Why do this problem?
challenges children to make sense of information by applying their knowledge of number properties. They are required to make and test conjectures, and this will encourage them to work in a systematic way.
You could introduce this problem by choosing a rule and inviting children to offer numbers. If the number fits your rule, write it up on the board under a heading 'I like these numbers' (or a tick). If the number doesn't fit your rule, write it on the other side under a heading 'I don't like these numbers' (or a cross). Try to remain silent during the activity so the only feedback the
children get is the position of their chosen numbers. You might insist that the children must all agree on the rule before someone is allowed to check it with you. You could challenge them to find the rule by choosing, for example, fewer than ten numbers.
Once they are familiar with the way the game works, show them the picture of Mr Gilderdale's class's game and set them off on the problem itself, perhaps working in pairs. You may need to bring them together for a 'mini plenary' at some stage, so they can share how they are getting on so far.
In a plenary, you could work through their suggested solutions on the board, encouraging them to justify their ideas.
Of course this game can be played again and again. You could start off a game on the board, which could continue over several days. In this way, learners can form a conjecture for your rule, but you will not confirm their conjecture, you will only place numbers in the appropriate column.
What do the numbers Mr Gilderdale likes have in common? What is the same about them?
What do the numbers Mr Gilderdale doesn't like have in common?
What number could you choose to test your idea?
Some children might like creating their own 'snapshots' of an imaginary game, so that the rule is ambigous.
Having a hundred square to mark the 'I like' numbers on might help some children see, and understand, a pattern.