Yuto, Joshua, Daichi and Tom's full solution to see more of their ideas. I wonder how they worked out that 900 would work?

Arryan from Eastcourt Independent School in England had different rules for their lights, but interestingly it was still at the number 900 that all the lights lit up:

For this problem, I started inputting ascending numbers and decided to focus on the yellow one. The first yellow light that came after zero was three, then it became six, then nine. After this discovery, I realised that the yellow-numbered numbers are multiples of three, starting from zero. This means all the yellow-lighted numbers are multiples of 3, which is the rule.

For red, it started with one and went to four. After four, it went to nine, and after nine, it went to 16. I realised that the difference between each number was increasing. I had now realised that each number was a square number. Now I figured out the pattern. Each number for the red light is a square number, which is the rule.

The blue light started with zero, then went to five, then to ten, and after ten was fifteen. Each number was a multiple of five. That was the rule: each blue-lighted number was a multiple of 5!

Finally, I started with the green light. It began at zero and went to four, then eight, and after that to twelve and sixteen. I found that each of these numbers is a multiple of four!

It is possible to have a number that lights up all the bulbs. This number is 900. It is divisible by three, four, and five and is the square of 30.

For reference, here are the nth terms for the bulbs:
Yellow Bulb: 3n-3
Red Bulb: n²
Blue Bulb: 5n-5
Green Bulb: 4n-4

Thank you as well to the following children who sent in similar solutions to this problem: Adhaan from Pristine Private School; PJ and Charlie from Kilmore in Ireland; Simon from Headstart International School in Phuket, Thailand; Heiko from Clifton Hill Primary in Australia; and Jashwithaa from WPS in Australia.