In this problem, students are presented with a mystery to be solved (what are the rules which govern the lights, and can all the lights be switched on at once?).
To satisfy their curiosity and explain the mystery, they will need to go in search of the information they require, and work in a systematic way in order to make sense of the results they gather.
The questions encourage students to think about the properties of numbers, including divisibility and remainders. It may help students gain a deeper understanding of linear sequences (and perhaps quadratic sequences, if they explore Level 2 and 3).
One way they could record their work is by creating a table with sequence rules along the top and down the side, and indicating with a tick or a cross in each cell whether both lights could light up simultaneously. When appropriate, they could also indicate numbers which successfully switch on both lights.
Leave some time for the class to come together to share examples of rules where more than one light could be switched on simultaneously, and examples where it was impossible, together with their reasoning.
What is true about any pair of rules where it is not possible to light up both lights?
If two sequences are described by the rules $an+b$ and $cn+d$, can you explain the conditions for determining whether the lights will ever switch on together?
The problem Shifting Times Tables offers an introductory challenge for exploring linear sequences.
The problem Remainders explores some properties of numbers which could be useful when thinking about this problem.
Students could use a 100 square as a visual way to record sequences and see where (if) they coincide.
Some students may wish to use ideas of modular arithmetic to prove their findings; reading this article first may help.