1. If a puzzle has cogs that are all multiples of 4, then the only numbers that can be made are multiples of 4. Yet it is possible to land on 61, an odd number. Explain how — and why — this occurs. What would happen if the large gear had 60 teeth, like a clock, rather than 65?
2. Ask students to construct helpful notations to keep track of their reasoning as they play the game. These will be versions of the equation km + ln = p.
3. In some puzzles with two cogs, the numbers are closely related, such as one is a factor of the other; in others, the greatest common denominator is 1. If players are encouraged to make up their own puzzles to play off-screen, they may find that they can make unsolvable puzzles using some of these pairs. This provides the ground work for later understanding of linear relationships.
4. A good way to prepare students for creating their own puzzles is to ask them to list the teeth reached when they count round the large gear clockwise in 5s, then when they count counter-clockwise in 5s. Then ask them to count clockwise in, 3s and then counter-clockwise in 3s, again listing the teeth reached. What do they notice compared to counting in 5s? After that, do the same thing with 4s, an even number. Again, what do they see? Has anything changed? From then on, let them explore on their own with different skip-counts, comparing clockwise and counter-clockwise lists. What numbers can be reached and which ones cannot?

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