Sliding Bus Puzzle TN

Teacher Notes

This activity is another great example of a puzzle whose solutions can be modeled by an algebraic sequence. The puzzle provides an engaging introduction and an incentive to generalise, which helps students with this traditionally difficult idea. The puzzle can be modeled by a linear sequence and broken down into a series of different linear sequences that combine to form the overall model. As such this activity has lots of scope for relating sequences to physical situations, breaking them down in to parts and seeing how algebraic manipulation links the different solutions together. This is a good, deep problem that only involves linear sequences.


  • I do recommend that this problem starts with students being physically engaged. Arrange a 3 by 3 grid of chairs and sit 8 students down and ask students to swap places as described in the problem outline. With bigger classes, you might have two or three groups doing this.
  • Students should keep trying to make sure they arrive at the smallest possible number of moves (11 in this in this first case)
  • Expand the grid to 4 x 4 and repeat the exercise (18 moves)
  • Then repeat with a 5 x 5 grid (25 moves)
  • At this point the teacher can consider whether or not they want to continue or ask students to work in smaller groups with counters (for example). The position to term rule is 7n – 10 where n is the size of the square grid.
  • Students should then think about breaking the pattern down in to different types of move as set out in the worksheet.


This problem has huge scope for change and/or extension. Here are just a few

  • The first space can appear in a different place
  • The finishing place could be different
  • The shape of the bus could be rectangular
  • The shape of the bus could be triangular


To get the full benefit of this activity it is probably worth spending 2 hours on it. Students need to investigate the problem and collect data and then break the problem down in to the different types of move.


The worksheet provides an obvious record of activity and can be either written on or completed digitally. This activity also lends itself to photo and video. The video on the activity page is a good example and can be used to quickly remind students of what they did and it's fun!

What to expect

One of the many aims of this type of activity is to explore generalising about sequences and so we should expect many of the same issues we get when trying to do this. Students are drawn to generalizing a ‘term to term’ rule and then incorrectly expressing this as ‘n+7’ (in this case). The physical involvement in this problem does help to steer students towards a ‘position to term’ rule. Having to move themselves encourages them to think more about the number of moves related to the size of the grid. 

Breaking the problem down into ‘different types of move’ is further encouragement to think about the physical situation. The different types are

  • Number of moves the person changing places actually has to move (2n + 2)
  • Number of moves required to make the first space available (n-2)
  • Number of moves required to make each subsequent space available (2 which must then be multiplied by the number of remaining moves, 2n + 1)

Each of these are related to the square grid size and the link is perhaps more obvious than it is for the general rule. This can be modeled nicely by looking at a particular case which is perhaps just too big to want to do all the moves (7 x 7, for example).

The really lovely moment in this activity is when students deduce that the total of those three individual expressions is 7n – 10. As such this activity becomes more than just another means of generating a linear sequence and an opportunity for a rich activity that links a number of abstract ideas together in a fun and engaging context.

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