A Little Light Thinking

This problem follows on from Charlie's Delightful Machine, so you may wish to take a look at that first.

When you enter a number in the interactivity below, a light turns on if your number belongs to one of the computer's chosen linear sequences. (A linear sequence is one that goes up by the same number each time.) If your number belongs to both sequences, both lights will switch on.

Your challenge is to work out the two sequence rules in a quick, efficient way, and to determine whether both lights can be switched on together. To generate a new pair of rules, click on "restart".




Here are some examples of possible sequence rules that the computer might choose:



Identify some pairs of sequences for which it is possible to turn both lights on.
What do you notice about their rules?

For each pair of sequences, find a number that would turn both lights on.
Can you find a way of generating lots of numbers, once you have found one that works?

Identify some pairs of sequences for which it is not possible to turn both lights on.
What can you say about their rules that convinces you that it is not possible?

And now for something a little more challenging...

If the 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?

Once you have a method for turning on pairs of lights, why not try to apply it for all four lights in Charlie's Delightful Machine?