In conjunction with the other tasks in this group, this task also offers the chance to focus on any of the five key ingredients that characterise successful mathematicians.
You could begin the lesson by dividing the board into two columns, one headed with a tick and the other headed with a cross.
Ask students to suggest numbers, and write each suggestion in the appropriate column according to a rule of your own choice. Possible rules could be prime numbers, odd numbers, multiples of 4...
Once a few students think they know what your rule might be, invite them to make guesses that will help the rest of the class guess the rule too.
"Some friends played a game like this with two-digit numbers, and they used these four rules:
Choose any two of the rules, and see if you can find some two-digit numbers that satisfy both of them."
Give students a little bit of time to find some examples, and then extend the problem...
"Now choose one of the other rules. Do any of your numbers satisfy that one too?"
Again, give students some thinking time. Then set the final challenge:
"Can you find a two-digit number that satisfies all four of the statements?"
Again, give students some time to work on this, then finish by bringing the class together to discuss how they know it's not possible to satisfy all four rules using two-digit numbers.
You could follow this up by asking students to consider how they would describe the numbers that satisfy pairs of statements:
"How could you describe the pattern of the numbers that satisfy both Alison's and Sam's statements?
How about the numbers that satisfy both Alison's and Matt's statements?
Can you describe patterns for other pairs of statements?"
Which rules have lots of numbers that satisfy them?
Which rules don't have very many numbers that satisfy them?
Where is it best to start if you're looking for numbers that satisfy more than one rule?
Invite students to relax the two-digit number restriction - can they satisfy all four rules now?
Charlie's Delightful Machine would follow on nicely from this task.