Here is a game I stole from Marty Weissman:

You start at zero on the number line. You are allowed "hop" forward or backwards by $6$ and "jump" forwards or backwards by $10$. Which numbers can you reach? Are there any numbers you cannot reach?

Variations:

1. Change $6$ and $10$ to other numbers.
2. Use "hop", "skip", and "jump" (3 numbers) instead of just two.
3. Now you are only allowed to jump forwards, never backwards. How does that change the numbers are you able to reach?

Here is a game I stole from Marty Weissman:

You start at zero on the number line. You are allowed "hop" forward or backwards by $6$ and "jump" forwards or backwards by $10$. Which numbers can you reach? Are there any numbers you cannot reach?

Variations:

1. Change $6$ and $10$ to other numbers.
2. Use "hop", "skip", and "jump" (3 numbers) instead of just two.
3. Now you are only allowed to hop or jump forwards, never backwards. How does that change the numbers are you able to reach?